harmonic estimation and source identification in …

HARMONIC ESTIMATION ANDSOURCE IDENTIFICATION IN

POWER DISTRIBUTION SYSTEMSUSING OBSERVERS

A THESIS SUBMITTED TO THE UNIVERSITY OF MANCHESTER

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

IN THE FACULTY OF ENGINEERING AND PHYSICAL SCIENCES

2015

ByAwajiokiche Ujile

School of Electrical and Electronic Engineering

Contents

Abstract 11

Declaration 12

Copyright 13

Acknowledgements 14

Publications related to this thesis 16

Acronyms 17

1 Introduction 211.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2 Motivations for this research . . . . . . . . . . . . . . . . . . . . . . 251.3 Aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Power distribution system dynamic modelling and optimum measurementplacement 312.1 Dynamic modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1.2 Literature review on distribution system dynamic modelling . 322.1.3 Distribution system dynamic modelling . . . . . . . . . . . . 33

2.1.3.1 Generator/grid modelling . . . . . . . . . . . . . . 352.1.3.2 Cable modelling . . . . . . . . . . . . . . . . . . . 372.1.3.3 Load modelling . . . . . . . . . . . . . . . . . . . 392.1.3.4 Transformer modelling . . . . . . . . . . . . . . . 40

2.2 Optimum measurement placement in power distribution systems . . . 43

2

2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.2.2 Literature review on optimum meter placement and observ-

ability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 442.2.3 Optimum meter placement method based on binary integer

programming . . . . . . . . . . . . . . . . . . . . . . . . . . 462.3 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Harmonic estimation 563.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Literature review on harmonic estimation . . . . . . . . . . . . . . . 573.3 Iterative observer design for harmonic estimation in power distribution

systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.1 Harmonic estimation on a single output . . . . . . . . . . . . 653.3.2 Simultaneous harmonic estimation in multiple outputs . . . . 71

3.4 Harmonic estimation algorithm based on the iterative observer . . . . 733.5 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.5.1 Single measurement harmonic estimation . . . . . . . . . . . 793.5.1.1 Normal operating condition . . . . . . . . . . . . . 793.5.1.2 Change in amplitude of harmonics with time . . . . 863.5.1.3 Harmonic estimation with variations in fundamental

frequency . . . . . . . . . . . . . . . . . . . . . . 883.5.1.4 Effect of decaying DC component and noise on har-

monic estimation . . . . . . . . . . . . . . . . . . 913.5.2 Harmonic estimation in multiple measurements . . . . . . . . 91

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4 Harmonic source identification 994.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2 Literature review on harmonic source identification . . . . . . . . . . 100

4.2.1 Harmonic State Estimation . . . . . . . . . . . . . . . . . . . 1024.2.1.1 Static HSE . . . . . . . . . . . . . . . . . . . . . . 1034.2.1.2 Dynamic HSE . . . . . . . . . . . . . . . . . . . . 104

4.3 Harmonic source identification using the fault observer based approach 1064.3.1 Identification of multiple harmonic sources . . . . . . . . . . 111

4.4 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3

4.4.1 Single harmonic source . . . . . . . . . . . . . . . . . . . . . 1164.4.2 Time varying harmonic injections . . . . . . . . . . . . . . . 1254.4.3 Change in amplitude of harmonic injection . . . . . . . . . . 1254.4.4 Sudden change in harmonic source . . . . . . . . . . . . . . . 1284.4.5 Identification of multiple harmonic sources with noisy mea-

surements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.4.5.1 First sub-system . . . . . . . . . . . . . . . . . . . 1324.4.5.2 Second sub-system . . . . . . . . . . . . . . . . . 135

4.4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5 Harmonic source identification with output delay 1395.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.2 Literature review on output delay systems . . . . . . . . . . . . . . . 1415.3 Harmonic source identification for single output delay systems . . . . 142

5.3.1 Observer design for single output delay systems . . . . . . . . 1425.3.2 Harmonic source identification in single output delay systems 144

5.4 Harmonic source identification with multiple output delays . . . . . . 1465.4.1 Observer design for multiple output delay systems . . . . . . 1465.4.2 Harmonic source identification for multiple output delay systems149

5.5 Fault observer algorithm for harmonic source identification with de-layed measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.6 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.6.1 Single output delay . . . . . . . . . . . . . . . . . . . . . . . 153

5.6.1.1 First sub-system . . . . . . . . . . . . . . . . . . . 1535.6.1.2 Second sub-system . . . . . . . . . . . . . . . . . 155

5.6.2 Multiple output delays . . . . . . . . . . . . . . . . . . . . . 1585.6.2.1 First subsystem . . . . . . . . . . . . . . . . . . . 1585.6.2.2 Second subsystem . . . . . . . . . . . . . . . . . . 159

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6 Conclusions and recommendations 1636.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.2 Recommendations for further study . . . . . . . . . . . . . . . . . . 167

Bibliography 170

4

Appendices 190

A IEEE 13-node distribution test feeder data 191A.1 Line connectivity data . . . . . . . . . . . . . . . . . . . . . . . . . . 196Word Count: 45, 200

5

List of Tables

1.1 IEEE 519-1992 current harmonic limits (<69 kV) [1] . . . . . . . . . 241.2 G5/4-1 recommended planning levels for harmonic voltages in systems

with voltages 6.6 kV, 11 kV and 20 kV [2] . . . . . . . . . . . . . . . 24

2.1 Harmonic spectrum for the nonlinear load at node 4 . . . . . . . . . . 53

3.1 IEEE 13-node distribution system load data . . . . . . . . . . . . . . 773.2 IEEE 13-node distribution system line data . . . . . . . . . . . . . . 773.3 Harmonic spectrum at node 46 . . . . . . . . . . . . . . . . . . . . . 773.4 Harmonic spectrum at node 75 . . . . . . . . . . . . . . . . . . . . . 783.5 Comparison of harmonic estimates using the FFT, Kalman-Bucy filter

and iterative observer algorithm for I45−46 . . . . . . . . . . . . . . . 853.6 Iterative observer harmonic estimates with fluctuating fundamental fre-

quency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.7 Harmonic estimates for the current measurements . . . . . . . . . . . 97

4.1 Harmonic spectrum for node 71 . . . . . . . . . . . . . . . . . . . . 1174.2 Shunt capacitance at system branches . . . . . . . . . . . . . . . . . 1174.3 Combinations and their respective nodes . . . . . . . . . . . . . . . . 1214.4 Mean squared error in all measurements for single harmonic source . 1224.5 Comparison between actual and estimated current injection at node 71 1244.6 Mean squared error in all measurements for time varying harmonic

injections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.7 Harmonic spectrum at node 75 . . . . . . . . . . . . . . . . . . . . . 1294.8 Harmonic spectrum at node 45 . . . . . . . . . . . . . . . . . . . . . 1334.9 MSE of all suspicious node combinations for the first sub-system . . . 133

5.1 Harmonic spectrum for node 46 . . . . . . . . . . . . . . . . . . . . 154

6

5.2 Comparison between actual and estimated harmonic current injectionsat node 46 for delayed first sub-system . . . . . . . . . . . . . . . . . 155

5.3 Harmonic spectrum for node 75 . . . . . . . . . . . . . . . . . . . . 1565.4 Comparison between actual and estimated harmonic current injections

at node 75 for delayed second sub-system . . . . . . . . . . . . . . . 158

A.1 Line data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

7

List of Figures

1.1 A harmonic signal with its individual components . . . . . . . . . . . 22

2.1 Power distribution feeder . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Nonlinear load representation . . . . . . . . . . . . . . . . . . . . . . 342.3 Single phase generator model . . . . . . . . . . . . . . . . . . . . . . 362.4 Three phase generator model . . . . . . . . . . . . . . . . . . . . . . 362.5 Three phase cable π model . . . . . . . . . . . . . . . . . . . . . . . 372.6 Three phase load model . . . . . . . . . . . . . . . . . . . . . . . . . 402.7 Three phase ∆-wye grounded transformer equivalent circuit . . . . . . 412.8 Sequential elimination method for optimum measurement placement . 502.9 One line diagram of the four-node test feeder . . . . . . . . . . . . . 51

3.1 Harmonic estimation using the iterative observer . . . . . . . . . . . 573.2 Classification of harmonic estimation methods [3] . . . . . . . . . . . 593.3 Power distribution system with harmonic disturbance . . . . . . . . . 623.4 Flowchart for iterative observer design . . . . . . . . . . . . . . . . . 753.5 Single-line diagram of IEEE 13-node distribution feeder . . . . . . . 763.6 IEEE 13-node feeder current measurements . . . . . . . . . . . . . . 783.7 Top: Harmonic disturbance-free current estimate at node 45 Bottom:

Residual from fundamental frequency estimation of I45−46 . . . . . . 803.8 Estimate of the third harmonic for I45−46 . . . . . . . . . . . . . . . . 813.9 Estimates of the fifth harmonic for I45−46 measurement . . . . . . . . 823.10 Residual after termination of iteration for I45−46 measurement . . . . 833.11 Comparison between the Kalman-Bucy filter estimates and the itera-

tive observer estimates for I45−46 at normal operating conditions . . . 843.12 Comparison of residuals between the Kalman-Bucy filter and iterative

observer estimates for the I45−46 measurement . . . . . . . . . . . . . 85

8

3.13 Comparison between the Kalman-Bucy and iterative observer estimatesfor change in amplitude of harmonic injections with time . . . . . . . 87

3.14 Estimation error with change in amplitude of harmonics with time . . 883.15 I45−46 measurement with fluctuating fundamental frequency . . . . . 893.16 Comparison of harmonic estimates with fluctuating fundamental fre-

quency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.17 I45 measurement with decaying DC component . . . . . . . . . . . . 923.18 Estimation error for decaying DC component in the measured signal . 933.19 Comparison between harmonic estimation methods in the presence of

measurement noise and decaying DC component . . . . . . . . . . . 943.20 Harmonic estimates for multiple measurements . . . . . . . . . . . . 953.21 Estimation error after fifth harmonic estimation . . . . . . . . . . . . 963.22 Estimation error for node 92 after estimating the seventh harmonic . . 963.23 Amplitude of harmonic estimates for I45−46 and I92−75 . . . . . . . . 97

4.1 Framework for fault observer based harmonic source identification . . 1014.2 Fault observer based harmonic source identification . . . . . . . . . . 1144.3 IEEE 13-node distribution system measurements . . . . . . . . . . . 1184.4 MSE for all measurements with single harmonic source . . . . . . . . 1204.5 Estimation error when node 71 is taken as the harmonic source . . . . 1234.6 Node 71 harmonic current injections . . . . . . . . . . . . . . . . . . 1244.7 Mean squared error for time varying harmonic injections at node 71 . 1264.8 Harmonic current injection at node 71 with sudden harmonic injection

at 0.5s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.9 Estimation error for timevarying harmonic injections . . . . . . . . . 1284.10 MSE for combination representing node 71 for change in harmonic

source at t = 0.5s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.11 MSE for combination representing node 75 for change in harmonic

source at t = 0.5s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.12 Estimation error for the combination representing node 71 . . . . . . 1314.13 Estimation error for the combination representing node 75 . . . . . . 1314.14 Sub-systems for the IEEE 13-node system . . . . . . . . . . . . . . . 1334.15 MSE of suspicous node combinations for first sub-system . . . . . . . 1344.16 Node 45 current injections . . . . . . . . . . . . . . . . . . . . . . . 1354.17 MSE of node combinations for the second sub-system . . . . . . . . . 1364.18 Harmonic current injections at node 75 . . . . . . . . . . . . . . . . . 137

9

5.1 MSE of all combinations for the first sub-system with delay at 0.1s . . 1535.2 Comparison between the actual and estimated measurements for the

first subsystem with delay at 0.1s for combination 2 . . . . . . . . . . 1545.3 Harmonic current injections for both delayed and delay-free system for

the first sub-system . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.4 MSE for second subsystem with delay at 0.1s . . . . . . . . . . . . . 1565.5 Comparison between the actual and estimated measurements with de-

lay at 0.1s for the third combination . . . . . . . . . . . . . . . . . . 1575.6 Harmonic current injections for both delayed and delay-free system for

the second sub-system . . . . . . . . . . . . . . . . . . . . . . . . . 1575.7 Delayed measurements for first sub-system. . . . . . . . . . . . . . . 1595.8 MSE of all node combinations with multiple output delays for first

sub-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.9 Residual for multiple time delays in the first sub-system . . . . . . . . 1605.10 Delayed measurements for second sub-system. . . . . . . . . . . . . 1615.11 MSE of all node combinations with multiple output delays for second

sub-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.12 Residual for multiple time delays in the second sub-system . . . . . . 162

A.1 IEEE 13-node distribution feeder . . . . . . . . . . . . . . . . . . . . 191

10

HARMONIC ESTIMATION AND SOURCE IDENTIFICATION IN POWER

DISTRIBUTION SYSTEMS USING OBSERVERS

Awajiokiche UjileA thesis submitted to the University of Manchester

for the degree of Doctor of Philosophy, 2015

Abstract

With advances in technology and the increasing use of power electronic componentsin the design of household and industrial equipment, harmonic distortion has becomeone of the major power quality problems in power systems. Identifying the harmonicsources and quantifying the contributions of these harmonic sources provides utilitycompanies with the information they require to effectively mitigate harmonics in thesystem.

This thesis proposes the use of observers for harmonic estimation and harmonicsource identification. An iterative observer algorithm is designed for performing har-monic estimation in measured voltage or current signals taken from a power distribu-tion system. The algorithm is based on previous observer designs for estimating thepower system states at the fundamental frequency. Harmonic estimation is only car-ried out when the total harmonic distortion (THD) exceeds a specified threshold. Inaddition, estimation can be performed on multiple measurements simultaneously. Sim-ulations are carried out on an IEEE distribution test feeder. A number of scenarios suchas changes in harmonic injections with time, variations in fundamental frequency andmeasurement noise are simulated to verify the validity and robustness of the proposediterative observer algorithm.

Furthermore, an observer-based algorithm is proposed for identifying the harmonicsources in power distribution systems. The observer is developed to estimate the sys-tem states for a combination of suspicious nodes and the estimation error is analysedto verify the existence of harmonic sources in the specified node combinations. Thismethod is applied to the identification of both single and multiple harmonic sources.The response of the observer-based algorithm to time varying load parameters andvariations in harmonic injections with time is investigated and the results show that theproposed harmonic source identification algorithm is able to adapt to these changes.

In addition, the presence of time delay in power distribution system measure-ments is taken into consideration when identifying harmonic sources. An observeris designed to estimate the system states for the case of a single time delay as wellas multiple delays in the measurements. This observer is then incorporated into theobserver-based harmonic source identification algorithm to identify harmonic sourcesin the presence of delayed measurements. Simulation results show that irrespective ofthe time delay in the measurements, the algorithm accurately identifies the harmonicsources in the power distribution system.

11

Declaration

No portion of the work referred to in this thesis has beensubmitted in support of an application for another degree orqualification of this or any other university or other instituteof learning.

12

Copyright

i. The author of this thesis (including any appendices and/or schedules to this the-sis) owns certain copyright or related rights in it (the “Copyright”) and s/he hasgiven The University of Manchester certain rights to use such Copyright, includ-ing for administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electroniccopy, may be made only in accordance with the Copyright, Designs and PatentsAct 1988 (as amended) and regulations issued under it or, where appropriate,in accordance with licensing agreements which the University has from time totime. This page must form part of any such copies made.

iii. The ownership of certain Copyright, patents, designs, trade marks and other in-tellectual property (the “Intellectual Property”) and any reproductions of copy-right works in the thesis, for example graphs and tables (“Reproductions”), whichmay be described in this thesis, may not be owned by the author and may beowned by third parties. Such Intellectual Property and Reproductions cannotand must not be made available for use without the prior written permission ofthe owner(s) of the relevant Intellectual Property and/or Reproductions.

iv. Further information on the conditions under which disclosure, publication andcommercialisation of this thesis, the Copyright and any Intellectual Propertyand/or Reproductions described in it may take place is available in the Uni-versity IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited inthe University Library, The University Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations) and in The Univer-sity’s policy on presentation of Theses

13

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Acknowledgements

I would like to express my profound gratitude to God for giving me the strength,

patience and ability to accomplish this work.

My appreciation goes to a number of people who supported me throughout the

course of my PhD. First of all, I would like to thank my supervisor, Dr. Zhengtao Ding,

for his supervision, help, advice, guidance and support throughout the course of this re-

search. His wisdom, understanding and consideration, even when there were setbacks,

went a long way to enable me complete my research. His exceptional dedication and

efforts in reviewing this thesis and all other publications were instrumental in ensuring

that my research is of the highest possible standard. My appreciation also goes to Dr.

Haiyu Li for his continuous collaborations, contributions and support throughout my

research.

Very special thanks to my family for all their support. To my dad, Dr. A. A. Ujile,

for constantly ensuring that I am up to date on my research; to my mum, for calling

me daily to encourage me and lift my spirits, even when there are setbacks; to my

siblings, Ichechiek, Itong and Tonejit, for constantly filling my days with humour and

laughter and encouraging me to be the best I can be; To my cousins, Aaron, Martha

and Solomon, for their prayers and support.

I would also love to thank my friends: Collins, for being there to encourage and

support me through every step of the way, Sophie for pushing me constantly, Tariye for

14

being there to talk about all sorts when I’m in need of a break, Onyinye, Great, Lynda,

Eugene, Isobo, Kaine, Okwy, Ijeoma, Emmanuel Sogein, Awo and Wilson, for all the

support and prayers.

My appreciation also goes to my colleagues; Dr. Mobolaji Osinuga and Dr. Zongyu

Zuo for all their support and technical guidance and Chunyan Wang, Tianqiao Zhao,

Ahmad Isira and Zhenhong Li for their assistance at various levels of my research.

Lastly, I would like to acknowledge the Rivers State Sustainable Development

Agency (RSSDA) for their financial support.

15

Publications related to this thesis

The following publications have been made during the course of this research

1. Ujile, A. and Ding, Z. "An iterative observer for harmonic estimation in power

distribution networks." In 2014 IEEE PES T&D Conference and Exposition, pp.

1-5. IEEE, 2014.

2. Ujile, A. Ding, Z. and Li, H. "An iterative observer approach to harmonic esti-

mation for multiple measurements in power distribution systems" Transactions

of the Institute of Measurement and Control (In press)

3. Ujile, A. and Ding, Z. "A Dynamic Approach to Identification of Multiple Har-

monic Sources in Power Distribution Systems" International Journal of Electri-

cal Power and Energy Systems (In review)

4. Ujile, A. and Ding, Z. “Identification of harmonic sources using a fault observer

approach”, IEEE Transactions on Power Delivery Special Issue on “Contempo-

rary Issues in Power Quality” (Extended abstract accepted)

16

Acronyms

Ah Amplitude of harmonic component

A State transition matrix

B Input matrix

Cl Capacitance

C Measurement matrix

E Harmonic disturbance matrix

G Conductance

Ll Inductance

L Observer gain for harmonic states

M Current injection matrix

P Binary connectivity matrix

R Resistance

Y Shunt admittance

µ Harmonic output signal

17

Acronyms 18

φh Phase angle at harmonic order h

L Power network observer gain

d Time delay

ei Harmonic injection index at bus i

e Estimation error

h Harmonic order

l Cable/line length

n Dimensions of system states

p Number of nodes in a power system

q Dimensions of the matrix S

s Number of suspicious buses

t Number of simultaneous harmonic sources

u Input

v Number of harmonic frequencies in a given signal

w Harmonic disturbance state

x System states

y Outputs

ADALINE Adaptive linear neuron

ARMA Autoregressive moving average

Acronyms 19

ASD Adjustable Speed Drives

AWNN Adaptive wavelet neural network

CZT Chirp-z transform

DFT Discrete Fourier transform

ESPRIT Estimation of signal parameters via rotational invariance technique

FFT Fast Fourier Transform

HHT Hilbert-Huang transform

HSE Harmonic state estimation

IGBT Insulated gate bipolar transistors

KCL Kirchhoff’s current law

KVL Kirchhoff’s voltage law

MOSFET Metal oxide semiconductor field effect transistor

MSE Mean squared error

MUSIC Multiple signal classification

PCC Point of common coupling

PMU Phasor measurement units

SCADA Supervisory control and data acquisition

Acronyms 20

SVD Singular value decomposition

TDD Total demand distortion

THD Total Harmonic Distortion

WLS Weighted least squares

WT Wavelet transform

Chapter 1

Introduction

1.1 Background

The increasing use of power electronic devices in the design of household and in-

dustrial equipment has resulted in an escalation in the level of harmonic injections in

power distribution systems. These power electronic devices consist of silicon based

components such as diodes, transistors, insulated gate bipolar transistors (IGBT) and

metal oxide semiconductor field effect transistor (MOSFET). Loads which consist of

these devices are termed nonlinear loads. A nonlinear load is one in which the volt-

age is not proportional to the current. Examples of nonlinear loads include fluorescent

lamps, synchronous machines, rectifiers, inverters, arc furnaces, power conversion us-

ing high voltage direct current (HVDC), wind and solar power converters in distribu-

tion systems and electric vehicles which utilise power rectification for battery charging.

These loads draw up current in a non-sinusoidal manner, thereby distorting the voltage

waveform which results in a non-sinusoidal periodic voltage waveform [4]. Due to the

non-sinusoidal voltage and current waveforms, the quality of power delivered to the

end user may become degraded. In addition, the systems can experience overheating

21

CHAPTER 1. INTRODUCTION 22

and equipment failure, false trips in fuses or sensitive loads, interference in telecom-

munication circuits and an overall degradation of the quality of power delivered to

consumers. The resulting distorted voltage or current waveform can be represented as

a sum of sinusoids at different frequencies.

A harmonic is a voltage or current signal whose frequency is an integer multiple

of the fundamental frequency [4]. The non-integer multiples of the fundamental fre-

quency are referred to as interharmonics. An inter-harmonic frequency which is less

than the fundamental frequency is called a sub-harmonic. Figure 1.1 shows the rep-

resentation of a harmonic signal and its individual harmonic frequencies. y(t) is the

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Fun

d

-1

0

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

3rd

-0.5

0

0.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

5th

-0.2

0

0.2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

7th

-0.1

0

0.1

Time(s)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

11th

-0.05

0

0.05

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

y(t)

-2

0

2

Figure 1.1: A harmonic signal with its individual components

distorted signal which consists of the fundamental frequency, third, fifth, seventh and

eleventh harmonics. The signal shown may be represented mathematically as

y(t) = sin(ωt +30)+0.3sin(3ωt +45)+0.2sin(5ωt−30)+0.1sin(7ωt +120)


+0.05sin(11ωt−60) (1.1)

where ω = 2π f , f is the fundamental frequency taken as 50 Hz.

As a result of the negative effects harmonics pose on distribution systems, utility

companies are becoming more interested in determining the source of these harmonic

injections as well as carrying out harmonic studies. Harmonic studies provide a quan-

tification of the voltage and current distortions in a given power system.

Various organisations have placed restrictions on the allowable harmonic content

in power systems. IEEE Std-519 [1] recommends that for systems below 69 kV, the in-

dividual voltage distortion should not exceed 3% of the fundamental frequency and the

voltage total harmonic distortion (THD) should not exceed 5% of the fundamental fre-

quency. Table 1.1 shows the recommended current harmonic distortion limits based on

IEEE Std-519 where ISC is the maximum short circuit current at the point of common

coupling (PCC), IL represents the maximum demand load current and h is the harmonic

order. The total demand distortion (TDD) is defined as the harmonic current distortion

expressed as a percentage of the maximum demand load current. In the United King-

dom, the Engineering Recommendation G5/4-1 [2] is used to place restrictions on the

overall voltage distortion levels in power systems. According to the recommendation,

for systems with voltages greater than 20 kV and less than 145 kV, the voltage THD

should not exceed 3%. The individual harmonic distortion limit is shown in Table 1.2.

To comply with these regulations, utility companies need to be aware of the harmonic

distortion levels in a given power distribution system to enable them take appropriate

action. This action may include the design of a filtering mechanism to extract the har-

monics or utility companies may impose a fine on the responsible parties. Therefore,

a method of determining the amplitude and phase of each harmonic order present in

the system is required. In addition, the sources of these harmonic injections need to


Table 1.1: IEEE 519-1992 current harmonic limits (<69 kV) [1]

ISC/IL h<11 11<h<17 17<h<23 23<h<35 35≤ h TDD

< 20 4.0 2.0 1.5 0.6 0.3 5.0

20 < 50 7.0 3.5 2.5 1.0 0.5 8.0

50 < 100 10.0 4.5 4.0 1.5 0.7 12.0

100 < 1000 12.0 5.5 5.0 2.0 1.0 15.0

> 1000 15.0 7.0 6.0 2.5 1.4 20.0

Table 1.2: G5/4-1 recommended planning levels for harmonic voltages in systems withvoltages 6.6 kV, 11 kV and 20 kV [2]

Odd harmonics Odd harmonics Even harmonics

(Non-multiples of 3) (multiples of 3)

Order Harmonic Order Harmonic Order Harmonic

h voltage (%) h voltage (%) h voltage (%)

5 3.0 3 3.0 2 1.5

7 3.0 9 1.2 4 1.0

11 2.0 15 0.3 6 0.5

13 2.0 ≥ 21 0.2 8 0.4

17 1.6 10 0.4

19 1.2 ≥ 12 0.2

23 1.2

25 0.7

>25 0.2+0.5(25h )

be determined. This is important because if the exact node where the harmonic injec-

tions occur is identified, the harmonic disturbances in the system can be dealt with at

their sources and the propagation of these harmonic disturbances throughout the power

system can be avoided.


1.2 Motivations for this research

There are three main objectives of carrying out harmonic studies in power distribu-

tion systems [5]:

1. To determine the characteristics of the harmonic injections. These characteristics

include the amplitude and phase of the individual harmonic frequencies being

injected into the power system.

2. To identify the sources of harmonic injections in the power system.

3. To design a suitable mechanism for extracting the injected harmonics from the

system and ensuring that the THD after harmonic extraction complies with reg-

ulatory requirements.

The most common methods of harmonic source identification and harmonic estima-

tion are static methods. Some of these static methods include singular value decom-

position (SVD) [6–8], weighted least squares (WLS) [9, 10] and sparsity maximisa-

tion [11,12]. Node voltages are estimated at each harmonic order and the nodes which

yield non-zero voltages at the harmonic frequencies are identified as the harmonic

sources. Although these static harmonic source identification techniques effectively

identify harmonic sources in transmission systems, they may not be accurate for dis-

tribution systems due to the constantly varying load and harmonic injections. Trans-

mission system are used to transmit power over long distances at high voltages while

distribution systems are used to transmit power at short distances and lower voltages.

Static methods carry out harmonic source identification at each harmonic order, and

this requires knowledge of the system parameters such as impedance and admittance

at each harmonic order. The measurement at each harmonic order is also required for

harmonic source identification.


Power distribution systems are susceptible to dynamic changes due to the time-

varying nature of loads. The harmonic currents individually injected by some of these

loads may not create a significant distortion on the harmonic levels in the distribution

system. However, when there are several of these nonlinear loads, the cumulative

effects of the harmonic distortion can be significant. As a result of the continuous

fluctuation in load demand, the harmonic injections, which are a result of the use of

nonlinear loads, may fluctuate as well. Thus, the need arises for dynamic and adaptive

measures to determine the source and characteristics of harmonic producing loads in

the system. This dynamic approach should be able to account for the time varying

nature of both linear and nonlinear loads in distribution systems. The most common

dynamic approach of harmonic estimation and source identification is the Kalman filter

[13, 14]. A drawback of the Kalman filter approach is that knowledge of the process

and measurement noise covariance is necessary to obtain an optimum estimate of the

distribution system states and perform harmonic estimation.

The application of a dynamic method to harmonic estimation and source identifi-

cation requires accurate modelling of system components and harmonic injections. A

state-space model is required to represent the dynamic model of the distribution sys-

tem. Various methods of obtaining a dynamic representation of a given power system

have been proposed in literature [15–17]. The state-space representation comprises of

a set of differential equations which describe the interaction between system states and

measurements. These measurements need to be optimally placed throughout the power

system to ensure complete observability. A power system is said to be observable if

the system states can be accurately estimated from the measurements [18].

This thesis is focused on the use of observers for power system harmonic estima-

tion and source identification. An observer is a system which provides state estimates

of a given system using available measurements. Observers are applied to harmonic


estimation and source identification due to their ease of implementation and reduced

computational burden. An iterative observer algorithm is proposed to carry out har-

monic estimation in multiple measurements. This algorithm is capable of determining

the individual harmonic orders present in a number of measurements simultaneously.

The proposed algorithm is designed to only carry out harmonic estimation when the

measurements are distorted. This is achieved by computing the THD and measure-

ments with THD exceeding a threshold are subjected to the iterative observer algo-

rithm. The threshold may be chosen based on the distortion limits in Tables 1.1 and

1.2. Thus, harmonic estimation is carried out for measurements which do not comply

with regulatory requirements.

Furthermore, an observer-based algorithm is proposed to identify multiple har-

monic sources in power distribution systems. Ideally, the number of harmonic sources

is unknown. A combination of system nodes is applied to the algorithm and the residual

from observer-based estimation is analysed for each combination. If the residual ap-

proaches zero, then the specified combination is identified as a harmonic source. This

dynamic method of identifying harmonic sources takes into account the time varying

nature of system loads and variations in harmonic sources with time. The proposed

observer-based algorithm is also modified to account for time delay in system mea-

surements. As a result of the time it takes the measurements to get to the control

station for processing, the measurements may become unsynchronised or delayed. An

observer is designed to take into account the delay in system measurements and this

observer is incorporated into the algorithm to identify harmonic sources.

1.3 Aims and objectives

This thesis focuses on two main aims:


1. To determine the amplitude and phase of each harmonic order in a measured

voltage or current signal.

2. To identify the harmonic sources in power distribution systems.

To achieve these aims the following objectives have been provided

1. Determination of the state-space dynamic model of the power distribution sys-

tem .

2. Determination of the optimum locations for meter placement to ensure observ-

ability of the system.

3. Formulation of an algorithm to perform harmonic estimation in voltage and cur-

rent measurements.

4. Identification of harmonic sources in power distribution systems when the mea-

surements are synchronised.

5. Identification of harmonic sources in power distribution systems with unsyn-

chronised measurements and delay in the outputs.

1.4 Outline of thesis

Besides chapter 1, the rest of this thesis is outlined as follows:

Chapter 2 is divided into two parts; firstly, a method of determining the state-space

matrices of a given power distribution system is presented. A brief literature survey

of existing methods of distribution system dynamic modelling is given. The mathe-

matical description of the system components is determined using Kirchhoff’s voltage

and current laws at all nodes and branches to create a set of differential equations de-

scribing the system. The second part of this chapter describes an optimum method


of placing measurement devices in power distribution systems. The concept of nu-

merical and topological observability is described and a survey of existing methods

of optimum meter placement is carried out. The method of binary integer linear pro-

gramming combined with sequential elimination is presented. Binary integer linear

programming ensures topological observability, while sequential elimination ensures

numerical observability of the power system. A case study is carried out on a four-node

test feeder for both optimum measurement placement and dynamic modelling.

Chapter 3 presents an algorithm for determining the amplitude and phase of each

harmonic order present in measured signals. An extensive review of existing literature

on current harmonic estimation methods is carried out. The power system is mod-

elled as a process at the fundamental frequency and harmonic injections are taken as

exogenous disturbances to the system. A dynamic representation of the harmonic dis-

turbance is derived and an observer for harmonic estimation in a single measurement

is designed. This observer operates on the principle that a previous observer design ex-

ists for estimating the power system states and the iterative observer algorithm builds

on this existing observer design. The iterative observer is able to estimate further har-

monic orders as and when they are injected into the power system with time. Simulta-

neous estimation of harmonic amplitudes and phase angles in multiple measurements

is also presented. In addition, the proposed iterative observer algorithm only carries

out harmonic estimation when the measured signal is distorted. This is determined by

computing the THD and comparing it against a set threshold, which is determined to

be in compliance with regulatory requirements. The algorithm is applied to the IEEE

13-node test feeder measurements and a number of scenarios are simulated to verify

the robustness and accuracy of the iterative observer algorithm. A comparison between

the results obtained using the iterative observer algorithm and the Kalman-Bucy filter

is presented.


Chapter 4 provides an observer-based algorithm for identifying harmonic sources

in power distribution systems. A survey of existing methods of identifying harmonic

sources based on single point and multiple point methods is carried out. The harmonic

disturbance matrix represents the relationship between harmonic injections and the

distribution system state variables. This matrix is computed for each node combination

and the observer is used to estimate the system states at the specified combination.

If the estimation error approaches zero, then the specified combination is taken as

the harmonic source and vice versa. For the case of multiple harmonic sources, the

system equations are divided into sub-systems to reduce the computational burden.

The algorithm is applied to the IEEE 13-node distribution system for the case of a

single harmonic source and multiple harmonic sources. In addition, the response of

the algorithm to changes in the amplitude of harmonic injections and variations in

harmonic sources with time are investigated.

Chapter 5 extends the harmonic source identification algorithm to cases where the

measurements are not synchronised. This may occur if there is a phasor measurement

units (PMU) outage or the measurements are delayed during transmission to a control

station for processing. Cases of a single output delay and multiple output delays are

considered. The observer design for both scenarios are incorporated into the fault

observer based algorithm for harmonic source identification. A case study is presented

at the end of the chapter for both cases of single and multiple output delays on the

IEEE 13-node test feeder.

Chapter 6 draws concluding remarks on the contributions of this thesis and pro-

vides suggestions for future research.

Chapter 2

Power distribution system dynamic

modelling and optimum measurement

placement

2.1 Dynamic modelling

2.1.1 Introduction

This chapter entails determination of the dynamic model of a given power system

as well as optimum placement of measurement devices in power distribution systems.

To obtain accurate results for harmonic studies, it is important to determine the dy-

namic model of the system. Knowledge of distribution system states is essential to

enable utility companies accurately identify the harmonic sources and perform har-

monic estimation. A typical power distribution system consists of a source, usually the

power grid, lines/cables, system loads and transformers. Figure 2.1 shows the structure

of a typical power distribution system. To obtain the state space model of the system,

the dynamic model of these individual components is determined [19]. These dynamic

31

CHAPTER 2. DYNAMIC MODELLING 32

Figure 2.1: Power distribution feeder

models are then interconnected to form the state space model of the entire distribution

system.

In order to effectively implement the proposed observer methods for harmonic es-

timation and source identification, the dynamic model of the power distribution system

is required. Distribution system dynamic modelling is essential because the time evo-

lution of the system is described using a set of differential equations. In this section,

the state-space model of a given power distribution system is obtained by applying the

modelling of individual system components to all nodes in the power system.

2.1.2 Literature review on distribution system dynamic modelling

A number of methods have been introduced in literature for obtaining the dynamic

model of a power distribution system. In [16], a clustering procedure is introduced such

that the input and output dynamic responses of the network are approximated. A grey-

box approach was used as a result of its capability to incorporate prior knowledge of

the power system parameters. Partial differential equations were proposed in [15] for


determining power system state space equations when the lines are represented using

distributed parameters. Due to the time consuming nature of numerical integration

methods, closed form solutions were used to compute the steady-state and transient

responses of the line models. Subsequently, the state-space model of a transmission

line with a fault arc using the lumped parameter model was determined in [20]. An

approach to state space modelling of power systems was introduced in the appendix

of [17]. Each component of the system was individually modelled and combined to

create the dynamic model of the entire power system.

2.1.3 Distribution system dynamic modelling

For accurate determination of the state-space equations of a given power distribu-

tion system which contains a harmonic disturbance, four variables are considered:

1. System inputs, u(t)

2. System outputs, y(t)

3. System states, x(t)

4. Harmonic disturbance states, w(t)

The system inputs may be considered as the power grid, which may be represented

as a voltage source with an impedance. Various sections of the system are modelled

and the resulting state space model is obtained from a combination of the individually

modelled components of the system.

The state-space representation of a power distribution system with harmonic dis-

turbances is given by

x = Ax+Bu+Ew (2.1)


y =Cx (2.2)

where x ∈ Rn is the state vector, u represents the system inputs, w is the harmonic

disturbance, y is the output and A, B, C, E are matrices of known dimensions.

Consider a typical power distribution system which consists of a grid connected

voltage source, overhead and underground cables, shunt capacitors, transformers, lin-

ear and nonlinear loads. The linear loads in the distribution network may be modelled

as series RL loads. On the other hand, nonlinear loads can be represented as linear

loads in parallel with current sources at the specified harmonic frequency as shown in

Figure 2.2, where ih is the harmonic current amplitude at harmonic order h. The state-

∑=

=v

hhii

2

Figure 2.2: Nonlinear load representation

space matrices of the distribution system may be calculated based on the following

assumptions:

1. Generators are modelled as a voltage source with an equivalent impedance. This

impedance is calculated using the short circuit capacity.

2. Lines and cables are represented by the π model.

3. Linear loads are modelled as constant RL impedances at the fundamental fre-

quency.

4. Nonlinear loads are modelled as a linear load in parallel with a current generator.


Each of the system components is modelled separately to determine the state space

model and the overall state space model of the system is obtained by aggregating the

state space model of the individual components.

2.1.3.1 Generator/grid modelling

Generally, distribution systems are connected to the medium voltage or high volt-

age grid and the voltages are stepped down using transformers to provide power to

consumers. In order to obtain the dynamic model of the grid for harmonic studies,

certain parameters are required such as:

• The system voltage, Vs.

• Single phase and three phase short circuit current

• X/R ratio

• The single phase and three phase short circuit capacity

These parameters are used to determine the source impedance parameters. The grid

may be modelled as a voltage source in series with an impedance, referred to as the

source impedance. A single phase equivalent of the generator model is shown in Fig

2.3, where Vs is the source voltage, R,L and C are the resistance, inductance and ca-

pacitance respectively and Vi is the voltage at node i.

For a three phase generator, the equivalent model may be represented as shown in

Figure 2.4, with no inter-phase mutual coupling between phases. The mathematical

formulation for each phase of the generator is given by

Vs =Vi + IsRs +LsdIs

dt+Vcs (2.3)

Is =CsdVcs

dt(2.4)


Figure 2.3: Single phase generator model

Vs Vs 1+ Vs 2+

Rs

Rs 1+

Rs 2+

Ls

Ls 1+

Ls 2+

Vi V i 1+ Vi 2+Cs

Cs 1+

Cs 2+

is

i s 1+

is 2+

Figure 2.4: Three phase generator model

where Vs is the grid voltage, Vi is the voltage at node i, which is the closest node con-

nected to the grid Vcs, the source capacitor voltage [21], Is represents the source current

and Rs,Ls,Cs are the source resistance, inductance and capacitance respectively. The

state equations representing the grid dynamics per phase are thereby expressed as

Is

Vcs

=

−RsLs− 1

Ls

1Cs

0

Is

Vcs

+ 1

Ls− 1

Ls

0 0

Vs

Vi

(2.5)


where Is is the time derivative of Is.

2.1.3.2 Cable modelling

Distribution system cables consist of inductance, resistance and capacitance which

represent the magnetic and electrostatic conditions of the line. These cables, including

overhead and underground cables, can be modelled using nominal π circuits. Figure

2.5 shows the three-phase nominal π model,

where Vs = Sending end voltage

R x33 L x33

C xsh 33, C xsh 33,

I b

Vs Vs 1+ Vs 2+ VrV r 1+Vr 2+

is

is 1+

is 2+

ir

ir 1+

ir 2+

Figure 2.5: Three phase cable π model

Vr = Receiving end voltage

R = Series resistance

L = Series inductance

Csh = Shunt capacitance

Ib = Branch current

Is = Sending end current

Ir = Receiving end current.


For distribution systems, cables are modelled using the π model. Given a line with

length l km, the series impedance is expressed as

z = r+ jx Ω/km (2.6)

where z is the series impedance of the cable in Ω/km. The impedance is obtained by

multiplying (2.6) by the line length, l. At each harmonic order, the total impedance of

the line is given by

Z(h) = R+ jhX (2.7)

Consider a given line section in the distribution feeder connecting nodes i and j. To

obtain the state-space equations for this line section, Kirchhoff’s voltage law (KVL) is

applied to the line section yielding:

Vi = Ii jRi j +Li jdIi j

dt+Vj (2.8)

Hence,

Ii j =−Ri j

Li jIi j−

1Li j

Vj +1

Li jVi (2.9)

Applying Kirchhoff’s current law (KCL) at node i, we have

Is = Ii j +Csh

2dVi

dt(2.10)

Hence,

Vi =2

CshIs−

2Csh

Ii j (2.11)

Applying KCL at node j:

Ii j = Ir +Csh

2dVj

dt(2.12)


Hence,

Vj =−2

CshIr +

2Csh

Ii j (2.13)

where Vi and Vj are the voltages at nodes i and j respectively, Ri j, Li j, Csh are the cable

resistance, inductance and shunt capacitance respectively. Is and Ir are the sending and

receiving end currents respectively.

In general, the state-space model of a power distribution system cable connecting node

i to node j is given by

Ii j

Vi

Vj

=

−Ri j

Li j1

Li j− 1

Li j0 0

− 2Csh

0 0 2Csh

0

2Csh

0 0 0 − 2Csh

Ii j

Vi

Vj

Is

Ir

(2.14)

2.1.3.3 Load modelling

Loads in a power system may be either residential, commercial or industrial. Trans-

mission systems may feed very large industrial loads while smaller industrial loads are

more likely to be served from the subtransmission network. Loads which are typi-

cal to distribution systems may be commercial or residential. The real and reactive

power components of these loads can be represented by an equivalent series or parallel

impedance. Figure 2.6 shows a three phase equivalent load model. The voltage at node

k is given by

Vk = IkRk +Lk Ik (2.15)

Ik =−Rk

LkIk +

1Lk

Vk (2.16)


Vk Vk 1+ Vk 2+Rk Lk

Rk 1+ Lk 1+

Rk 2+ Lk 2+

Ik

I k 1+

Ik 2+

Figure 2.6: Three phase load model

where Ik is the time derivative of Ik. Hence, the state space model for loads per phase

at node k is given by Ik

Vk

=

−RkLk

1Lk

0 2Ck

Ik

Vk

(2.17)

The differential equations representing the voltage at node k, Vk can be obtained from

cable modelling or transformer modelling, depending on the closest component adja-

cent to that node.

2.1.3.4 Transformer modelling

Consider a transformer connecting two nodes in a given power distribution system.

Figure 2.7 shows a three phase ∆-wye grounded transformer equivalent circuit. The

resistance and inductance per phase is given by

RT =

Ri,i 0

0 R j, j

(2.18)


ii i j

Vj Vj 1+ Vj 2+Vi Vi 1+ Vi 2+

ii 1+ i j 1+

i i 2+ i j 2+

Figure 2.7: Three phase ∆-wye grounded transformer equivalent circuit

LT =

Li,i Li, j

L j,i L j, j

(2.19)

where Li,i represents the self inductance and Li, j represents the mutual inductance. For

a three-phase transformer model, the resistance and inductance are given by

RT =

Ri,i 0 0 0 0 0

0 R j, j 0 0 0 0

0 0 Ri+1,i+1 0 0 0

0 0 0 R j+1, j+1 0 0

0 0 0 0 Ri+2,i+2 0

0 0 0 0 0 R j+2, j+2

(2.20)


LT =

Li,i Li, j 0 0 0 0

L j,i L j, j 0 0 0 0

0 0 Li+1,i+1 Li+1, j+1 0 0

0 0 L j+1,i+1 L j+1, j+1 0 0

0 0 0 0 Li+2,i+2 Li+2, j+2

0 0 0 0 L j+2,i+2 L j+2, j+2

(2.21)

The transformer state-space model is therefore given by

ii

i j

ii+1

i j+1

ii+2

i j+2

= L−1

T RT

ii

i j

ii+1

i j+1

ii+2

i j+2

+L−1

T

vi− vi+1

v j

vi+1− vi+2

v j+1

vi+2− vi

v j+2

(2.22)

where ii and vi represents the primary current and voltage at node i respectively and i j

and v j represent the secondary current and voltage at node j respectively. If the core

losses are neglected, the first term in (2.22) is negligible and only the second term is

used.

To determine the state-space model of a given power distribution system, KVL and

KCL are applied on all nodes and branches of the system and the state-space models

of these individual components are utilised.


2.2 Optimum measurement placement in power distri-

bution systems

2.2.1 Introduction

An optimum measurement placement scheme is required for accurate estimation

of power system state variables. The aim of this scheme is to ensure that the optimum

number of measurement devices are used and that they are placed in strategic locations

within the system to ensure complete observability. A system is said to be observable

if a set of measurements is sufficient to estimate all the system states [18]. The term

‘observability’ is a measure of how well the system states can be estimated from the

outputs or measurements. The observability can be verified by ensuring that the ob-

servability matrix of the system has full rank. This matrix not only depends on the

measurement matrix, but also the state transition matrix.

Power system observability analysis may be determined by using these criteria

[18]:

1. Numerical observability

2. Topological observability

A power system is said to be numerically observable if the measurement model is suf-

ficient to solve for the state variables; a closely related term is algebraic observability.

Algebraic observability is verified if the rank of the measurement matrix is equiva-

lent to the number of state variables to be estimated [18]. If a system is algebraically

observable, this does not guarantee numerical observability, however, numerical ob-

servability guarantees algebraic observability.

The topological observability of a power system is guaranteed if there exists a

spanning tree of full rank. The topological approach utilises graph theory to determine


the observability of the power system based on the type and location of measurements.

Observability analysis is an important aspect of power system state estimation because

the accuracy of state estimates depends on it. If inadequate measurements are utilised

or the measurements are not placed at the optimum locations, the proposed estimator

may not accurately reconstruct all the system states using the measurement locations

provided.

2.2.2 Literature review on optimum meter placement and observ-

ability analysis

Various methods have been proposed for determining the optimum locations to

place measurements in a power system to ensure that the system is observable. The

minimum condition number of a measurement matrix based on sequential elimination

was introduced in [22]. The condition number of a matrix is defined as the ratio of the

largest singular value to the smallest. A matrix is said to be singular if the condition

number is infinite and ill-conditioned if the condition number is too large [23]. Hence,

the combination of measurement points which yields the minimum condition number is

chosen as the set of measurements. Singular value decomposition (SVD) was proposed

in [23] for optimum placement of meters in power systems. One advantage of SVD is

the elimination of the need for observability analysis prior to state estimation. This

method may be used to detect the unobservable regions of the system. A genetic

algorithm (GA) based meter placement approach was proposed in [24] to solve the

optimum placement of meters for static harmonic state estimation. In [25], a method

of determining the lines to place the measurement devices was developed. Line current

measurements were used instead of node measurements due to the increased sensitivity

of line currents to harmonic injections. The locations of the harmonic sources were


assumed to be known a priori. A seeker optimisation algorithm was employed in [26]

for identifying the optimum locations to place power quality meters for harmonic state

estimation (HSE). The weighted sum of relative errors for HSE was minimised for

each harmonic order.

These meter placement algorithms presented have been applied to transmission

systems. However, distribution systems only have a limited number of measurements

available due to the cost and the large number of nodes present. Placement of harmonic

meters in distribution systems was explored in [27]. An optimisation algorithm based

on dynamic programming was proposed to determine the optimum number and loca-

tions to place measurement devices in electric distribution systems to ensure complete

observability. Particle swarm optimisation was also presented in [28] for solving the

meter placement problem in distribution systems. This approach may be extended to

contingencies such as loss of measurements or branch outages.

In recent years, the use of PMU has been researched extensively for power system

monitoring and control. A PMU is a device which uses state-of-the-art digital signal

processors to measure voltage or current waveforms typically at a rate of 48 samples

per cycle [29]. PMUs provide real time synchronised phasor measurements for power

system studies such as state estimation and fault location. Conventional supervisory

control and data acquisition (SCADA) measurement systems do not have the capability

to provide real time synchronised measurements. Another advantage of PMUs over

SCADA measurement systems is PMUs are able to provide the current measurements

at the branches incident to a node in addition to the node voltage when the PMU is

placed at that node.

PMU placement methods in power systems have been widely explored. One of

the most common methods of PMU placement is integer linear programming [30, 31].

In [32], an integer quadratic approach was used to determine the minimum set of PMUs


to ensure complete observability on a single island. The outage of a single line or PMU

was factored into the design process and measurement redundancy was reduced. One

drawback of this approach is that it is time consuming and not applicable to large

power systems. Although both integer linear and integer quadratic programming have

been useful for solving the optimum PMU placement problem, this approach only

guarantees topological observability. Numerical observability is not guaranteed with

these two methods.

2.2.3 Optimum meter placement method based on binary integer

programming

In this section, binary integer linear programming is used to determine the opti-

mum number and locations to place measurement devices in power distribution sys-

tems. The installation of PMUs at the distribution level is not cost effective due to the

large number of nodes involved. In addition, the methods proposed in this thesis are

mainly carried out in the time domain. Hence, the results obtained from PMU mea-

surements may not be applicable to the methods proposed as PMUs provide phasor

measurements. Regardless, the mode of operation of PMUs is adopted to solve the

meter placement problem in distribution systems. The identification of nodes to take

voltage measurements is followed by the addition of a branch current measurement on

at least one branch which is adjacent to the identified node. This way, topological ob-

servability can be satisfied and the resulting measurement matrix is analysed to verify

the observability of the system using the proposed set of measurements.

To ensure both topological and numerical observability, the method of meter place-

ment proposed in [33] is adopted. The method proposed in this literature was applied

to placement of PMUs. However, as a result of the time domain approach used in


this thesis, PMUs are not used because of measurements which are taken in phasor

mode. Instead, voltage and current measurements are used at the system nodes and

branches. A two-stage method is proposed in this thesis. The first stage involves de-

termining the minimum number of voltage measurements required to make the system

topologically observable. The second stage involves adding the current measurements

on the lines incident to the nodes where voltage measurements are taken. Then the

resulting measurement set is checked for numerical observability. In the event that nu-

merical observability is not achieved, a sequential elimination algorithm is proposed.

To achieve optimum measurement placement, the following assumptions are made:

Assumption 1. 1. If the voltage and current at the sending end of a line are mea-

sured, the voltage at the receiving end of the line can be calculated using Ohm’s

law and the line impedance parameters.

2. If the voltage at both ends of a line are measured, the line current can be deter-

mined using the impedance

Obtaining voltage measurements at a node can be seen as a binary decision variable

such that

u j =

1 if a voltage measurement is taken at node j

0 otherwise

(2.23)

Consider a p-node system, with the cost of placing a meter at node i is ci. The optimi-

sation problem is formulated as:

Minimizep

∑i=1

ciui (2.24)

subject to the constraints:

PU ≥ 1 (2.25)


where ui is a binary decision variable, ci represents the cost of obtaining a voltage

measurement at node i which is taken as 1, P ∈ Rp×p is a binary connectivity matrix

with entries ki j such that

ki j =

1 If i = j or i and j are connected

0 otherwise

(2.26)

The solution to this minimisation provides an indication of the minimum number of

voltage measurements required and the optimum locations to place them to ensure

topological observability of the p-node system.

For the second stage of the optimum meter placement method, the resulting ob-

servability matrix from the first stage is checked to verify if it is full rank. This observ-

ability test is carried out using the Popov-Belevitch-Hautus(PBH) rank test [34]. The

observability matrix based on this method is given by

O =

C

A−λI

(2.27)

In the event that the rank of O is not full, then additional measurements need to be

included to guarantee numerical observability. Each row of the measurement matrix

represents a node voltage or branch current measurement. The method of sequential

elimination entails the continuous addition of each of the system node voltage and

branch current measurements to the measurement matrix and analysing the resulting

observability matrix to determine the additional measurement required to offset the

rank deficiency.

The method of sequential elimination is as follows:

1. Create a set consisting of the system measurements which ensure topological


observability, Pin.

2. Create another set of measurements, Pnew which consist of the rest of the system

measurement points not contained in Pin.

3. Append both measurement sets, Ptot = [Pin,Pnew].

4. Create the measurement matrix consisting of the measurement set Ptot .

5. Remove each row of the new measurement set and compute the rank of O

6. If rank(O) is full, then remove the measurement set which gives full rank, else

restore the original measurement previously removed and continue iteration until

all additional measurement sets whose removal yield full rank have been elimi-

nated.

The flowchart for sequential elimination is shown in Figure 2.8, where l is the total

number of measurements including Pnew.

2.3 Case study

This section determines the state-space model of a benchmark distribution system

and optimum placement of measurements in the system. Consider the four-node test

feeder given in [35]. This feeder was originally designed to carry out testing of all

possible three phase transformer connections. The one-line diagram of the feeder is

shown in Figure 2.9. The point of connection of the feeder to the grid is represented as

a voltage source behind an impedance. This source is a 12.47 kV line-to-line infinite

bus. The transformer is rated at 6 MVA, 12.47 kV Wye - 4.16 kV Delta, Z = 1+ j6%.

The system is assumed to be balanced with delta connected load parameters given


Ptot

Ptot

ο i

ο i

!

" li =

Figure 2.8: Sequential elimination method for optimum measurement placement

as 1800 kW, 871.8 kVar. For the cables, a four-wire configuration is used and the

sequence impedances are given by

z1 = 0.3060+ j0.6272Ω / mile (2.28)


3 421

InfiniteBus

Load34

[I ]12 [I ]

2000 ft. 2500 ft.

Figure 2.9: One line diagram of the four-node test feeder

z0 = 0.5919+ j2.9855Ω / mile (2.29)

where z1 is the positive sequence impedance and z0 is the zero sequence impedance.

The line lengths are 2000ft and 2500ft for lines 1-2 and 3-4 respectively.

To determine the optimum locations to place meters in this system, the binary con-

nectivity matrix is given by

P =

1 1 0 0

1 1 1 0

0 1 1 1

0 0 1 1

(2.30)

Binary integer linear programming is applied to determine the optimum meter place-

ment positions. The binary decision variable U , which is the solution to the minimisa-

tion in (2.24), is computed as:

U =

0

1

1

0

(2.31)

This indicates that the voltage measurements should be taken at nodes 2 and 3 and cur-

rent measurements at the corresponding branches. By inspection, we may observe that

if a voltage measurement is taken at node 2, the voltage at node 1 can be determined


using Ohm’s law, given the branch impedance. In like manner, the voltage measure-

ment at node 3 provides an indication of the voltage at node 4, given the branch current

measurements and branch impedance. This shows that the system is topologically ob-

servable with this set of measurements. To guarantee numerical observability, the rank

of the resulting observability matrix is examined.

For a three-phase model of the four node system the states include:

1. Three phase-phase load currents at node 4

2. Three branch currents from node 1-2

3. Three line-line voltages at node 1

4. Line-neutral voltage at node 1



7. Three branch currents from node 3-4





12. Transformer primary currents for phase 2 and 3

13. Three transformer magnetising currents

14. Three transformer secondary currents


15. Three source currents

Thus, the total number of states in the three-phase four wire distribution system is 36.

The phase 1 transformer primary current is a dependent state hence it is not included

in the states. A total of 12 measurements are taken from the three-phase system. This

includes three voltage and three current measurements at nodes 2 and 3 respectively.

The rank of the observability matrix using this measurement set is 36, which is equal

to the number of states. Hence, the system is observable.

The load at node 4 is a nonlinear load with harmonic spectrum shown in Table 2.1.

The nonlinear load is represented as a linear load in parallel with a current generator

as shown in Figure 2.2. The spectrum in Table 2.1 applies to phase A. For phase B and

Table 2.1: Harmonic spectrum for the nonlinear load at node 4

Harmonic order Amplitude(%) Phase ()

1 100.00 -24.77

3 14.95 273.40

5 6.22 339.00

7 2.19 137.70

9 1.05 263.2

11 0.12 39.80

C, the same amplitude is used and the phase angles are varied by ±120.

The system is assumed to be balanced. Consider a single phase model of the power

distribution system. The states of the single phase model are given by

1. Source current

2. Current at the primary winding of the transformer

3. Current at the secondary winding of the transformer

4. Transformer magnetising current


5. Voltage at node 1

6. Branch current from node 1-2



9. Branch current from node 3-4


Since the system is balanced, the state space representation for the single-phase model

can be concatenated to form the state space model for the three-phase model. For the

three-phase four-wire configuration, there are additional states representing the line-

neutral voltage for each node. The state space matrices for a single-phase model of

this feeder are given by

A =

−4.42e1 0 0 0 −2.24e3 0 0 0 0 0

0 −4.93e7 −1.65e7 4.93e7 0 0 3.81e3 0 0 00 −1.48e8 −4.93e7 1.48e8 0 0 0 3.42e4 0 00 3.14e2 1.05e2 −3.14e2 0 0 0 0 0 0

2.58e8 0 0 0 0 −2.58e8 0 0 0 00 0 0 0 1.32e3 −2.47e2 −1.32e3 0 0 00 −2.58e8 0 0 0 2.58e8 0 0 0 00 0 −2.06e8 0 0 0 0 0 −2.06e8 00 0 0 0 0 0 0 1.06e3 −2.47e2 −1.06e30 0 0 0 0 0 0 0 2.06e8 0

(2.32)

B =

[2.24e3 0 0 0 0 0 0 0 0 0

]T

(2.33)

C =

0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 1 0 0

0 1 0 0 0 0 0 0 0 0

0 0 −1 0 0 0 0 0 0 0

(2.34)


The rank of the observability matrix is 10, which is equivalent to the number of states

for the single phase equivalent model. This indicates that the proposed meter place-

ment method guarantees both topological and numerical observability for the single

phase system. To determine the state-space model of the three-phase system, the same

measurement points are used. Voltage measurement at a node implies measurements

for all three phases at that node.

2.4 Summary

This chapter has provided a method of determining the state space model of a given

power distribution system. In addition, a binary integer linear programming approach

was used to determine the optimum locations to place the measurement devices in the

power system. The resulting meter placement positions guarantee both topological

and numerical observability of the distribution system. This approach of determining

the state-space model of a power distribution system and optimum placement method

is instrumental in the design of state observers for both harmonic estimation and har-

monic source identification. Results were obtained for a four node balanced power

distribution system.

Chapter 3

Harmonic estimation

3.1 Introduction

Harmonic estimation involves determination of the amplitude and phase compo-

nents of the harmonic frequencies present in a distorted voltage or current signal. This

is vital in harmonic analysis because it provides an indication of the level of penetra-

tion of harmonic disturbances in the power distribution or transmission system. This

chapter presents an iterative observer algorithm to carry out harmonic estimation in

power distribution system measurements. An important feature of the proposed itera-

tive observer design is that harmonic estimation is only implemented when the mea-

surement is distorted. The level of distortion is determined by calculating the THD

of the measured signal. This feature is an advantage because computation time and

cost are conserved by not carrying out estimation if it is not required. The framework

for the iterative observer based harmonic estimation is shown in Figure 3.1, where w

represents the harmonic disturbance and u and y represent the system inputs and out-

puts respectively. This chapter is organised as follows: firstly, a review of existing

methods of harmonic estimation is explored. Secondly the proposed iterative observer

56

CHAPTER 3. HARMONIC ESTIMATION 57

u

w

y

w∧

Figure 3.1: Harmonic estimation using the iterative observer

method of harmonic estimation is proposed. A case study is also presented to verify

the validity, accuracy and robustness of the proposed iterative observer algorithm.

3.2 Literature review on harmonic estimation

A wide range of methods have been used in literature for the estimation of general

periodic disturbances. These disturbances are periodic in nature, but non-sinusoidal.

In [36], an adaptive regulation approach to estimation of sinusoidal disturbances where

the disturbance is represented as an exogenous input which is not available for mea-

surement is presented. A Youla parameterisation of stabilising controllers, also known

as Q parameterisation, is applied to the system to achieve regulation.

Estimation of the harmonic components of a measured signal has been extensively

explored in literature and a plethora of methods have been introduced. The discrete

Fourier transform (DFT) which is implemented using the fast Fourier transform (FFT)

has been the most common method of harmonic estimation for decades [37–40]. A


transformation is carried out on the signal from the time domain to the frequency do-

main and the harmonic components can then be estimated. An advantage of the FFT

is its simplicity and ease of implementation. As a result of the time-varying nature of

the harmonic loads in distribution systems, the FFT may be unsuitable for harmonic

estimation due to aliasing, leakage and picket fence effects [41–44]. In addition, the

FFT does not take into account the dynamic nature of distribution system loads and

estimation can only be carried out on one measurement at a time. Due to the loss of

time information, transients cannot be estimated using the FFT.

Apart from the FFT, other methods have been proposed in literature for estimation

of harmonics in measured signals. In general, these methods may be classified into

parametric and non-parametric methods [3, 45, 46]. Non-parametric methods distin-

guish the signal into time-dependent-frequency or frequency dependent components.

In parametric methods, non steady-state power system harmonics are formed by mod-

elling the harmonic disturbances and estimating the disturbance parameters of interest

in the model. Figure 3.2 shows a summary of the classification of harmonic estimation

methods. From the figure, four main non-parametric methods are presented: the DFT,

wavelet transform (WT), Hilbert-Huang transform (HHT) and the chirp-z transform

(CZT). The wavelet transform [47–49] carries out harmonic estimation by decompos-

ing signals using multiple time-frequency resolutions. Although this method is easy

to implement with a filter bank, the accuracy is compromised due to the arbitrary se-

lection of the mother wavelet [3]. Also, additional tools are required for interpretation

of the transformed parameters. The HHT was applied in [50, 51] to carry out har-

monic and interharmonic estimation to overcome the shortcomings of the FFT and the

wavelet transform method [52]. The signals are decomposed using empirical mode de-

composition method and a series of intrinsic mode functions (IMF) are obtained. The

CZT proposed in [53] utilises a synchronisation technique such that the fundamental


Harmonic estimation

methods

Non-parameteric method Parametric method

Frequency

domain analysis

Time-frequency

domain analysis

Sinusoidal

models

Stochastic

models

• Discrete Fourier

Transform (DFT)

• Wavelet transform

• Hilbert-Huang

transform

• Chirp z-transform

• ESPRIT

• MUSIC

• Kalman

Filter

• Autoregressive

model

• ARMA

• Prony

Figure 3.2: Classification of harmonic estimation methods [3]

frequency of the signal is detected and applied to set the sampling frequency.

Parametric methods of harmonic estimation include SVD [54–56], estimation of

signal parameters via rotational invariance technique (ESPRIT), multiple signal classi-

fication (MUSIC), autoregressive moving average (ARMA), autoregressive model and

Prony’s method. Prony’s method has been widely used in carrying out harmonic esti-

mation [57,58]. In [59], a Prony-based method combined with downsampling was used

for harmonic and inter-harmonics detection in a measured signal. To prevent the leak-

age effect, SVD was used to determine the Prony’s model coefficients. One drawback

of Prony’s method is the computational burden in the root finding process. A modi-

fied Prony-based method was presented in [60] to address this problem. In [61, 62],

ESPRIT was introduced for estimation of harmonics. ESPRIT uses sinusoidal or com-

plex exponential models to decompose the covariance matrix into noise and signal

subspace. Sliding window ESPRIT was also introduced in [63] for non-stationary data

and in [64], a combination of sliding window ESPRIT and DFT were introduced for

power system waveform distortion assessment. ESPRIT is first used to estimate the

fundamental frequency and inter-harmonics, then DFT was applied for more accurate


windowing. Parametric methods of harmonic estimation provide better accuracy and

a higher frequency resolution [65]. However, they are sensitive to noise and require

higher computational cost.

Many neural network techniques have been proposed for estimation of harmonics

in a measured signal [66–70]. The adaptive linear neuron (ADALINE) was presented

in [71, 72] for harmonic estimation. In [73], a two-stage ADALINE was presented

for both harmonics and inter-harmonic estimation. This method is easy to implement

and may be used for online tracking of time varying signals. It is also robust against

noise and provides better convergence. However, the results are inaccurate if the signal

contains harmonics which are not included while modelling the ADALINE. Although

this problem is solved in [73], the computation time required is large. In [74], an

adaptive wavelet neural network (AWNN)-based approach to harmonic estimation in

single phase systems was presented. A combination of ESPRIT and AWNN was also

presented in [42]. The AWNN provides fast harmonic estimates while the ESPRIT

method was used to handle time-varying signals with improved accuracy.

So far, all these methods of harmonic estimation are static methods. The dynamic

nature of the signals which are a result of the time-varying nature of distribution sys-

tem loads were not considered using these methods. In order for utility companies to

have knowledge of the changing harmonic injections which result from the constantly

varying loads, a dynamic method of harmonic estimation is required.

The Kalman filter is a commonly used dynamic method for estimation of harmonics

in a measured signal. The Kalman filter was first introduced for harmonic analysis

in [75]. Over the years, the Kalman filter has been extensively applied to harmonic

analysis in distorted signals [76–79]. Various modifications have been made to the

Kalman filter and some of these modifications include the adaptive Kalman filter [80,

81], extended Kalman filter [82–84], unscented Kalman filter [85, 86] and ensemble


Kalman filter [87]. In [88], Prony’s method was combined with a Kalman filter to

estimate the amplitude, phase and frequencies of each harmonic order present in the

signal to be analysed. The Kalman filter approach is advantageous due to its ability

to track the time-varying characteristics of the harmonic signal, its robustness against

noise and the recursive method applied. However, prior knowledge of the process and

measurement noise covariances is required to correctly implement this approach. In

addition, the Kalman filter often suffers from ‘filter dropping off’ and may become

insensitive to variations in state variables.

One drawback of previously used methods of harmonic estimation is that the si-

multaneous estimation in multiple measurements has not been explored. Estimation

is carried out one measurement at a time and estimation on multiple measurements

simultaneously has not been investigated. If, however, there is a need to estimate the

harmonics from another measurement taken from a separate point in the power system,

estimation needs to be carried out again at that point to obtain the required information.

Although the possibility of modifying the Kalman filter to simultaneously carry out

harmonic estimation in multiple measurements exists, the computation time required

is very large.

Observers have been widely used in literature for sinusoidal disturbance estimation

[89–91]. However, the case of harmonic estimation in power distribution systems using

observers has not been explored.


3.3 Iterative observer design for harmonic estimation

in power distribution systems

An iterative observer algorithm is presented for harmonic estimation in voltage or

current measurements. This involves the determination of the amplitude and phase an-

gle of each harmonic order present in the measured signal. Figure 3.3 shows a model

of the power distribution system with harmonic injections. These harmonic injections

Figure 3.3: Power distribution system with harmonic disturbance

may be modelled as a disturbance to the system and their characteristics (amplitude

and phase) are estimated from available measurements. This approach is based on the

assumption that an observer has been previously designed to estimate the power sys-

tem states without the consideration of the harmonic disturbance [92]. The operating

principle of the proposed iterative observer is based on the method of observer design

for estimation of external disturbances proposed in [93] and the asymptotic estimation

and rejection of periodic disturbances in [94].

An important feature of the proposed iterative observer is its ability to carry out

harmonic estimation based on information from prior observer designs. Estimation is

carried out one harmonic at a time until all harmonic orders present in the signal have


been estimated and the estimation error, e, satisfies the condition

limt→∞

e(t)→ 0 (3.1)

The THD of the estimation error, e, is continuously analysed after each harmonic or-

der is estimated. Estimation of subsequent harmonic orders depend on the THD of the

estimation error for the previous observer design. If the THD is below the set thresh-

old, the iteration is terminated and this implies that all harmonic orders present in the

signal have been successfully estimated. If, however, the THD is above the threshold,

then the iteration is not terminated. Thus, the iterative observer is able to estimate

only the harmonic orders present in the measured signal at a given time based on in-

formation from previous estimates. Other harmonic estimation methods do not carry

out further harmonic estimation if and when additional harmonics become present.

They only carry out estimation on the harmonic frequencies they are initially designed

to estimate. The iterative observer is capable of performing estimation of additional

harmonic frequencies if and when they become available.

Consider the linearised dynamic model of the power distribution system in (3.2).

x = Ax+Bu+Ew

y =Cx(3.2)

where x∈Rn is the state, u is the system input, y∈Rm is the output, A,B,C are matrices

with known dimensions, E is the harmonic disturbance matrix, which is unknown and

w ∈ R is the harmonic disturbance given by

w(t) =v

∑h=2

Ah sin(ωht +φh) (3.3)


where h = 2 . . .v, Ah,φh ∈ R and ωh = 2π f h, f is the fundamental frequency and φh is

the phase angle at harmonic order h. For each harmonic order, h, we have the harmonic

disturbance given by

µ(t) = Ah sin(ωht +φh) (3.4)

where µ is the harmonic output signal. Let w1 = µ. Taking derivatives with time, we

have

w1 = ωhAh cos(ωht +φh) = ωhw2 (3.5)

and

w2 =−ωhAh sin(ωht +φh) =−ωhw1 (3.6)

The harmonic disturbance may therefore be represented as a linear exosystem with

dynamics given by

wh = Shwh (3.7)

µh = gT wh (3.8)

where wh ∈ R2, gT = [1 0], Sh =

0 ωh

−ωh 0

, ωh = 2π f h.

Remark 1. Nonlinear loads in a power distribution system may cause the voltage and

current signals to become periodic and nonsinusoidal. The Fourier transform of these

signals yield the individual harmonic components which can be modelled as two states

for each harmonic order, as shown in (3.9).

w1,h

w2,h

=

Ah sin(ωht +φ)

Ah cos(ωht +φ)

(3.9)


In addition, h = 1, which is the fundamental frequency is not considered to be a har-

monic disturbance as this is the base frequency upon which the distribution system

operates. Every other frequency is taken as a disturbance to the system.

Assumption 2. The harmonic disturbance is not directly measured. Hence, the state

space representation in (3.2) has no indication of a harmonic disturbance in the output.

The harmonic disturbance only directly affects the system states.

Assumption 3. The pair (gT ,Sh) is observable for all harmonic orders, h.

Remark 2. The state vector consists of node voltages and branch currents. These

states are determined by applying KCL and KVL at all nodes and branches in the

power distribution system to determine the set of differential equations which define

the dynamic model as shown in Chapter 2.

The exact states being affected by the harmonic disturbance is unknown. As a

result, E is unknown and not available for harmonic estimation.

3.3.1 Harmonic estimation on a single output

An iterative observer for harmonic estimation in a single measurement taken from

a power distribution system is presented. The system is represented as a dynamic

model as shown in (3.2) where y ∈Rm, m = 1. Suppose the ith harmonic order is to be

estimated with dynamics given by

wi = Siwi (3.10)


The distribution system dynamic model and exogenous harmonic disturbance dynamic

model are therefore augmented to yield:

x

wi

=

A 0

0 Si

x

wi

+B

0

u (3.11)

The Luenberger observer [95] is applied to the augmented system in (3.11)

˙x

˙wi

=

A 0

0 Si

x

wi

+B

0

u+

L

Li

(y−Cx−gT wi) (3.12)

Splitting (3.12) into the process and exogenous disturbance estimates, we have

˙x = Ax+Bu+L(y−Cx−gT wi)

˙wi = Siwi +Li(y−Cx−gT wi)

(3.13)

The state estimates presented in (3.13) has been designed with an inclusion of the sys-

tem states and the exogenous disturbance states. Suppose an observer has been previ-

ously designed to estimate the system states without the consideration of the harmonic

disturbance. This observer is expressed as

˙x = Ax+Bu+ L(y−Cx) (3.14)

where L is designed such that (A− LC) is Hurwitz. The observer gain may be de-

signed based on pole placement methods as depicted in [96]. The residual y−Cx, is

orthogonal to the harmonic states to be estimated, w.

If at a certain instant, the system states were estimated but harmonic injection oc-

curs in the system subsequent to state estimation, the state variable estimate may be


expressed as a sum of the system state estimates and the exogenous disturbance esti-

mates.

x = x+Viwi (3.15)

where Vi is a weighting matrix. Taking derivatives of (3.15) with respect to time, we

have

˙x = Ax+Bu+(L+ViLi)(y−Cx)+(ViSi−ViLiCVi−ViLigT )wi

= Ax+Bu+(L+ViLi)(y−Cx)+Vi(Si−Li(CVi +gT ))

(3.16)

However, from (3.13):

˙x = Ax+Bu+L(y−Cx−gT wi) (3.17)

Hence

˙x = Ax+Bu+L(y−Cx)+(AVi−LCVi−LgT )wi (3.18)

Comparing (3.16) and (3.18):

L = L+ViLi (3.19)

ViSi−ViLiCVi−ViLigT = AVi−LCVi−LgT

ViSi− (A− LC)Vi =−LgT(3.20)

(3.20) represents a Sylvester equation of the form Fz+ zH = Q, with z calculated as

follows:

(I⊗F +H⊗ I)vec(z) = vec(Q)

vec(z) = (I⊗F +H⊗ I)−1vec(Q)

(3.21)


where I is the identity matrix and vec(z) represents the vectorization of z. The same

method used in (3.21) is used to calculate Vi. Equation (3.20) is only applicable when

a single harmonic order is to be estimated. The estimate for the ith harmonic order is

therefore given by.

˙wi = Siwi +Li(y−Cx− (CVi +gT )wi) (3.22)

where Li is designed such that Si−Li(CVi +gT ) is Hurwitz.

Suppose an additional harmonic order is to be estimated, we define a set of positive

integers H = hi for i = 1 . . .v. For estimation of the initial harmonic where i = 1,

(3.22) is used. However, for hi+1, we have

˙wh,i+1 = Sh,i+1wh,i+1 +Lh,i+1(y−Cx−gT wh,i+1) (3.23)

Due to the addition of a harmonic order to the harmonic estimation process, the system

state estimate becomes

x = x+v

∑i=1

Vh,iwh,i (3.24)

Taking derivatives of (3.24) with time

˙x = ˙x+v

∑i=1

Vh,i ˙wh,i

= Ax+Bu+(L+v

∑i=1

Vh,iLh,i)(y−Cx)+v

∑i=1

Vh,i(Sh,i−Lh,i(CVh,i +gT ))wh,i

(3.25)

However, from (3.13), we have

˙x = Ax+Bu+L(y−Cx)+v

∑i=1

((A−LC)Vh,i−LgT )wh,i (3.26)


Comparing (3.25) and (3.26)

L = L+v

∑i=1

Vh,iLh,i (3.27)

In general, harmonic estimation in a given measurement taken from a power system is

carried out as follows:

For i = 1,

Vh,1Sh,1− (A− LC)Vh,1 =−LgT (3.28)

The harmonic estimate is given by:

˙wh,1 = Sh,1w1 +Lh,1(y−Cx− (CVh,1 +gT )wh,1) (3.29)

and for i = 2 . . .v

Vh,iSh,i− (A− (L+v−1

∑j=1

VjL j)C)Vh,i =−(L+v−1

∑j=1

VjL j)gT (3.30)

Therefore the harmonic estimate is given by

˙wh,i = Sh,iwh,i +Lh,i(y−Cx−v

∑i=2

(CVh,i +gT )wh,i) (3.31)

where Lh,i for i = 1 . . .v is designed such that Sh,i−Lh,i(CVh,i +gT ) is Hurwitz.

The results obtained for harmonic estimation are summarised in Theorem 1

Theorem 1. For a given power distribution system represented as the dynamic model

in (3.2), containing harmonics which are represented as a linear exosystem given

in (3.8), estimation of the harmonic content can be produced using (3.22) for Hi =


h2 . . .hv. Additional harmonics Hv+1, can be estimated by using the iterative ob-

server of the form presented in (3.32).

˙wh,v+1 = Sh,v+1wh,v+1 +Lh,v+1(y−Cx−v

∑i=2

(CVh,i +gT )wh,v+1) (3.32)

Proof. The residual, y−Cx−∑vi=2(CVh,i +gT )wh,v+1 approaches zero asymptotically

when all harmonic frequencies have been estimated. Consider the time derivatives of

the estimation error

e = x− ˙x (3.33)

Substituting (3.24) and (3.32) we have

e = Ax+Bu−Ax−Av

∑i=2

Vh,iwh,v+1−Bu−L(y−Cx−Cv

∑i=2

Vh,iwh,v+1

= (A−LC)(x− x−v

∑i=2

Vh,iwh,v+1)

= (A−LC)e

(3.34)

Equation (3.34) shows that the estimation error approaches zero asymptotically when

the system state variables are expressed as shown in (3.24).

For i = v + 1, the estimation error for harmonic estimates is given by ev+1 =

wh,v+1− wh,v+1. Taking the time derivative of this error yields

eh,v+1 = (Sh,v+1−Lh,v+1gT )eh,v+1−Lh,v+1(y−gTv

∑k=2

wh,k) (3.35)

for k 6= h, where (Sh,v+1 − Lh,v+1gT ) is Hurwitz and (y− gT∑

vk=2 wh,k) is orthog-

onal to the harmonic frequency, h. As a result of the asymptotic convergence of

y−gT∑

vk=2 wh,k to zero for i= 2 . . .v, this result holds true for i= v+1. This completes


the proof.

3.3.2 Simultaneous harmonic estimation in multiple outputs

A multiple point approach to harmonic estimation is presented. Synchronised mea-

surements are taken from the system to ensure complete numerical and topological

observability [18] and then harmonic estimation is carried out on these measurements

to determine the amplitude and phase of the harmonic components present in the mea-

sured signals. The location of the measurement points are chosen to ensure that the

system is observable [10, 33, 97, 98]. The observability criteria ensures the capability

of the iterative observer based harmonic estimation to provide a representation of the

propagation of harmonics from their sources to other parts of the system. This infor-

mation is important to utility companies because it may give a rough estimate of the

parts of the system which contain a higher concentration of harmonic injections. The

harmonic content at a node is an aggregate of the harmonic contributions from adjacent

nodes [99].

Application of the iterative observer to simultaneous harmonic estimation in multi-

ple measurements requires modification of certain parameters in the iterative observer

algorithm. Evidently, the A,B,C matrices which constitute the state space representa-

tion remain unchanged. However, the observer gain Lh,i is modified according to the

number of measurements taken and the number of harmonic orders to be estimated.

For i = 1 . . .v, where v represents the number of harmonics present in the measured


signals,

Sh,i =

S1h,i 0 . . . 0

0 S2h,i . . . 0

...... . . . ...

0 0 . . . Smh,i

(3.36)

where m represents the number of measurements and Sh,i ∈ R2m×2m. The observer

gain, Lh,i ∈ R2m×m is designed such that Sh,i−Lh,iCh,i is Hurwitz. The harmonic mea-

surement matrix, gT ∈ Rm×2m, is given by

gT =

1 0 0 . . . 0

0 1 0 . . . 0

......

... . . . ...

0 0 0 1 0

(3.37)

One important feature of the iterative observer approach is that estimation is only car-

ried out when required. This implies that the iterative observer algorithm provides

estimates of the harmonic content in the measurements if and only if the harmonic dis-

turbance is present in the measurements. The THD of the measurements are analysed

and if they are below a threshold, estimation is not carried out for that particular mea-

surement and vice versa. the parameters, Sh,i ,gT and Lh,i are adjusted according to the

number of measurements for which estimation will be carried out.


3.4 Harmonic estimation algorithm based on the itera-

tive observer

In this section, the iterative observer designed in section 3.3 is presented as an al-

gorithm. The proposed algorithm enables harmonic estimation to be carried out only

when the set criteria is met. In this case, the criteria used for determination of estima-

tion is the THD. The THD of residuals obtained from harmonic estimation is checked

to ensure that the utility companies and customers are in compliance with the standards

such as IEEE Std-519 [99]. According to these standards, the THD of each individual

harmonic should not exceed 3% of the fundamental frequency. Hence, THD below

3% may be set as the threshold for termination of iteration to comply with the given

standards. The algorithm for harmonic estimation based on the iterative observer is

as shown in algorithm 1. The threshold, T , is chosen to satisfy regulatory stan-

dards. This algorithm presents a systematic approach to estimation of harmonics in

a measured voltage or current signal taken from the power distribution system. It is

important to note that this algorithm allows harmonic estimation only when there is

a distortion in the measurements. The absence of a distortion terminates iteration,

thereby saving time and costs. In addition, the definition of the THD shows that it is a

ratio of the harmonic content with respect to the fundamental frequency. As such, the

residual is added to the fundamental frequency estimate, which in this case is the term

Cx. This is to ensure that the computed value of the THD is taken with respect to the

fundamental frequency. The flowchart representing the iterative observer algorithm is

shown in Figure 3.4.


Algorithm 1 Iterative observer algorithm1: Obtain voltage or current measurements, y.2: Design an observer for the system without the consideration of the harmonic dis-

turbance (If a previous design does not already exist for the system considered)

˙x = Ax+Bu+ L(y−Cx)

3: if THD of (y−Cx)> T then4: Define a set of positive integers H = hi for i = 1, . . . ,v5: For i = 1:

A− LC)Vh,1 +Vh,1Sh,1 =−LgT

˙wh,1 = Sh,1w1 +Lh,1(y−Cx− (CVh,1 +gT )wh,1)

6: For i = 2 . . .n7: repeat8:

−(A− (L+v−1

∑j=2

VjL j)C)Vh,i +Vh,iSh,i =−(L+v−1

∑j=2

VjL j)gT

˙wh,i = Sh,iwh,i +Lh,i(y−Cx−v

∑j=2

(gT +CVj)w j)

9: The residual,

rh,i = y−Cx−v

∑j=1

(gT +CVj)w j

10: Calculate the THD of rh,i +Cx.11: i = i+112: until THDrh,i < T13: Re-estimate previously estimated harmonics from i = 1, . . . ,v14: end if


Start

Obtain

measurement

s, y

Does system state

observer exist?No

Design system

state observer

Yes

Check %THD of

residual

Is %THD > T

Yes

Define a set of

positive integers

H = hi for I =

1….v

Calculate

Vh,1

Estimate

wh,1

Is %THD of

residual >T

End

Yesi = i + 1

Calculate Vh,i

for i ≠1

Calculate wh,i

for i≠ 1

Is %THD of

residual h,i > T

Yes i = i+1No

No

No

Figure 3.4: Flowchart for iterative observer design

3.5 Case study

The validity of the proposed harmonic estimation algorithm is verified by carrying

out simulations on a benchmark IEEE distribution system. Consider the IEEE 13-node

power distribution test feeder, which is test system 2 in [99]. Figure 3.5 shows the

single-line diagram of the model. The distribution system parameters are as follows:

• Short circuit capacity: 1100 MVA, 82 lagging

• Substation transformer at node 50-31: 5 MVA, 115 kV delta - 4.16 kV wye

grounded, impedance, Z = 1+j8% at 50Hz


Figure 3.5: Single-line diagram of IEEE 13-node distribution feeder

• Transformer at node 33-34:

500 KVA, 4.16 kV delta - 480 V wye, impedance, Z = 1.1 + j2.0%

A voltage source behind an impedance is connected at node 50 to represent the grid

and transformers are represented by a two-winding model. The fundamental frequency

is taken to be constant at 50Hz. Linear loads, which are connected at constant RL

impedances are connected at nodes 34, 46, 71, 75, 52 and 911. The load data is shown

in table 3.1. Cables are modelled using their π models with length dependent param-

eters. The line data is shown in table 3.2 where R, Ll and Cl represent the resistance,

inductance and capacitance respectively which are all expressed per unit length. State

space matrices of this test feeder are obtained using the method proposed in Chap-

ter 2. The choice of measurement points is carried out using binary integer linear

programming and sequential elimination as proposed in chapter 2 to ensure complete

observability. Nonlinear loads are modelled as linear loads in parallel with a current

source. Two harmonic sources are taken to be present in the system at nodes 46 and

75. The harmonic spectra for both harmonic sources are shown in Table 3.3 and 3.4


Table 3.1: IEEE 13-node distribution system load data

Load P(kW) Q(kVar)

L46 230.22 131.97

L34 42.63 20.18

L71 383.70 219.95

L75 486.02 189.07

L911 170.53 80.74

L52 127.90 85.79

Table 3.2: IEEE 13-node distribution system line data

From node To node R(Ω/km) L(mH/km) Cl(nF/km) Length(km)

31 32 0.2161 2.007 12.358 0.6096

32 45 0.4684 2.330 11.198 0.1524

32 33 0.4684 2.330 11.198 0.1524

32 71 0.2161 2.007 12.358 0.6096

71 84 0.8226 2.685 9.233 0.2438

92 75 0.5285 0.799 187.270 0.1524

71 150 0.2161 2.007 12.358 0.3048

84 52 0.6093 1.018 0.775 0.2438

45 46 0.8226 2.685 11.198 0.0914

84 911 0.8226 2.685 312.250 0.0914

respectively. Applying the meter placement algorithm, six measurement points are

Table 3.3: Harmonic spectrum at node 46

Harmonic order Amplitude(%) Phase(deg)

Fundamental (50Hz) 100.00 -30.68

3rd(150Hz) 14.98 178.3

5th(250Hz) 5.22 243.9

identified at nodes 31, 33, 45, 92, 71 and 84. Branch currents at these measurement

points are used for harmonic estimation. These branch current measurements include



Harmonic order Amplitude(%) Phase(deg)

Fundamental(50Hz) 100.00 -22.86

3rd(150Hz) 13.96 216.7

5th(250Hz) 4.68 282.3

7th(350Hz) 1.70 81.0

I31−32,, I33−34, I45−46, I92−75, I32−71 and I71−84. The measurements are shown in Fig-

ure 3.6. The measurements taken at a specific node do not exactly portray the harmonic

Time(s)0 0.05 0.1

I 31-3

2(A

)

-5000

0

5000

Time(s)0 0.05 0.1

I 33-3

4(A

)

-20

0

20

Time(s)0 0.05 0.1

I 45-4

6(A

)

-200

0

200

Time(s)0 0.05 0.1

I 32-7

1(A

)

-5000

0

5000

Time(s)0 0.05 0.1

I 71-8

4(A

)

-50

0

50

Time(s)0 0.05 0.1

I 92-7

5(A

)

-50

0

50

Figure 3.6: IEEE 13-node feeder current measurements

content injected at that node. They represent a combination of the harmonic content at

the nodes adjacent to the nodes where the measurements are taken. The iterative ob-

server algorithm is applied to the measurements to determine the amplitude and phase

of the injected harmonic components. A threshold of 0.5% is set as the termination

criteria for iteration. This is below the maximum allowable THD in systems below 69

kV for each individual harmonic as specified in [1]. For the case study presented, the


iterative observer algorithm is first applied to a single measurement and then the case

of simultaneous harmonic estimation in all six measurements is explored. The results

obtained are compared to those obtained from the conventional FFT and the Kalman

filter.

3.5.1 Single measurement harmonic estimation

The iterative observer based harmonic estimation algorithm is applied to the mea-

surement at node 45 to identify the amplitude and phase of the harmonic frequencies

present in the measurement. A number of scenarios are simulated to verify the accu-

racy and robustness of the proposed approach.

3.5.1.1 Normal operating condition

For normal operating conditions, the measurements are synchronised and there are

no load variations. The first step of the iterative observer algorithm is the design of an

observer for the harmonic disturbance-free system. The observer gain, L is designed

such that A− LC is Hurwitz.

The THD of the residual after estimation of the system states without the harmonic

disturbance, with respect to the fundamental frequency is analysed to ensure that it is

compliant with the set threshold for termination of iteration. In this case, the THD is

8.514%, therefore, iteration continues. Figure 3.7 shows the estimated disturbance-

free output and the residual. The residual is a representation of the sum of harmonic

amplitudes in the measured signal. The amplitude of the fundamental frequency es-

timate is 134.2A. Prior to estimation of the third harmonic, the Sylvester equation is

solved for the fundamental frequency as shown in (3.28). This is because subsequent

computations of the solution to the Sylvester equations for each harmonic order re-

quires a sum of the solutions for previously estimated harmonics as shown in (3.30).


Time(s)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

I 45-4

6(A

)

-200

-100

0

100

200

Time(s)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Res

idua

l(A)

-100

-50

0

50

Figure 3.7: Top: Harmonic disturbance-free current estimate at node 45Bottom: Residual from fundamental frequency estimation of I45−46

The poles for harmonic estimation for all harmonic orders, h, are placed at

ph =

−2π f +2π f hi

−2π f −2π f hi

(3.38)

The observer gain at the fundamental frequency is computed as:

L1 =

628.3185

314.1593

(3.39)

As a result of the condition for termination of iteration not being met, the third har-

monic estimation is carried out. The third harmonic state transition matrix is given

by

S3 =

0 942.4778

−942.4778 0

(3.40)


and the observer gain is computed as

L3 =

628.3185

104.7198

(3.41)

using the pole structure in (3.38), where h = 3. The estimate of the third harmonic is

shown in Figure 3.8. The figure shows that after an initial transient, the third harmonic

Time(s)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

I 3(A

)

-20

-15

-10

-5

0

5

10

15

Figure 3.8: Estimate of the third harmonic for I45−46

is estimated with an amplitude of 10.789A. The THD of the residual after estimation

of the third harmonic is 2.4695%. This is above the set threshold for termination of

iteration, therefore, iteration continues to the fifth harmonic. The fifth harmonic state


transition matrix is given by

S5 =

0 1570.80

−1570.80 0

(3.42)

and the observer gain is computed as

L5 =

628.32

62.83

(3.43)

using the pole structure in (3.38), where h = 5. The fifth harmonic estimate is shown

in Figure 3.9. The amplitude of the fifth harmonic estimate is 4.3796A. The THD

Time(s)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

I 5(A

)

-15

-10

-5

0

5

10

Figure 3.9: Estimates of the fifth harmonic for I45−46 measurement

of the residual after estimation of the fifth harmonic is 0.0187%, which is below the


threshold, hence, iteration is terminated. This implies that there is no more harmonic to

be estimated in the signal. The residual after termination of iteration is shown in Figure

3.10. The figure shows that after estimation of the fifth harmonic, the estimation error,

Time(s)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

e45

-46(A

)

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Figure 3.10: Residual after termination of iteration for I45−46 measurement

e45−46 satisfies the condition

limt→∞

e45−46→ 0

The results of harmonic estimation using the iterative observer algorithm for nor-

mal operating conditions are compared to those obtained using the Kalman filter and

FFT. For this study, due to the use of continuous time for the iterative observer algo-

rithm, the Kalman-Bucy filter, which is the continuous time form of the Kalman filter

is used to compare the results for ease of comparison. In addition, the observer gain

used for estimation is equivalent to the Kalman-Bucy gain.


One advantage of the proposed iterative observer algorithm for harmonic estima-

tion is that estimation is only carried out when necessary, i.e. when there is a harmonic

distortion in the measured signal. This may result in an improvement in computation

time and cost. The Kalman-Bucy filter carries out estimation on all harmonic orders it

is designed to estimate without considering the distortion levels in the measurement. In

addition, the Kalman-Bucy filter requires knowledge of the process and measurement

noise covariances, which may be unknown. Figure 3.11 shows a comparison between

the harmonic estimates using the Kalman-Bucy filter and iterative observer. The figure

0 0.05 0.1 0.15 0.2 0.25 0.3

I 1(A

)

-200

0

200

0 0.05 0.1 0.15 0.2 0.25 0.3

I 3(A

)

-20

0

20

0 0.05 0.1 0.15 0.2 0.25 0.3

I 5(A

)

-10

0

10

Iterative observerKalman-Bucy filter

0 0.05 0.1 0.15 0.2 0.25 0.3

I 7(A

)

-505

0 0.05 0.1 0.15 0.2 0.25 0.3

I 9(A

)

-2

0

2

Time(s)0 0.05 0.1 0.15 0.2 0.25 0.3

I 11(A

)

-2

0

2

Figure 3.11: Comparison between the Kalman-Bucy filter estimates and the iterativeobserver estimates for I45−46 at normal operating conditions

shows that for the same gain used for both methods, the amplitude of estimation for

each harmonic order is approximately equivalent. In addition, the iterative observer

only carries out harmonic estimation when the amplitude of the harmonic order to be

estimated is non-zero. The Kalman-Bucy filter goes on to estimate the seventh, ninth

and eleventh harmonics, irrespective of their non-existence in the measured signal. Ta-

ble 3.5 compares the amplitude and phase angles of the harmonic estimates for the

Kalman-Bucy filter, FFT and the iterative observer algorithm, where Amp represents

amplitude. A comparison between the residuals for Kalman-Bucy filter estimates and


Table 3.5: Comparison of harmonic estimates using the FFT, Kalman-Bucy filter anditerative observer algorithm for I45−46

Harmonic order, h FFT Kalman Iterative observer

Amp(A) Ph() Amp(A) Ph() Amp(A) Ph()

1 134.100 145.6 134.2 145.6 134.2 146.5

3 10.540 175.6 10.792 174.9 10.797 177.2

5 3.648 242.3 3.725 239.0 3.812 242.4

7 0.134 11.7 3.447e-4 18.4 - -

9 0.107 12.0 2.475e-3 -25.1 - -

11 0.080 14.3 4.372e-3 56.3 - -

the iterative observer estimates after estimation of the fifth harmonic is shown in Fig-

ure 3.12. From the figure, we deduce that using the same gain for both methods, the

Time(s)0 0.05 0.1 0.15 0.2 0.25 0.3

e45

-46(A

)

-80

-60

-40

-20

0

20

40

60

80

100


Figure 3.12: Comparison of residuals between the Kalman-Bucy filter and iterativeobserver estimates for the I45−46 measurement

residual approaches zero asymptotically. In addition, the iterative observer method

provides less transient oscillations irrespective of the observer gain being equivalent to

the Kalman-Bucy gain.


3.5.1.2 Change in amplitude of harmonics with time

In actual power distribution systems, the amplitude of harmonic injections may

fluctuate with time. This stems from the continuous variation in the use of power

electronic devices at various points in the system. Customers constantly use nonlinear

loads as and when needed and as a result the harmonic amplitudes may fluctuate with

time. Suppose the current measurement at branch 45-46 is represented as

I45−46 =v

∑h=1


where Ah represents the amplitude of the harmonic current, h is the harmonic order, v

is the number of harmonic components in the signal, φh represents the phase angle at

the harmonic order, h and ωh = 2π f h. The harmonic currents at node 45 are as a result

of the harmonic load at node 46. From the single-line diagram of the power system

in Figure 3.5, it is apparent that node 45 and 46 are adjacent to each other hence,

the harmonic injection at node 46 is most likely to have an impact on the current

measurement taken at branch 45-46. Suppose the amplitudes of harmonic currents

injected at node 46 are time-varying as shown in (3.45) and (3.46).

A3 =

0 t < 0.2s

8.3% 0.2s < t < 0.4s

14.98% t > 0.4s

(3.45)

A5 =

0 t < 0.2s

5.22% 0.2s < t < 0.4s

10.5% t > 0.4s

(3.46)


At t < 0.2s, there is no harmonic injection at node 46. However, at the interval

0.2s< t < 0.4s, both the third and fifth harmonic currents are injected while at t > 0.4s,

there is an increase in the amplitude of the third and fifth harmonic injections. The

iterative observer algorithm is applied to the current measurement to determine the

harmonic current amplitudes and phase angles with the variations in nonlinear loads.

Figure 3.13 shows a comparison between the Kalman-Bucy estimates and the iterative

observer estimates when there is a change in the amplitude of harmonics with time.

From the figure, it is apparent that at t < 0.2s, there is no harmonic current in the mea-

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 1(A

)

-200

0

200Iterative observerKalman-Bucy filter

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 3(A

)

-20

0

20

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 5(A

)

-10

0

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 7(A

)

-2

0

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 9(A

)

-1

0

1

Time(s)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 11(A

)

-1

0

1

Figure 3.13: Comparison between the Kalman-Bucy and iterative observer estimatesfor change in amplitude of harmonic injections with time

surement, hence the harmonic estimates are zero. The Kalman-Bucy estimates exhibit

a transient response initially before settling to zero, whereas, the iterative observer

estimates exhibit no transient response. At the interval 0.2 < t < 0.4, both the itera-

tive observer and Kalman-Bucy filter accurately estimate the third and fifth harmonics

and at t > 0.4s, both methods adapt to the amplitude change accordingly. As was the


case for normal operating condition, the iterative observer terminates iteration after

the fifth harmonic is estimated, while the Kalman-Bucy filter continues estimation un-

til the eleventh harmonic, irrespective of the non-existent seventh, ninth and eleventh

harmonics in the signal. Figure 3.14 shows the estimation error for both methods of

harmonic estimation. The figure shows that the estimation error approaches zero with

Time(s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

e45

-46(A

)

-40

-20

0

20

40

60

80

100

Kalman-Bucy filterIterative observer

Figure 3.14: Estimation error with change in amplitude of harmonics with time

small transients at 0.2s and 0.4s.

3.5.1.3 Harmonic estimation with variations in fundamental frequency

Variations in the fundamental frequency with time are very common in power dis-

tribution systems. Although the fundamental frequency is assumed to be fixed at 50Hz,

it is likely to fluctuate with time. Suppose there is a change in the fundamental fre-

quency in the range of ±2% in the system. The fundamental frequency variation is as


shown in (3.47).

f =

0 t < 0.2s

−2% 0.2s < t < 0.4s

+2% t > 0.4s

(3.47)

At t < 0.2s there is no variation in the fundamental frequency. It remains fixed at

50Hz. However, at the interval 0.2 < t < 0.4 there is a decrease in the fundamental fre-

quency by 2%. Since the fundamental frequency is taken to be 50Hz, the fundamental

frequency at 0.2 < t < 0.4 is 49Hz and at t > 0.4s, there is a 2% increase in the fun-

damental frequency which becomes 51Hz. The harmonic spectrum for the harmonic

injection at node 46 is as given in Table 3.3. Figure 3.15 shows the measured signal

with deviations in the fundamental frequency at 0.2 and 0.4s. The iterative observer

Time(s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 45-4

6(A

)

-150

-100

-50

0

50

100

150

Figure 3.15: I45−46 measurement with fluctuating fundamental frequency

algorithm is applied to the measured current signal I45−46 to determine the harmonic

injections with fluctuating fundamental frequency. There is no prior knowledge of


the frequency fluctuations in the system, hence, the iterative observer algorithm is de-

signed using the system base frequency which is taken as 50Hz. Figure 3.16 shows the

result of harmonic estimation using the iterative observer algorithm. From the figure, it

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 1(A

)

-200

0

200

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 3(A

)

-20

0

20

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 5(A

)

-10

0

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 7(A

)

-5

0

5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 9(A

)

-2

0

2

Time(s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 11(A

)

-2

0

2


Figure 3.16: Comparison of harmonic estimates with fluctuating fundamental fre-quency

is apparent that the iterative observer algorithm accurately estimates the harmonic fre-

quencies in the measured signal irrespective of the fluctuating fundamental frequency.

Table 3.6 shows the harmonic estimates using the iterative observer with variations in

the system fundamental frequency. From these estimates, it can be deduced that de-

spite the variations in fundamental frequency with time, the amplitude of the harmonic

estimates are accurate. The table shows that the amplitude estimates are approximately

equivalent for all fundamental frequency variations. This shows that the proposed iter-

ative observer is capable of providing accurate harmonic estimates in the presence of


Table 3.6: Iterative observer harmonic estimates with fluctuating fundamental fre-quency

h0 < t ≤ 0.2s 0.2s < t ≤ 0.4s t > 0.4s

Amp(A) Ph() Amp(A) Ph() Amp(A) Ph()

1 134.20 -33.7 134.20 -33.6 134.20 -33.5

3 10.79 -2.9 10.28 28.6 10.98 181.5

5 3.73 63.2 3.56 236.1 3.67 250.2

fundamental frequency variations.

3.5.1.4 Effect of decaying DC component and noise on harmonic estimation

Consider the current measurement at branch 45-46 with decaying DC component

and measurement noise given by:

y = I45−46 + IDCe−αt + γ(t) (3.48)

where IDC = 26.84A and α = 10. The noise γ is Gaussian distributed with zero mean

and 5% variance. This measurement with a decaying DC component and noise is

shown in Figure 3.17. Figure 3.18 shows the estimation errors for both the Kalman-

Bucy filter and iterative observer. The figure shows that the iterative observer responds

to the decay in the measured signal and the estimation error approaches zero asymptot-

ically. In addition, the harmonic currents are accurately estimated as shown in Figure

3.19.

3.5.2 Harmonic estimation in multiple measurements

Consider the power distribution system shown in Figure 3.5. Six current mea-

surements are taken and these measurements are shown in Figure 3.6. The iterative

observer algorithm is applied to all six measurements simultaneously to determine the


Time(s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

I 45-4

6(A

)

-150

-100

-50

0

50

100

150

200

Figure 3.17: I45 measurement with decaying DC component

amplitude and phase of each harmonic component present in the measurements. The

criteria for termination of iteration is also applied simultaneously to all measurements

to determine the existence of harmonics therein. The threshold for termination of iter-

ation is set at 0.5%, which is below the limits set by [1] for the maximum allowable

THD for individual harmonics in systems below 69 kV. The THD may be set according

to the requirements of the utility company. Measurements with THD greater than the

set threshold are subjected to the iterative observer algorithm to estimate the harmonic

distortion levels.

For each measurement taken, the %THD is computed as 0.176%, 0.099%, 0.233%,

15.523%, 0.308% and 14.260% for the measurements I31−32, I32−71, I33−34, I45−46,

I71−84 and I92−75 respectively. The measurements with %THD greater than the thresh-

old of 0.5% are I45−46 and I92−75, therefore harmonic estimation is only carried out for

these two measurements. The observer gain used for harmonic estimation at the third


Time(s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

e45

-46(A

)

-50

0

50

100

Kalman-Bucy filterIterative observer

Figure 3.18: Estimation error for decaying DC component in the measured signal

harmonic order is given by

L3 =

316.2278 0

1.8745 0

0 316.2278

0 1.8745

After the third harmonic has been estimated for both measurements, the THD becomes

4.8663% and 4.5714%. Hence iteration continues and the fifth harmonic is estimated.


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

I 1(A

)

-200

0

200

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

I 3(A

)

-20

0

20

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

I 5(A

)

-10

0

10

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

I 7(A

)

-5

0

5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

I 9(A

)

-2

0

2

Time(s)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

I 11(A

)

-1

0

1


Figure 3.19: Comparison between harmonic estimation methods in the presence ofmeasurement noise and decaying DC component

The observer gain for estimation of the fifth harmonic is given by

L5 =

316.2278 0

1.7902 0

0 316.2278

0 1.7902

The THD after the fifth harmonic is estimated for both measurements is 0.0139% for

I45−46 and 1.4964% for I92−75. Therefore iteration is terminated for I45−46 and con-

tinues for I92−75. Figure 3.20 shows the harmonic estimates for both measurements.

After estimation of the fifth harmonic, the estimation error is as shown in Figure 3.21.


Time(s)0 0.05 0.1 0.15 0.2

I 1(A

)

-50

0

50

Time(s)0 0.05 0.1 0.15 0.2

I 3(A

)

-10

-5

0

5

10 I45-46

I92-75

Time(s)0 0.05 0.1 0.15 0.2

I 5(A

)

-2

0

2

4

Time(s)0 0.05 0.1 0.15 0.2

I 7(A

)-0.5

0

0.5

Figure 3.20: Harmonic estimates for multiple measurements

The figure shows that after the fifth harmonic is estimated, the error approaches zero

asymptotically for the measurement I45−46 and the error for the measurement I92−75

does not approach zero. This indicates that there is an additional harmonic/harmonics

to be estimated in that measurement. The seventh harmonic is therefore estimated for

the measurement I92−75. This is accurate since the harmonic injection spectra for node

75 indicates injection of the seventh harmonic while for node 46, only the third and

fifth harmonics are injected. These nodes are adjacent to the nodes where the measure-

ments I92−75 and I45−46 are taken. The observer gain used for estimation of the seventh

harmonic is given by

L7 =

316.2278

1.7131

The THD after estimation of the seventh harmonic in I92 is 0.0137%. Estimation of

the seventh harmonic gives the estimation error shown in Figure 3.22. It is clear that

the estimation error approaches zero, thereby indicating the complete estimation of all


Time(s)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

e45

-46(A

)

-40

-20

0

20

Time(s)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

e92

-75(A

)

-10

-5

0

5

Figure 3.21: Estimation error after fifth harmonic estimation

Time(s)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

e92

-75(A

)

-40

-30

-20

-10

0

10

20

Figure 3.22: Estimation error for node 92 after estimating the seventh harmonic

harmonics present in the measured signal.

Figure 3.23 shows the amplitude estimates for both distorted measurements. The

data in Table 3.7 also shows the amplitude and phase of all estimated harmonic fre-

quencies for both distorted measurements.


Time(s)0 0.2 0.4 0.6

I(A

)

0

20

40

60

80Fundamental frequency

Time(s)0 0.2 0.4 0.6

I(A

)

0

5

10

15Third harmonic

I45-46

I92-75

Time(s)0 0.2 0.4 0.6

I(A

)

0

1

2

3

4Fifth harmonic

Time(s)0 0.2 0.4 0.6

I(A

)0

0.2

0.4

0.6Seventh harmonic

Figure 3.23: Amplitude of harmonic estimates for I45−46 and I92−75

Table 3.7: Harmonic estimates for the current measurements

Harmonic order, hI92−75 I45−46

Amp(A) Ph() Amp(A) Ph()

1 23.91 -22.8 74.72 -30.7

3 3.21 35.8 10.96 -2.6

5 1.07 101.8 3.80 62.6

7 0.41 258.7 - -

3.6 Summary

This chapter has explored the estimation of harmonics in a measured signal taken

from a power distribution system. The proposed iterative observer algorithm has been

proven to provide accurate harmonic estimates in the presence of noise, decaying DC

components and changes in the amplitude of harmonic injections with time. The re-

sults obtained for the various scenarios were compared to those obtained from the

Kalman-Bucy filter estimates and the FFT. One advantage of the proposed method is


that it is dynamic and responds to the time evolution of the distribution system. In ad-

dition, knowledge of the process and measurement noise covariances are not required

to carry out estimation, unlike the Kalman-Bucy filter. The observer gain is determined

using pole-placement methods. The iterative observer algorithm was also applied for

simultaneous harmonic estimation in multiple measurements taken throughout a power

distribution system. The algorithm was only applied to cases where the THD exceeded

the set threshold. Simultaneous harmonic estimation is important in cases where a lot

of measured signals need to be analysed simultaneously to provide a mapping of the

harmonic propagation throughout the distribution system. This approach is instrumen-

tal because the harmonic injections tend to flow from their sources to other parts of

the power system, hence, this flow may be mapped if the harmonic estimates can be

determined simultaneously for all available measurements.

Chapter 4

Harmonic source identification

4.1 Introduction

A great deal of attention has been placed on identifying harmonic sources in power

distribution systems. An important part of improving the quality of power delivered

to customers involves determining the source of harmonic injections and employing

a suitable mitigating technique. With advancement in technology and continuous de-

sign of renewable energy methods, there is an increase in the harmonic content in

power distribution systems. Although these advancements provide more efficient so-

lutions to researchers and industrialists, the harmonic disturbances they cause need to

be dealt with. The harmonic currents injected by a single user may create an impact

on the quality of power delivered to other users. As a result, it is important to carry

out system-wide identification of harmonic sources to accord utilities with the neces-

sary information to take appropriate action. This action may include the design of an

appropriate mechanism to extract or nullify the effects of the harmonic injections or

penalising those responsible.

Due to the continuous variation of power distribution system loads, a dynamic

99

CHAPTER 4. HARMONIC SOURCE IDENTIFICATION 100

method of harmonic source identification is required. This chapter focuses on a time

domain method of identifying harmonic sources in power distribution systems. The

system states are estimated using a fault observer approach and the results are anal-

ysed to determine the harmonic sources. Ideally, the number of harmonic sources is

unknown to utility companies due to the unpredictability of the loads used by the con-

sumers. The approach proposed in this chapter does not require prior knowledge of

the number of harmonic sources to be implemented. Both the number and locations of

the harmonic disturbances are estimated using the fault observer algorithm presented.

The framework of the fault observer based harmonic source identification is shown

in Figure 4.1. The measurement error is analysed to determine the harmonic source

and the estimated state variables are used to evaluate the current injections. In most

harmonic source identification methods, the states are chosen as either node voltages

or injection currents. Harmonic source identification is carried out for each harmonic

frequency separately. In contrast, the time domain method proposed in this chapter

uses both node voltages and branch currents as state variables. The harmonic current

injections are computed separately to determine the amplitudes and phase angles of

the harmonics injected. In addition, the fault observer method does not require compu-

tation at specific harmonic frequencies. The proposed algorithm may be implemented

on-line or offline.

4.2 Literature review on harmonic source identification

A number of methods have been previously used for harmonic source identifica-

tion. The methods proposed in literature may be divided into two categories: single-

point methods and multi-point methods. In single-point methods, the harmonic source


Figure 4.1: Framework for fault observer based harmonic source identification

is determined by taking measurements at a single point in the distribution system, usu-

ally the PCC, and the harmonic source is determined to be either upstream or down-

stream from the PCC. This method apportions responsibility for the harmonic source

to either the customers or the utility. One single point harmonic source identification

method is the direction of the harmonic active power [100]. Linear loads do not gen-

erate active power, hence, the presence of active power indicates the existence of a

nonlinear load. However, in [101], the power direction method was proven to be inef-

fective in determining the harmonic source from the PCC. The reactive power method

has also been applied to identification of harmonic sources in power systems [102].

This method is based on the direction of the reactive power to determine the dominant

harmonic source. A single-point approach based on the nonactive power was proposed

in [103, 104]. A comparison between different nonactive power quantities was used

to determine if the harmonic source is upstream or downstream from the metering

section.


In addition to these methods, other single-point methods of harmonic source iden-

tification proposed in literature include critical impedance methods [105–107] and

neural networks [108, 109]. For critical impedance methods, the harmonic voltages

between the customer and utility are compared and used to determine the harmonic

source. A drawback of this method is that the assumption that the harmonic impedance

is uniformly distributed throughout the power system is not applicable in practical sys-

tems. Also, knowledge of the system and harmonic impedance is required to obtain

the direction of the harmonic source.

Although the single point approach is effective in providing a general overview

of the source of harmonic disturbances in terms of the customer and the utility, it

does not specify the exact node where the harmonic disturbance occurs. For systems

with a large number of simultaneous harmonic sources at any given time, the harmonic

distortion measured at the PCC is an aggregate of all simultaneous harmonic sources in

the system [99]. These harmonic injections tend to flow from their sources towards the

grid, hence, they may offset each other due to phase angle differences. Therefore, the

harmonic distortion at the PCC may not be an accurate representation of the harmonic

injections at the system nodes. As a result, the multiple point approach is employed in

this research for harmonic source identification.

The most common method of multi-point harmonic source identification is HSE.

HSE involves estimating the states in a power system from available measurements

[18]. The states may be taken as node voltages, branch currents or current injections

depending on the method used to solve the HSE problem.

4.2.1 Harmonic State Estimation

HSE is the process of estimating the amplitude and phase angles of node voltages

at the fundamental frequency and higher order harmonics. Power system HSE is the


reverse harmonic power flow problem, which uses harmonic measurements at selected

nodes in the system to determine the location and amplitude of harmonic sources. Iden-

tification of harmonic sources using HSE was first carried out in [110]. The sources

of harmonic injections are determined from limited measurements. The chosen state

variables are estimated at each harmonic order to determine the nodes with harmonic

contributions to the system. Two different forms of HSE have been reported in litera-

ture: static HSE and dynamic HSE.

4.2.1.1 Static HSE

One of the most common methods of static HSE is SVD [111, 112]. An advantage

of SVD is that the system does not need to be observable to carry out HSE. A solution

to the problem can be obtained regardless of the partial observability of the system.

In [8], WLS and SVD are used to solve the HSE problem. WLS was also proposed

in [113] for HSE when the system parameters are uncertain. Measurements taken us-

ing PMUs and a representation of the state variables in rectangular coordinates are

applied to obtain a solution to HSE. A major factor to be considered when carrying out

HSE is the placement of meters in the power system. Sensitivity analysis and the min-

imum variance criterion were applied in [114] to determine the optimum locations to

place the measurement devices to carry out HSE. The sequential method significantly

reduces the number of measurement devices required to carry out HSE, however, it

does not take into account the time varying nature of harmonic loads. In [115], a two

stage approach was used to determine the location of multiple harmonic sources in a

power system. First, WLS was used to estimate the possible nodes which represent

harmonic sources and subsequently, the Euclidean norm is applied to ascertain the ex-

act nodes with harmonic injections. Sparsity maximisation has also been applied to

static HSE [11, 12]. One drawback of static HSE is that the dynamic nature of loads is


not taken into consideration while performing harmonic source identification.

4.2.1.2 Dynamic HSE

In dynamic HSE, a representation of the dynamic model of the power system is

used to carry out HSE. This dynamic model provides utilities with information on the

time evolution of the system, thereby enabling them react accordingly in the event

of changes in the power system. An advantage of the dynamic method of HSE over

the static method is the ability to track the harmonic content over a certain time. In

recent times, dynamic HSE has been of interest to researchers. Dynamic HSE was

first carried out in [13] using a Kalman filter and DFT. Various modifications of the

Kalman filter have since been used for harmonic state estimation in literature. These

include the adaptive Kalman filter [14], extended Kalman filter [116] and the robust

extended Kalman filter [117,118]. The Kalman filter method of harmonic source iden-

tification is dynamic and capable of identifying, analysing and tracking the harmonic

injections with respect to time. In practical scenarios, the possibility of having har-

monic sources at every node in the system is high. The Kalman filter based dynamic

methods are capable of tracking the variations in harmonic injections with time and

adapt accordingly. The adaptive Kalman filter proposed in [14] for dynamic harmonic

state estimation and harmonic injection tracking uses two process noise covariant, Q,

models for harmonic state estimation. This eliminates the need to determine an opti-

mum Q model. In [17, 119], a time domain approach to harmonic state estimation is

carried out using the Kalman filter Poincaré map.

A disadvantage of HSE based methods is that knowledge of system parameters

at each harmonic order of interest is required to accurately determine the harmonic

sources. In addition, the state variables are computed at each harmonic order which

may increase the computational burden.


Apart from HSE, other methods have also been applied to the identification of

multiple harmonic sources in power systems. The method of independent compo-

nent analysis for harmonic source identification was presented in [120, 121]. In [120],

independent component analysis and mutual information theory were proposed. Inde-

pendent component analysis was used to determine the current profile of the harmonic

current injections while mutual information theory was applied to identify the har-

monic sources. A pitfall of this method is that the impedance of the system at each

harmonic frequency needs to be determined before harmonic source identification can

be carried out. The theory of statistical inference was proposed in [122] to evaluate

the impact of multiple harmonic producing loads. Binary particle swarm optimisation

was applied in [123] to determine the optimum locations to place measurement de-

vices. The location of the harmonic source was then determined using the direction

of harmonic power flow to rank the nodes into suspicious and non-suspicious nodes.

This method assumes the availability of phasor voltage measurements at all nodes in

the system. This assumption is impractical because it is not cost effective to measure

all node voltages in the power system at various harmonic frequencies.

Harmonic source localisation in distribution systems was proposed in [9,124–128].

In [125], a Bayesian approach was applied to harmonic source identification. A dy-

namic representation of the system was formulated and the forcing term, which repre-

sents the harmonic current injections, was determined from measured and pseudomea-

sured data. A cascade correlation network for multiple harmonic source identification

was given in [129].

Irrespective of the extensive research in the area of multiple harmonic source identi-

fication, there still exists some concerns in terms of dynamic harmonic source identi-

fication and the capability of the methods to adapt to the constantly varying loads in

distribution systems.


4.3 Harmonic source identification using the fault ob-

server based approach

This section explores the identification of multiple harmonic sources in power dis-

tribution systems when the system parameters are known and harmonic disturbance

parameters are unknown. In real applications, utility companies are unaware of the ex-

act number of harmonic sources. Various residential or industrial customers constantly

use loads as and when needed. This may create a challenge for utilities to determine

the exact system nodes which inject harmonic disturbances into the power system. An

algorithm is presented to determine the source of harmonic disturbances in a power

distribution system using a fault observer-based approach.

Consider a linearised form of the power distribution system given in state-space

form as shown in (4.1).

x = Ax+Bu+Ew

y =Cx+Du(4.1)

where x ∈ Rn represents the system states, u is the system input, y∈ Rm is the output,

A,B,C,D are known matrices, E represents the harmonic disturbance matrix and w∈R

is the harmonic disturbance given by

w(t) =v

∑h=2


where for h = 2 . . .v, Ah,φh ∈ R represent the amplitude and phase of each harmonic

order respectively and ωh = 2π f h, f is the fundamental frequency and φh is the phase

angle at harmonic order h. In practical distribution systems, the disturbance matrix, E

is unknown. This matrix represents the relationship between the harmonic disturbance


and the distribution system state variables. It describes how the harmonic disturbance

injections interact with the system state variables.

Assumption 4. The harmonic source is not directly measured. As a result, the state

space representation in (4.1) has no term which reflects the harmonic injections in the

system measurements.

Remark 3. The direct measurement of a harmonic source creates another term in the

output equations and the coefficient of this term is unknown.

The states of the distribution system are taken to be the node voltages and branch

currents. A time-domain method is presented for identifying the harmonic sources.

This method is effective due to the time-varying nature of the loads in distribution

systems. The proposed method utilises a fault observer approach to determine the

exact nodes where the harmonic injections occur.

The dynamics of the harmonic disturbance may be described as a linear exosystem

given by

w = Sw

µ = gT w(4.3)

where S = diag(S2,S3 . . .Sh), Sh =

0 ωh

−ωh 0

, w= [w2,w3 . . .wh], h is a set of integers

H = hl, l = 2 . . .v. l = 1 is the fundamental frequency and is not considered to

be a harmonic disturbance to the system. Thus, the augmented form of the power


distribution system dynamic model is given by

x

w

=

A E

0 S

x

w

+B

0

u

y =[C 0

]x

w

+Du

(4.4)

Estimation of the augmented system states in (4.4) using state observers yields

˙x

˙w

=

A E

0 S

x

w

+B

0

u+

L

L

(y−Cx−Du) (4.5)

where L is designed such that A− LC is Hurwitz and L = [L, L]T , A =

A E

0 S

and

C = [C 0].

The augmented system given in (4.5) provides an estimate of the power system

states and harmonic disturbance states. These augmented states consist of node volt-

ages, branch currents and harmonic disturbances in all nodes and branches of the dis-

tribution system.

To obtain accurate state estimates, a correct representation of E is required. In prac-

tical systems, this matrix is unknown because prior knowledge of how the harmonic

disturbances interact with system states is not available. Therefore, a fault observer

is applied to the harmonic source identification procedure to determine the matrix, E.

If S ∈ Rq and A ∈ Rn, then E ∈ Rn×q. Assuming the harmonic disturbance enters the

system through the kth state. The kth row of the harmonic disturbance matrix becomes

Ek =

[ei ei . . . ei

](4.6)


where k represents the state which is directly affected by the harmonic disturbance, ei

is the harmonic injection index at node i. The state k represents the voltage at node i.

Definition 1. The harmonic injection index, ei at a given node i is the coefficient of the

harmonic disturbance injected into the states at that particular node.

The harmonic injection index is computed to determine the exact states which are

affected by the harmonic injection. This index constitutes the elements of the matrix, E

and is calculated from the shunt admittance of all lines adjacent to the specified node.

The shunt admittance is given by

Y = G+ jBs (4.7)

where Y is the shunt admittance, G is the conductance and Bs is the susceptance. The

shunt capacitance is then calculated from the capacitive susceptance given by

Bs = 2π fCs (4.8)

where f is the fundamental frequency and Cs is the shunt capacitance. The harmonic

injection index at a given node is calculated as twice the inverse of the sum of shunt

capacitances at all the branches adjacent to that node. For a p node power system, the

harmonic injection index at node i is computed as:

ei =2

∑pr=1Ci j

(4.9)

where Ci j is the shunt capacitance for line i− j and is calculated using (4.8).

System nodes are partitioned into suspicious and non-suspicious nodes. Suspi-

cious nodes are nodes which are connected to loads while non-suspicious nodes have

no loads, hence there is no chance of them being harmonic sources. To obtain an


accurate representation of the harmonic disturbance matrix, E, the row of E which

corresponds to the suspicious node voltage is replaced with (4.6) for each suspicious

node. State estimation is then carried out using (4.5). If the estimation error for node i

approaches zero asymptotically, then E is taken to be accurate and the suspicious node

i is identified as the harmonic source.

For a p node distribution system, Kirchhoff’s current law is applied at all the nodes

to calculate the harmonic current injection. The current injection at node i is given by

Ii =p

∑j=1

Ii j +∑ IL,i (4.10)

where Ii j represents the branch current from node i to node j and IL,i is the load current

at node i.

The node current injection matrix is used to compute the harmonic current injec-

tions at all the nodes in a given distribution system. The direction of current flow is

a crucial factor when determining the node current injections. Harmonic currents are

taken to flow into the given node. Nodes which have no load are taken to be zero

injection nodes because the possibility of those nodes being a harmonic source is neg-

ligible. Hence, there is no relationship between the harmonic current injections and

the state variables at that node. Consequently, the row of the current injection matrix

corresponding to a zero load node is populated with zeros. The relationship between

the node current injections and the augmented state variables is given as

Iin j = Mx (4.11)

where [x,w]T is the augmented state variable and M ∈ Rn×r is the current injection

matrix, with r being the total number of augmented states. The current injection matrix


is computed as

M(a,b) =

1 If current flows from node a to the receiving end of the bth state

OR if the load current at the bth state is connected to node a

−1 If current flows from the sending end of the bth state to node a

0 Otherwise(4.12)

4.3.1 Identification of multiple harmonic sources

In power distribution systems, determining the harmonic sources at a given time

instant may be challenging due to the simultaneous occurrence of multiple harmonic

producing loads present in the system. Hence, a combinational method is presented

for harmonic source identification using a time domain dynamic model. Zero injection

nodes, which are nodes with no load, are eliminated from the harmonic source identi-

fication process to improve computation time and reduce redundancy. Given a p node

power distribution system with s loads, there exists a maximum of s simultaneous har-

monic sources. In other words, there exists a likelihood that at a certain time, t, all load

nodes feeding customers may become harmonic sources simultaneously. As a result,

accurate determination of E is crucial for estimation of the harmonic sources irrespec-

tive of the number and locations of the harmonic sources. For the p node system, with

t harmonic sources, E ∈ Rn×tq, where n represents the number of states in the system

and q depicts the dimensions of S. As the number of harmonic sources changes, E and

S change as well, hence S ∈ Rtq×tq populates the diagonal of the matrix S.

In general, for a given power distribution system, the algorithm for determining

multiple harmonic sources is as follows:


1. Determine the state space matrices for the distribution system A,B,C,D using the

method proposed in Chapter 2.

2. Partition the system nodes into suspicious and non-suspicious nodes.

3. Calculate the harmonic injection index, ei, for each suspicious node i using (4.9)

4. Select the states which correspond to node voltages for suspicious nodes.

5. For each suspicious node, i, create Ei by populating the row of E which represents

each state selected in step 4 with the harmonic injection index at that node.

Ei =

0 0 . . . 0

...... . . . ...

ei ei . . . ei

0 0 . . . 0

(4.13)

6. for t = 1 to s, t represents the number of simultaneous harmonic sources and s is

the number of suspicious nodes do

7. Determine the combinations for harmonic sources, z =(

st

)8. for b = 1 to f , f is the number of combinations for t harmonic sources, do

9. Create Et ∈ Rn×st , which is a matrix with diagonal consisting of E for all

possible harmonic source combinations.

10. Create S ∈ Rqt×qt as diag([S . . .S])

11. Design a state observer given by

˙x

˙w

=

A Et

0 S

x

w

+B

0

u+L(y−Cx−Du) (4.14)


where L is designed such that A− LC is Hurwitz [96], A =

A Et

0 S

and

C =

[C 0

]12. if rt = y−Cx−Du and limt→∞ rt = 0 then

13. There are t harmonic sources in the system located at the nodes in the

combination, z(b).

14. else

15. The combination z(b) does not represent the set of nodes which are har-

monic sources.

16. end if

17. end for

18. Determine the current injection matrix using (4.11) and calculate the harmonic

current injections at the identified harmonic sources.

19. end for

The proposed harmonic source identification algorithm is carried out by determining

E for each assumed combination of harmonic sources. State estimation on the aug-

mented system is then carried out and the estimation error is analysed to verify if the

assumed combination of harmonic sources is accurate. If the error approaches zero

asymptotically, then the assumed combination is the harmonic source and if the con-

trary is the case, the combination does not represent the harmonic sources in the power

distribution system.

Remark 4. The condition that verifies the number and locations of harmonic sources

by checking the estimation error applies to all measurements taken throughout the

system. Hence, the estimation error for all measurements must approach zero for the

nodes to be correctly identified as the harmonic sources.


For large systems with a large number of suspicious nodes, the system can be

divided into sub-systems and the fault observer algorithm is applied to each sub-system

to determine the harmonic source. This reduces the number of combinations, thereby

improving computation time and cost. The proposed fault observer based harmonic

source identification is shown in Figure 4.2.

∫

)(ty)(tx

0

B

S

EA

0

∫ [ ]0C

)(ty∧

L

Figure 4.2: Fault observer based harmonic source identification


4.4 Case study

The validity of the proposed combinational fault observer approach to identification

of harmonic sources in power distribution systems is verified by carrying out simula-

tions on the IEEE 13-node distribution feeder which is test system 2 in [99] with short

circuit capacity given as 1100 MVA, 82 lagging, balanced. The single-line diagram

of the 13-node power distribution feeder is shown in Figure 3.5. The grid is modelled

as a voltage source (115 kV) behind an impedance. Transformers are represented by

a two winding model and lines are modelled using a π model with length dependent

parameters. The load and line data are the same as shown in Tables 3.1 and 3.2 respec-

tively.

Linear loads are modelled as an RL impedance while nonlinear loads are modelled

as linear loads in parallel with an AC current source. The system states consist of node

voltages, line currents and load currents.

The meter placement algorithm proposed in Chapter 2 is applied to this system to

identify the optimum number and locations to place measurements in the system. The

binary connectivity matrix for this system is given by

P =

1 1 0 0 0 0 0 0 0 0 0 0 0 01 1 1 0 0 0 0 0 0 0 0 0 0 00 1 1 1 0 1 0 1 0 0 0 0 0 00 0 1 1 1 0 0 0 0 0 0 0 0 00 0 0 1 1 0 0 0 0 0 0 0 0 00 0 1 0 0 1 1 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 0 0 0 0 00 0 1 0 0 0 0 1 1 0 1 0 0 10 0 0 0 0 0 0 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 0 0 0 00 0 0 0 0 0 0 1 0 0 1 1 1 00 0 0 0 0 0 0 0 0 0 1 1 0 00 0 0 0 0 0 0 0 0 0 1 0 1 00 0 0 0 0 0 0 1 0 0 0 0 0 1


with the system nodes given as [50,31,32,33,34,45,46,71,92,75,84,911,52,150].

The solution to the binary integer linear programming problem is given by

U =

[0 1 0 1 0 1 0 1 1 0 1 0 0 0

]T

Therefore, the measurement points are taken at nodes 31, 33, 45, 71, 92 and 84.

Hence, node voltages and branch currents are measured at these nodes and the adja-

cent branches respectively. Twelve synchronised measurements are taken throughout

the system. These measurements include V31, V33, V45, V71, V92, V84, I31−32, I45−46,

I32−71, I92−75, I33−34 and I71−84. This set of measurements makes the observability

matrix full rank, hence, topological and numerical observability are guaranteed.

Three types of residential loads are simulated in this study: Fluorescent lamps,

adjustable speed drives (ASD) and other composite residential loads [99]. From Table

3.1, we can deduce that nodes 45, 46, 71, 75, 92, 52 and 911 are the suspicious nodes

because they contain loads. All other nodes in the system are non-suspicious nodes

because they do not contain loads and are therefore excluded from the harmonic source

identification process. A number of scenarios are simulated to verify the validity and

robustness of the proposed harmonic source identification method.

4.4.1 Single harmonic source

Consider the scenario where there is a single harmonic source at node 71 with

harmonic spectra shown in Table 4.1. The harmonic amplitudes are expressed as a

percentage of the fundamental frequency. The measurements taken from the system

are shown in Figure 4.3.

Applying the harmonic source identification algorithm, the number of harmonic


Table 4.1: Harmonic spectrum for node 71

Harmonic Amplitude(%) Phase()1 100 00 -31.183 6.42 243.805 3.33 -260.707 0.63 178.90

sources is 1, hence, t = 1. A step-by-step application of the algorithm yields the fol-

lowing results:

First, the state space matrices of the system are determined. Then the nodes are par-

titioned into suspicious and non-suspicious nodes. The suspicious nodes are 45, 46,

71, 75, 92, 52 and 911. For a single harmonic source with 7 suspicious nodes, there

are 7 combinations of possible harmonic sources. The next step involves calculating

the harmonic injection index at each suspicious node. The shunt capacitances at all

branches are computed as shown in Table 4.2. Computation of the harmonic injection

Table 4.2: Shunt capacitance at system branches

From node To node Cl(nF)

31 32 7.534

32 33 1.707

32 45 1.707

45 46 1.024

32 71 7.534

71 92 28.540

92 75 28.540

71 84 2.251

84 911 28.540

84 52 0.189

71 150 3.767


Time(s)0 0.05 0.1

V31

(V)

-5000

0

5000

Time(s)0 0.05 0.1

V33

(V)

-5000

0

5000

Time(s)0 0.05 0.1

V92

(V)

-1000

0

1000

Time(s)0 0.05 0.1

V71

(V)

-1000

0

1000

Time(s)0 0.05 0.1

V84

(V)

-1000

0

1000

Time(s)0 0.05 0.1

V45

(V)

-5000

0

5000

Time(s)0 0.05 0.1

I 32-7

1(A

)

-5000

0

5000

Time(s)0 0.05 0.1

I 31-3

2(A

)

-5000

0

5000

Time(s)0 0.05 0.1

I 33-3

4(A

)

-20

0

20

Time(s)0 0.05 0.1

I 71-8

4(A

)

-50

0

50

Time(s)0 0.05 0.1

I 92-7

5(A

)

-50

0

50

Time(s)0 0.05 0.1

I 32-4

5(A

)

-100

0

100

Figure 4.3: IEEE 13-node distribution system measurements

index at each node yields:

e31 = 2.654×108

e32 = 1.082×108

e33 = 1.172×109

e34 = 0


e45 = 7.326×108

e46 = 1.954×109

e71 = 4.752×107

e92 = 3.504×107

e75 = 7.008×107

e84 = 6.456×107

e911 = 7.008×107

e52 = 1.059×1010

e150 = 5.310×108

The harmonic injection index at node 34 is taken as zero due to the inductive nature of

the transformer adjacent to it. After the harmonic injection index has been calculated,

the next step is identifying the states which correspond to suspicious node voltages.

These states include the 20th, 24th, 22nd , 29th, 35th, 33rd and 31st states for the nodes

45, 46, 71, 92, 75, 911 and 52 respectively. Hence, the harmonic injection index at

each suspicious node is used to populate the row which represents each of these states

and the estimation error from state estimation using the fault observer is analysed and

the results determine if the suspicious node is a harmonic source. The analysis of the

estimation error is carried out by determining the mean squared error (MSE) for the

estimation error. The MSE is expressed as

MSE =1N

N

∑m=1|ym− ym|2 (4.15)

where N is the total number of time samples, y represents the estimated measurement,

y is the true measurement.


A threshold is set at 0.05 for identifying the harmonic sources. If the MSE is below

the threshold, this is an indication that all estimation errors approach zero asymptot-

ically and the corresponding node combination is taken to be the harmonic source.

However, if the MSE is above the threshold, then the node combination is not the

harmonic source.

Figure 4.4 shows the MSE for the 13-node power distribution feeder with a single

harmonic source. Logarithmic scales are used to show the results for ease of readabil-

Node combinations1 2 3 4 5 6 7

V,A

10-10

10-5

100

105

V31

V33

V92

V71

V84

V45

I32-71

I31-32

I33-34

I71-84

I92-75

I32-45

X: 4Y: 0.1192

Threshold at 0.05

Figure 4.4: MSE for all measurements with single harmonic source

ity due to the variations between values of the MSE. The node combinations and the

corresponding nodes they represent are shown in Table 4.3. From Figure 4.4, it is clear

that the third node combination is the only instance where the MSE for all measure-

ments are below the threshold. The third combination represents node 71. The tick

in the figure shows the MSE for V92 at combination 4. This is the largest value of the

MSE at that combination. This point appears to be very close to the threshold however,

the MSE at this point is 0.1192 which is not as close to the threshold as it appears. The

appearance of proximity to the threshold is as a result of the logarithmic scales used to


Table 4.3: Combinations and their respective nodes

Combination Node1 452 463 714 925 756 9117 52

present the results. Table 4.4 shows the MSE values for the measurements with voltage

measurements in volts (V) and current measurements in Amperes (A). The table also

shows node 71 to be the harmonic source as all MSEs for the measurements are be-

low the 0.05 threshold for this combination. The requirement for identifying a specific

node combination as the harmonic source is that the MSE for all measurements should

be below the threshold. If, for instance, the MSE for some measurements satisfies this

condition but the others do not, then the specified node combination is not identified as

a harmonic source. Figure 4.5 shows the estimation error when node 71 is taken as the

harmonic source. The figure shows that the estimation error for all measurements ap-

proaches zero asymptotically, thereby satisfying the condition limt→inf rt = 0 as shown

in step 12 of the harmonic source identification algorithm.

After node 71 has been identified as the harmonic source, the amplitude and phase


Table 4.4: Mean squared error in all measurements for single harmonic source

Suspicious Nodes

Meas 45 46 71 92 75 911 52

V31(V ) 6.61e-6 6.42 3.41e-9 5.76e-9 0.04 4.24e-3 4.59e-3

V33(V ) 0.03 1.33e5 1.07e-6 4.79e-5 2.28e3 2.53e2 2.74e2

V92(V ) 4.13e-4 4.17e2 4.12e-6 0.12 6.75e7 1.13e3 1.22e3

V71(V ) 2.69e-2 5.63e3 1.77e-4 7.98e-3 3.80e5 4.43e4 4.80e4

V84(V ) 3.56e-4 3.81e2 3.57e-6 1.23e-4 1.07e4 7.44e6 7.63e6

V45(V ) 2.25 8.18e6 6.01e-7 3.12e-5 1.38e3 1.51e2 1.63e2

I32−71(A) 0.15 1.10e5 9.11e-6 1.49e-3 4.42e5 4.79e4 4.95e4

I31−321(A) 0.25 2.19e5 8.54e-7 1.93e-4 1.27e3 133.26 144.60

I33−34(A) 0.11 5.84e4 5.89e-7 1.93e-4 795.44 83.24 89.85

I71−84(A) 2.38e-3 2.68e2 8.84e-6 1.19e-4 3.34e4 3.77e6 3.84e6

I92−75(A) 2.38e-3 264.30 8.71e-6 7.19e-3 3.58e7 3.57e3 3.81e3

I32−45(A) 0.01 1.60e7 2.39e-7 7.09e-5 291.00 30.70 33.31

of each harmonic component being injected at that node is determined using the ex-

pression in (4.11). The current injection matrix for the 13-node feeder is given by

Is =

0 0 0 0 0 0 0 0 0 0 0 −1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 1 1 0 1 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 1 00 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 1 0 1 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0

(4.16)


Time(s)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

V,A

-4000

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500V

31

V33

V92

V71

V84

V45

I32-71

I31-32

I33-34

I71-84

I92-75

I32-45

Figure 4.5: Estimation error when node 71 is taken as the harmonic source

Ih =

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

T

(4.17)

Iin j =

[Is Ih

]x

w

(4.18)

where the system state, x ∈ R35×1, and the harmonic state, w ∈ R6×1. Each row of

the current injection matrix shows the relationship between the current injected at the

corresponding node and the state variables of the distribution feeder. The relationship

in (4.18) is used to determine the amplitude and phase of the exact harmonic injec-

tions at the identified node which is the harmonic source. Figure 4.6 shows the steady


state harmonic current injections at node 71. The iterative observer algorithm pro-

Time(s)0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Ih71

-4

-3

-2

-1

0

1

2

3

4

Figure 4.6: Node 71 harmonic current injections

posed in chapter 3 may be applied to the harmonic current injection at node 71. The

results in Table 4.5 shows a comparison between the actual current injected at node 71

as shown in the spectra in Table 4.1 and the estimated current injections whose am-

plitudes and phase angles were obtained using the iterative observer algorithm. The

Table 4.5: Comparison between actual and estimated current injection at node 71

h Actual Estimated

Amp(%) Ph() Amp(A) Amp(%) Ph() Amp(A)

1 100 -31.2 41.30 - - -

3 6.42 243.8 2.65 6.13 243.6 2.53

5 3.33 -260.7 1.38 2.95 98.0 1.22

7 0.63 178.9 0.26 0.51 175.7 0.22

current injection at node 71 only shows the sum of the harmonic currents injected at


that node excluding the fundamental frequency. This table verifies the accuracy of the

harmonic source identification method proposed because the estimated current injec-

tions are close to the actual harmonic spectrum injected at node 71.

4.4.2 Time varying harmonic injections

The time varying nature of harmonic loads in power distribution systems is a factor

to be considered when determining the location of harmonic sources. The proposed

harmonic source identification method is capable of adapting to the rapidly changing

load profiles connected to distribution systems. Two scenarios are simulated to show

the response of the proposed fault observer method to changes in the system load

profiles.

4.4.3 Change in amplitude of harmonic injection

Suppose the amplitudes of the harmonic currents injected at node 71 are time vary-

ing as shown in (4.19).

A3 =

0 t < 0.5s

6.42% t > 0.5s

A5 =

0 t < 0.5s

3.33% t > 0.5s

A7 =

0 t < 0.5s

0.63% t > 0.5s

(4.19)

At t < 0.5s, there is no harmonic disturbance injection in the distribution feeder. The

amplitude of the 3rd , 5th and 7th harmonic frequencies are zero. However, at t >


0.5s, a nonlinear load is connected at node 71 with amplitudes given in (4.19) where

the harmonic amplitudes are expressed as percentages of the fundamental frequency,

which is 100%. The MSE is computed for all the system measurements subsequent to

applying the fault observer algorithm to identify the harmonic source irrespective of

the time varying nature of the harmonic injections. Figure 4.7 shows the MSE for all

7 combinations when there is a time varying harmonic source at node 71. The figure

Node combinations1 2 3 4 5 6 7

V,A

10-10

10-8

10-6

10-4

10-2

100

102

104

106

108

1010

1012 V31

V33

V92

V71

V84

V45

I31-32

I32-45

I32-71

I33-34

I71-84

I92-75

Threshold at 0.05

X: 4Y: 0.1199

Figure 4.7: Mean squared error for time varying harmonic injections at node 71

shows that at combination 3, the MSE are all below the threshold, thus identifying node

71 as the harmonic source. The data in Table 4.6 shows the MSE at all combinations

used for harmonic source identification.

Figure 4.8 shows the harmonic current injection at node 71 with time varying har-

monic injections. From the figure, it is clear that there is no harmonic injection into the

feeder at t < 0.5s. At t > 0.5s, the harmonic currents are injected into the system and

the proposed fault observer algorithm is able to adapt to variations in harmonic injec-

tions with time. The amplitudes of the estimated harmonic current injection for each

harmonic order at t > 0.5s are 2.54A, 1.23A and 0.21A for the third, fifth and seventh


Table 4.6: Mean squared error in all measurements for time varying harmonic injec-tions

Suspicious Nodes

Meas 45 46 71 92 75 911 52

V31(V ) 6.37e-6 2.22e2 3.38e-11 5.97e-9 11.99 0.13 0.15

V33(V ) 0.03 4.73e6 1.33e-6 5.33e-5 6.99e5 8.12e3 9.27e3

V92(V ) 4.49e-4 1.43e4 5.11e-6 0.12 2.0810 3.62e4 4.14e4

V71(V ) 0.03 1.91e5 2.20e-4 8.85e-3 1.17e8 1.42e6 1.62e6

V84(V ) 3.86e-4 1.31e4 4.43e-6 1.38e-4 3.28e+6 2.25e8 2.56e8

V45(V ) 2.24 2.91e8 7.48e-7 3.44e-5 4.23e5 4.80e3 5.49e3

I31−32(A) 0.25 7.80e6 1.05e-6 2.04e-4 3.88e5 4.32e3 4.91e3

I32−45(A) 0.01 5.73e8 2.97e-7 7.45e-5 8.92e4 9.94e2 1.13e3

I32−71(A) 0.15 3.92e6 1.13e-5 1.47e-3 1.36e8 1.45e6 1.66e6

I33−34(A) 0.11 2.08e6 7.33e-7 2.02e-4 2.43e5 2.72e3 3.07e3

I71−84(A) 2.52e-3 9.24e3 1.09e-5 1.43e-4 1.03e7 1.14e8 1.29e8

I92−75(A) 2.51e-3 9.12e3 1.08e-5 7.11e-3 1.10e10 1.12e5 1.28e5

harmonics respectively. These values are close to the actual values of the harmonic

spectra and this indicates the capability of the fault observer algorithm to identify the

harmonic source and provide accurate information on the amplitude of the harmonic

currents being injected to the specified node. Figure 4.9 shows the estimation error

for the third combination which identifies node 71 as the harmonic source. The figure

shows that after a very short initial transient, the estimation errors for all measurements

approaches zero until t = 0.5s, where there is a very small transient before the errors

return to zero. The transient at 0.05s indicates the change in dynamics of the system

due to the harmonic injection at that point.


Time(s)0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

V,A

-20

-15

-10

-5

0

5

10

15

20

25

30

Figure 4.8: Harmonic current injection at node 71 with sudden harmonic injection at0.5s

Time(s)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

V,A

-4000

-3000

-2000

-1000

0

1000V

31

V33

V92

V71

V84

V45

I31-32

I32-45

I32-71

I33-34

I71-84

I92-75

Figure 4.9: Estimation error for timevarying harmonic injections

4.4.4 Sudden change in harmonic source

Distribution system customers, both residential and industrial, constantly vary the

type of loads they use. This may result in variations in the amplitudes and phase of

harmonic loads or the continuous or interrupted use of nonlinear loads. Some of these


loads, especially for industrial customers, may be mostly used at certain times of the

day. A typical example is the increased use of fax machines, photocopiers, uninter-

rupted power supplies or other nonlinear loads during working hours and a reduction

in the use of these loads after working hours. At a certain instant t, the harmonic

source may be prevalent at a specific node and after a while the harmonic source may

change to another node. It is important for utility companies to be able to track these

changes in the harmonic sources with time. The response of the proposed fault ob-

server based harmonic source identification method to changes in harmonic sources is

explored. For the power distribution system given in Figure 3.5, suppose at t < 0.5s

the harmonic source is at node 71 and at t > 0.5s, the harmonic source at node 71 is

switched off and the load at node 75 is a nonlinear load. The load at node 75 consists of

15% fluorescent, 20% ASD and 15% other residential loads. The harmonic spectra for

both nonlinear loads are shown in Tables 4.1 and 4.7 for nodes 71 and 75 respectively.

The harmonic source identification algorithm is used to determine the exact source of


Harmonic order Amplitude(%) Phase ()

1 100.00 -22.65

3 13.95 216.70

5 4.68 282.30

7 1.70 81.0

harmonic injections and tracks these sources with time. Figures 4.10 and 4.11 show

the MSE for node combinations 71 and 75 respectively.

From Figure 4.10, it is apparent that at t < 0.5s, the MSE for all measurements

is less than the threshold which is set at 0.05. This indicates that at the interval 0 <

t < 0.5s, node 71 is identified as the harmonic source. However, at the interval 0.5s <

t < 1s the MSE exceeds the threshold for some of the measurements therefore at that


Time(s)0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

V,A

0

0.05

0.1

0.15

0.2

0.25

0.3V31

V33

V92

V71

V84

V45

I31-32

I32-45

I32-71

I33-34

I71-84

I92-75

Threshold at 0.05

Figure 4.10: MSE for combination representing node 71 for change in harmonic sourceat t = 0.5s

Time(s)0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

V,A

0

0.2

0.4

0.6

0.8

1

V31

V33

V92

V71

V84

V45

I31-32

I32-45

I32-71

I33-34

I71-84

I92-75

Threshold at 0.05

Figure 4.11: MSE for combination representing node 75 for change in harmonic sourceat t = 0.5s

interval, node 71 ceases to be the harmonic source. Furthermore, the results in Figure

4.11, which represent the MSE when node 75 is taken as the harmonic source, show

that at t < 0.5s, the MSE exceeds the threshold, while at the interval 0.5s < t < 1s,

the MSE for all measurements are below the threshold. This confirms node 75 as the

harmonic source at this interval. The estimation errors for both cases of harmonic


sources are shown in Figures 4.12 and 4.13 for the combinations representing nodes

71 and 75 repectively. The errors show that for node combination 71, the estimation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4000

−3500

−3000

−2500

−2000

−1500

−1000

−500

0

500

T(s)

V,A

0.49 0.495 0.5 0.505 0.51 0.515 0.52

−3

−2

−1

0

Figure 4.12: Estimation error for the combination representing node 71

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4000

−3500

−3000

−2500

−2000

−1500

−1000

−500

0

500

T(s)

V,A

0.4 0.45 0.5 0.55 0.6−6

−4

−2

0

2

Figure 4.13: Estimation error for the combination representing node 75

error approaches zero for the interval 0 < t < 0.5s and at t > 0.5s, the error does not

approach zero asymptotically. The reverse is the case for Figure 4.13, where the error

does not approach zero for the interval 0< t < 0.5s, but approaches zero when t > 0.5s.


4.4.5 Identification of multiple harmonic sources with noisy mea-

surements

The possibility of having a single harmonic source in a distribution system is un-

likely as there are a numerous customers who may simultaneously use harmonic pro-

ducing loads. Hence, it is important for utilities to be aware of all harmonic producing

loads to enable them carry out the desired action for all sources accordingly. The

number of harmonic sources or their locations may be unknown by the utility com-

panies. Therefore, various combinations of possible harmonic sources are simulated

and the estimation errors are analysed to determine the combinations which represent

harmonic sources. Random noise with Gaussian distribution is added to the measure-

ments. For the case study presented, assume two harmonic sources are present in the

system simultaneously at nodes 45 and 75 with spectra given in Tables 4.1 and 4.7

respectively.

Due to the size of the distribution system and the number of suspicious nodes,

the number of possible node combinations is given by 2n− 1. In this case, there are

127 possible combinations and in the case of a larger system with more suspicious

nodes, the number of combinations may be larger. To reduce the computational burden,

the IEEE 13-node distribution system is divided into 2 sub-systems and the algorithm

is applied on both sub-systems to determine the harmonic sources.The split 13-node

system is shown in Figure 4.14.

4.4.5.1 First sub-system

The first sub-system consists of the nodes 50, 31, 46, 45, 32, 33 and 34. The

measurements taken from this sub-system are V31, V33, V45, I31−32, I33−34 and I45−46.

This set of measurements makes the observability matrix full rank. A Gaussian noise


Figure 4.14: Sub-systems for the IEEE 13-node system

of zero mean is added to the measurements. The suspicious nodes in the first sub-

system are 45 and 46, therefore the total number of combinations required for the first

sub-system using 2n−1 is 3.

For the first sub-system, the nonlinear load is taken to be at node 45 with spectrum

shown in Table 4.8. A threshold is set at 0.1 for identifying the harmonic source using


Harmonic order Amplitude(%) Phase()

1 100.00 -37.29

3 14.98 245.88

5 5.22 311.48

7 1.80 110.18

the fault observer algorithm. The results in Table 4.9 show the MSE for the first sub-

system. Figure 4.15 shows the MSE of all node combinations.

Table 4.9: MSE of all suspicious node combinations for the first sub-system

Node combination V31 V33 V45 I31−32 I33−34 I45−46

1 0.06 0.06 0.06 0.06 0.05 0.06

2 0.06 0.45 0.77 0.27 0.07 54.14

3 0.06 0.15 0.24 0.10 0.06 14.08

From the results, it is apparent that the first combination is the only instance where

the MSE is below the threshold for all the measurements. This is an indication that this


Node combinations1 2 3

MS

E(V

,A)

10-2

10-1

100

101

102

V31

V33

V45

I31-32

I33-34

I45-46

Threshold at 0.1

Figure 4.15: MSE of suspicous node combinations for first sub-system

combination represents the harmonic source in the distribution system. The second and

third combinations do not have all the MSE below the threshold, hence they are not

identified as the harmonic source. For this sub-system, the first combination represents

node 45. Therefore, node 45 is identified as the harmonic source. The current injection

matrix for this subsystem is given by

M =

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 −1 0 1 0 1 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 −1 0 1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0

(4.20)


The harmonic current injection at node 45 is determined as shown in Figure 4.16 with

amplitudes computed as 14.99% for the third harmonic, 5.21% for the fifth harmonic

and 1.81% for the seventh harmonic. These numbers are approximately equivalent

to the harmonic spectrum given in Table 4.8, thus, the proposed method of harmonic

source identification for the first sub-system not only accurately identifies the harmonic

source at node 45, it also provides an estimate of the amplitude of the harmonic injec-

tions at that node regardless of the noisy measurements.

0 0.05 0.1 0.15 0.2 0.25 0.3−25

−20

−15

−10

−5

0

5

10

15

20

25

Time(s)

I(A

)

Figure 4.16: Node 45 current injections

4.4.5.2 Second sub-system

The second sub-system consists of the nodes 32, 911, 84, 71, 92, 75, 52 and 150.

Node 32 is the point of connection to the first sub-system. A voltage source behind

an impedance is used to represent the source at node 71. The voltage is obtained from


load flow solutions carried out for the entire 13-node system and the voltage at node

32 is used as the source. To ensure observability of the sub-system while reducing

redundancy, the measurements taken are V71, V92, V84, I32−71, I71−84 and I92−75. This

measurement set makes the observability matrix full rank. The suspicious nodes in the

second sub-system are 71, 92, 75, 911 and 52. Hence, the number of combinations

required for the second sub-system is 25− 1, which gives 31 combinations. This re-

duces the total number of combinations for the entire distribution system from 127 to

34, resulting in a reduction in computational burden.

Assuming there is a harmonic source at node 75 with spectrum shown in Table 4.7

with the threshold set at 0.05, the fault observer algorithm is applied to the second sub-

system to determine the exact nodes where harmonic injections occur. The MSE for all

31 combinations of suspicious nodes in this sub-system are shown in Figure 4.17. From

Node combinations1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

MS

E(V

,A)

10-4

10-3

10-2

10-1

100

101

102

103

V92

V71

V84

I92-75

I71-84

I32-71Threshold at 0.05

Figure 4.17: MSE of node combinations for the second sub-system

the figure, it is apparent that the third node combination is the only instance where the

MSE for all measurements is below the threshold. This indicates that this combination

represents the harmonic source. For this sub-system, the third combination represents

node 75. Figure 4.18 shows the harmonic current injections at node 75. The amplitudes

of harmonic currents injected at node 75 are computed as 13.54%, 4.24% and 1.41%


Time(s)0.05 0.1 0.15 0.2 0.25 0.3

I 75(A

)

-8

-6

-4

-2

0

2

4

6

8

Figure 4.18: Harmonic current injections at node 75

for the third, fifth and seventh harmonics respectively.

4.4.6 Discussion

Unlike static harmonic source identification methods, the fault observer method of

harmonic source identification is capable of adapting to both load and harmonic distur-

bance changes. The network is continuously monitored and information on changes to

harmonic injections or varying harmonic sources are fed back to utility companies to

enable them take appropriate action. This action may be the design of a harmonic mit-

igating technique and/or issuing an appropriate penalty to the customers responsible

for the harmonic pollution. Furthermore, the proposed fault observer method does not

carry out harmonic source identification at each harmonic order of interest, contrary

to previously used methods of harmonic source identification. The exact parameters


of harmonic injection, which include the amplitude and phase, are obtained from the

computation of the node current injections. This provides the possibility of improved

computation time and cost.

Higher order harmonics may be estimated by adjusting the harmonic disturbance

dynamic model. This is done by utilising the specified harmonic order, h in the har-

monic disturbance state transition matrix, S.

4.5 Summary

This chapter has explored the use of a fault observer algorithm to identify harmonic

sources in power distribution systems. The dynamic model of the system is represented

as a state space model and the harmonic disturbance matrix is used to determine the

states which represent system nodes with harmonic injections. A combination of sus-

picious nodes are used to ascertain the harmonic source by estimating the MSE after

each combination is taken as the harmonic source. Combinations which give an MSE

below the specified threshold are identified as the harmonic sources. Several scenarios

have been simulated to verify the accuracy of the proposed fault observer approach.

An advantage of this method is its capability to adapt to changes in the topology and

loads in the distribution system.

Chapter 5

Harmonic source identification with

output delay

5.1 Introduction

Time delay is a cause for concern in a number of applications such as signal pro-

cessing and power systems. This could be as a result of measurement data being

transmitted over a distance due to the use of wide area measurement systems. These

measurements may take some time to arrive at the control station or substation for pro-

cessing. The time taken to transmit measurement data to the control centre for analysis

is called communication delay or latency [130]. Delay in the measurements can result

in inaccurate results of power system studies such as harmonic estimation and power

system state estimation.

In the 1990s, phasor measurement units (PMU) were developed to provide synchro-

nised time stamped measurement data in power systems [131]. These measurements

include voltages, currents or phase and they are synchronised using GPS satellites.

PMUs provide accurate and synchronised measurements for power system analysis and

139

CHAPTER 5. HSID WITH OUTPUT DELAY 140

protection. However, the information provided from PMUs need to be transported to

the control centre for analysis or processing. This information may experience delays

or the failure of a PMU could result in the measurements becoming unsynchronised.

This chapter presents a time-domain approach to state estimation and identifica-

tion of harmonic sources in power distribution systems with unsynchronised measure-

ments. The problem is formulated as an observer design for systems with delay in

the output. This approach utilises a set of differential equations which represent the

dynamic model of the power system to determine the state estimates and identify the

harmonic sources when the measurements are not synchronised. This is instrumental

in the event of a failure or unavailability of PMUs or loss of synchronisation as a result

of communication delay. The method of state estimation using observers adopts the

reduction method used for controller design with input delay proposed in [132, 133].

However, the observer proposed for identifying harmonic sources utilises the concept

of fault observers where accurate modelling of the disturbance parameters yields a

zero steady-state error and vice versa. Delay in the system measurements is taken into

account when designing the observer for identifying harmonic sources. Two cases are

considered: the case of a single delay and the case of multiple delays in measurements.

A new state vector is defined and a delay-free system is developed. The observer is

then designed based on the model of the differential equations representing the new

state vector.

For the identification of harmonic sources, the power distribution system is mod-

elled as a set of differential equations with delayed output. The harmonic disturbance

state is augmented with the power system states and an observer is designed for the

augmented state for delayed outputs.


5.2 Literature review on output delay systems

Power system state estimation with measurement delay is an important area of re-

search due to the need to transmit measurement data to the control station to carry out

studies such as fault detection, system wide state estimation or harmonic source iden-

tification. Observer design for output delayed systems has been a subject of research

over the years. The observer has been applied to the design of an observer based output

feedback controller for time delay systems. In [134, 135] chain observers were pro-

posed for state estimation when the time delay is piecewise constant. Observer-based

controllers have also been applied to systems with output delay as shown in [136]. The

case of multiple input and output delays was presented in [137,138]. In [139], a unified

Smith predictor based H∞ control strategy was proposed for systems with delays in the

measurement. The power system was considered to be a dead-time system with delay

in transmitting the signals from remote locations to the controller site. The impact of

time delay on wide area power systems was investigated in [140] using a supervisory

power system stabiliser(SPSS). The control design was based on H∞ gain scheduling

theory.

State estimation in distribution systems with unsynchronised phasor measurements

was dealt with in [141]. A method of state estimation applicable to three phase unsyn-

chronised phasor measurements in an unbalanced distribution system was presented. A

synchronisation operator was added to the measurement equation to make it synchro-

nised. This approach, however, is only applicable to measurements in phasor mode.

Although harmonic state estimation in power systems with synchronised measure-

ments has been widely investigated [29,31], the case of unsynchronised measurements

has not been extensively explored.


5.3 Harmonic source identification for single output de-

lay systems

5.3.1 Observer design for single output delay systems

Consider an output delay linear system given by

x(t) = Ax(t)+Bu(t) (5.1)

y(t) =Cx(t−d) (5.2)

where x(t) is the state, A,B,C are system matrices, u(t) is the input and the output, y(t)

is a function of the state at the time t−d and d is the time delay which is constant. The

undelayed output is denoted at y(t). The system state at current time t is described as

x(t) = eAdx(t−d)+∫ t

t−deA(t−s)Bu(s)ds (5.3)

We define a new state variable, η(t), given by

η(t), x(t−d) (5.4)

The aim of this section is the design of an observer to estimate the states of the linear

system in (5.2) where the pair (A,C) is observable. The method proposed for observer

design is similar to the truncated predictor method proposed in [132, 133]. Taking

derivatives of (5.4), we have

η(t) = x(t−d) (5.5)


Thus,

η(t) = Aη(t)+Bu(t−d) (5.6)

ζ(t) =Cη(t) ∀t ≥ d (5.7)

where ζ(t) is the output for the new state variable. This formulates the problem as a

linear system with delay in the input. An observer is designed to estimate the states of

the new system and this observer is given by:

˙η(t) = Aη(t)+Bu(t−d)+L(ζ(t)−Cη(t)) (5.8)

where

L = PCT (5.9)

and P is the positive definite solution to the algebraic Riccati equation

AP+PAT +Q−PCTCP = 0 (5.10)

Q is a covariance matrix. The observer given in (5.8) provides state estimates for the

system at time t−d. To obtain the system state estimates at time t, x(t), we have

x(t) = eAdη(t)+

∫ t



5.3.2 Harmonic source identification in single output delay sys-

tems

The state space model of a given power system consisting of harmonic disturbances

with delay in the output may be represented as

x(t) = Ax(t)+Bu(t)+Ew(t)

y(t) =Cx(t−d) (5.12)

where x(t) ∈ Rn represents the system states, y(t) is the delayed output, w(t) is the

harmonic disturbance, A,B,C are system matrices of known dimensions and d> 0 is

the output delay. Initial conditions for the system in (5.12) is assumed to be x0(θ),∀θ∈

[−d,0]. The harmonic disturbance is modelled as an exogenous system with dynamics

as shown in (4.3). This section provides a method of estimating the distribution system

states using observers in the presence of harmonic disturbances and delay in the output.

To obtain the distribution system state estimates, the following assumptions are used:

Assumption 5. The pair (C,A) is observable.

Assumption 6. The time delay, d, is constant.

The system states are therefore augmented with the harmonic disturbance states to

yield: η(t)

w(t)

=

A E

0 S

η(t)

w(t)

+B

0

u(t−d) (5.13)

τ(t) =[C 0

]η(t)

w(t)

(5.14)


Let A =

A E

0 S

, x =η(t)

w(t)

, B =

B

0

and C =

[C 0

]. The observer for the aug-

mented system is therefore given by

˙z(t) = Az(t)+ Bu(t−d)+ L(τ(t)−Cz(t)) (5.15)

where L is the observer gain, which is designed such that A− LC is Hurwitz and z(t)

represents the augmented system state estimate given by z(t) =

η(t)

w(t)

.

The state estimates at time t are required to determine the harmonic sources. To

obtain estimates of the augmented states at the current time, χ(t), the following ex-

pression is applied

χ(t) = eAd z(t)+∫ t


where χ(t) =

x(t)

w(t)

.

Hence,

χ(t) = eAd x(t)+∫ t


where x is the delayed augmented state. The state estimates are subsequently used to

obtain the harmonic current injections by multiplying them by the current injection

matrix. The harmonic current injections are therefore given by

Iin j = Mχ (5.18)

where M is determined using (4.12).

Theorem 2. Given a power distribution system with state space model given by (5.12).

The output of the system is delayed by d > 0 and harmonic disturbances exist in the


system. The state estimates for the augmented system consisting of the distribution

system states and the harmonic disturbance states are obtained using the observer for

the states at time t−d given in (5.15) and the state estimates at the current time t are

obtained using (5.16).

Proof. The observer error dynamics are defined as:

e(t) = x(t)− z(t) (5.19)

Taking derivatives with time yields

e(t) = ˙x(t)− ˙z(t) (5.20)

= Ax(t)+ Bu(t−d)− Az(t)− Bu(t−d)−L(τ(t)−Cz(t)) (5.21)

= (A−LC)e(t) (5.22)

Thus, the error approaches zero asymptotically and the system is stable since (A−

LC) is Hurwitz.

5.4 Harmonic source identification with multiple out-

put delays

5.4.1 Observer design for multiple output delay systems

Consider the linear system with multiple output delays with dynamics given by

x(t) = Ax(t)+Bu(t) (5.23)

y(t) =m

∑j=1

Cx(t−d j) (5.24)


where A,B,C j, j ∈ I[1,m] are constant matrices, d j, j ∈ I[1,m] represents the output

delay and m is the number of measurements. Without loss of generality, the delay

profile may be represented as

0≤ d1 < d2 < .. .dm = d (5.25)

The state for the system (5.24) at time t is given by

x(t) = eAdx(t−d)+∫ t


Hence, the undelayed system output may be defined as

y(t) =m

∑j=1

CeAd jx(t−d)+m

∑j=1

C∫ t

t−d j

eA(t−s)Bu(s)ds (5.27)

Lemma 1. The time delay system in (5.24) can be reduced to the delay-free system

using the expression in (5.27) as shown [138]

x(t) = Ax(t)+Bu(t)

y(t) =Cx(t) (5.28)

A predictor-based observer is thus designed to estimate the system states and this

observer is given as

˙x(t) = Ax+Bu(t)+L(y(t)−Cx(t)) (5.29)

with initial condition assumed to be x(0) and A−LC is Hurwitz. A general method of

state estimation with delay in the output is summarised in the following theorem:

Theorem 3. Assuming the system is observable and L is designed such that A−LC

is Hurwitz, an observer x(t) which provides estimates of the system states at time t is


given as

˙x(t) = Ax+Bu(t)+L(y(t)−Cx(t)) (5.30)

For the system in (5.24) representing multiple delays in the output, the observer pre-

sented in (5.30) provides stable observation error dynamics such that limt→∞ e(t)→ 0.

Proof. The estimation error is defined as

e(t) = x(t)− x(t) (5.31)

Taking derivatives of the estimation error with time yields

e(t) = x(t)− ˙x(t) (5.32)

= Ae(t)−L(Cx(t)−Cx(t)) (5.33)

= (A−LC)e(t) (5.34)

A Lyapunov function is defined ∀t ≥ d and is given by

V = e(t)T Pe(t) (5.35)

where P > 0 is the solution to the Lyapunov equation

(A−LC)T P+P(A−LC) =−Q (5.36)

Taking the time derivative of V along the trajectory (5.34) yields

V (e(t)) = eT (t)[(A−LC)T P+P(A−LC)]e(t) (5.37)

≤−eT (t)Qe(t) (5.38)


Thus, the exponential stability of the observer error is guaranteed.

5.4.2 Harmonic source identification for multiple output delay sys-

tems

The distribution system dynamic model with multiple delays in the output and

harmonic disturbance is given by

x(t) = Ax(t)+Bu(t)+Ew(t)

y(t) =m

∑j=1

Cx(t−d j) (5.39)

The augmented system is therefore given by

x(t)

w(t)

=

A E

0 S

x(t)

w(t)

+B

0

u(t) (5.40)

y(t) =[C 0

]x(t)

w(t)

(5.41)

The observer for the augmented system with multiple output delays is thus given by

˙z(t) = Az(t)+ Bu(t)+L(y(t)−Cz) (5.42)

where z is the observed state for the augmented system.


5.5 Fault observer algorithm for harmonic source iden-

tification with delayed measurements

In this section, the fault observer algorithm is used to determine harmonic sources

in distribution systems for cases where there is a delay in the measurements. The

fault observer proposed in Chapter 4 is modified to take into account both single

and multiple time delays in the measured signals. The modified algorithm is as fol-

lows:

1: Determine the state space matrices for the distribution system

2: Partition the system nodes into suspicious and non-suspicious nodes.

3: Calculate the harmonic injection index, ei, for each suspicious node i using (4.9)

4: Select the states which correspond to node voltages for suspicious nodes.

5: For each suspicious node, i, create Ei by populating the row of E which represents

each state selected in step 4 with the harmonic injection index at that node.

Ei =

0 0 . . . 0

...... . . . ...

ei ei . . . ei

0 0 . . . 0

(5.43)

6: for t = 1 to s, t represents the number of simultaneous harmonic sources and s is

the number of suspicious nodes do

7: Determine the combinations for harmonic sources, z =(

st

)8: for b = 1 to f , f is the number of combinations for t harmonic sources, do

9: Create Et ∈ Rn×st , which is a matrix with diagonal consisting of E for all

possible harmonic source combinations.


10: Create S ∈ Rqt×qt as diag([S . . .S])

11: if there is a single delay in the measurements then

12: Design a state observer given by

˙z(t) = Az(t)+ Bu(t−d)+ L(τ(t)−Cz(t)) (5.44)

where L is designed such that A−LC is Hurwitz.

13: Calculate the distribution system state estimate, x(t) using (5.16).

14: if rt is the residual and limt→∞ rt = 0 then

15: There are t harmonic sources in the system located at the nodes in the

combination, z(b).

16: else

17: The nodes in z(b) are not harmonic sources.

18: end if

19: else

20: For multiple output delays, the state observer is given by:

˙z(t) = Az(t)+ Bu(t)+L(y(t)−Cz) (5.45)

21: if rt is the residual and limt→∞ rt = 0 ∀t ≥max(d j), j = 1 . . .m then

22: There are t harmonic sources in the system located at the nodes in the

combination, z(b).

23: else

24: The nodes in z(b) are not harmonic sources.

25: end if

26: end if


27: end for

28: Determine the current injection matrix and calculate the harmonic current injec-

tions at the nodes identified as the harmonic sources.

29: end for

The fault observer algorithm is modified to take into account the delay in the measure-

ments.

5.6 Case study

Consider the IEEE 13-node distribution feeder used as a case study in chapter 4.

The same system parameters are also used in this case study. Twelve measurements

are taken from the system and these measurements include: V31, V33, V92, V71, V81,

V45, I31−32, I45−46, I33−34, I92−75, I71−84 and I32−71. This measurement set makes

the observability matrix full rank. There are two harmonic sources in the distribution

system, at nodes 46 and 75, with spectra shown in Tables 5.1 and 4.7 respectively.

As discussed in chapter 4, the size of the system and the number of suspicious loads

results in a large number of combinations. Hence, the system may be split into two sub-

systems and the fault observer algorithm is applied to both subsystems to determine the

harmonic source. The sub-systems are shown in Figure 4.14. The mean squared error

for each combination is analysed to determine the harmonic source by checking if the

residual from state estimation approaches zero asymptotically.

Two scenarios are simulated to verify the validity of the proposed harmonic source

identification algorithm with time delay in the outputs. First of all, the case of a single

output delay is considered and subsequently, the case of multiple measurement delays

is explored.


5.6.1 Single output delay

5.6.1.1 First sub-system

The first sub-system consists of the nodes 50, 31, 32, 33, 45, 46 and 34. We

assume that the measurements are delayed by 0.1s. The suspicious nodes are node

45 and 46. The threshold for identifying the harmonic source is set at 0.05s. The

load at node 46 is taken to be a harmonic source with spectrum shown in Table 5.1.

Figure 5.1 shows the MSE for all node combinations in the first sub-system. The


MS

E(V

,A)

10-10

10-5

100

V31

V33

V45

I31-32

I33-34

I45-46

Threshold at 0.05

Figure 5.1: MSE of all combinations for the first sub-system with delay at 0.1s

figure shows that for the second node combination, which represents node 46, the

MSE is less than the threshold for all measurements, thus, node 46 is identified as the

harmonic source. Figure 5.2 shows a comparison between the actual and estimated

measurements. From the figure, it is apparent that for the combination representing

node 46, the fault observer accurately estimates the system states and measurements

and the residual approaches zero asymptotically.

The harmonic currents injected at node 46 are calculated using (5.18) and are

shown in Figure 5.3. The figures show that the calculated harmonic current injec-



h Amplitude(%) Phase ()

1 100.00 -90.87

3 11.98 245.88

5 4.17 311.48

7 1.44 110.18

0 0.05 0.1 0.15 0.2 0.25 0.3

V,A

×104

-2

0

2Actual measurements

0 0.05 0.1 0.15 0.2 0.25 0.3

V,A

×104

-2

0

2Estimated measurements

Time(s)0.05 0.1 0.15 0.2 0.25 0.3

V,A

-1

0

1Error

V31

V33

V45

I31-32

I33-34

I45-46

Figure 5.2: Comparison between the actual and estimated measurements for the firstsubsystem with delay at 0.1s for combination 2

tions for the system with delay is equivalent to the harmonic current injection for the

delay-free system. This implies that irrespective of the delay in the measurements, the

fault observer algorithm accurately estimates the system states. These state estimates

then provide an indication of the harmonic source and are used to calculate the har-

monic currents being injected at the identified source. Table 5.2 shows a comparison

between the actual and estimated harmonic injections for the time delay system. From

the table, we can deduce that in addition to identifying the harmonic source, the fault

observer algorithm provides an accurate representation of the amplitude and phase of

the harmonic current injections at the identified harmonic source irrespective of the

time delay in the measurements.


0 0.05 0.1 0.15 0.2 0.25 0.3−25

−20

−15

−10

−5

0

5

10

15

20

25

Time(s)

I 46(A

)

(a) Time delayed system

0 0.05 0.1 0.15 0.2 0.25 0.3−25

−20

−15

−10

−5

0

5

10

15

20

25

Time(s)

I 46(A

)

(b) Delay-free system

Figure 5.3: Harmonic current injections for both delayed and delay-free system for thefirst sub-system

Table 5.2: Comparison between actual and estimated harmonic current injections atnode 46 for delayed first sub-system

h Actual Estimated

Amplitude(%) Phase () Amplitude(%) Phase ()

3 11.98 245.88 11.95 246.10

5 4.17 311.48 4.14 311.60

7 1.44 110.18 1.44 109.90

5.6.1.2 Second sub-system

The second sub-system consists of the nodes 71, 92, 75, 52, 911, 84 and 150. The

source for this subsystem is represented as a voltage source behind an impedance as

was done in chapter 4. The same measurement points are also used for the delayed

system. A time delay of 0.1s is applied to the measurements and the threshold is set

at 0.05. A harmonic source is present at node 75 with spectra shown in Table 5.3.

The fault observer is applied to the sub-system with delayed measurements to deter-

mine the harmonic source. Figure 5.4 shows the MSE for all possible combinations of

suspicious nodes. From the figure, it is apparent that the MSE for all measurements is

below the threshold at the third combination. Hence, the third combination, which rep-

resents node 75, is identified as the harmonic source. Figure 5.5 shows a comparison



h Amplitude(%) Phase ()

1 100.00 -22.60

3 13.96 216.70

5 4.68 282.30

7 1.70 81.00


MS

E(V

,A)

10-10

10-5

100

105

V71

V84

V92

I92-75

I71-84


Figure 5.4: MSE for second subsystem with delay at 0.1s

between the actual and estimated measurements for the third combination. The figure

shows that the proposed fault observer accurately estimates the delayed measurements

including the time delay, and the residual approaches zero asymptotically.

The harmonic current injections are calculated using (5.18) and the results are

shown in Figure 5.6. The figures show a comparison between the computed harmonic

current injections at node 75 for the delayed and the delay-free system. It is apparent

that the current injections for the delayed system is equivalent to the current injections

for the delay-free system. This implies that irrespective of the presence of time delay

in the system, the fault observer algorithm accurately provides state estimates of the

distribution system, which are used to calculate the harmonic injections at the node

which is identified as the harmonic source. Table 5.4 shows a comparison between the


0 0.05 0.1 0.15 0.2 0.25 0.3

V,A

×104

-1

0

1Actual measurements

V71

V92

V84

I92-75

I71-84

I32-71

0 0.05 0.1 0.15 0.2 0.25 0.3

V,A

×104

-1

0

1Estimated measurements

Time(s)0 0.05 0.1 0.15 0.2 0.25 0.3

V,A

×104

-1

0

1Error

Figure 5.5: Comparison between the actual and estimated measurements with delay at0.1s for the third combination

0.05 0.1 0.15 0.2 0.25 0.3−8

−6

−4

−2

0

2

4

6

8

Time(s)

I 75(A

)

(a) Time delayed system

0.05 0.1 0.15 0.2 0.25 0.3−8

−6

−4

−2

0

2

4

6

8

Time(s)

I 75(A

)

(b) Delay-free system

Figure 5.6: Harmonic current injections for both delayed and delay-free system for thesecond sub-system

actual and estimated harmonic current injections at node 75. The table verifies the ca-

pability of the fault observer to identify the harmonic current injections at the identified

harmonic source.


Table 5.4: Comparison between actual and estimated harmonic current injections atnode 75 for delayed second sub-system

h Actual Estimated

Amplitude(%) Phase () Amplitude(%) Phase ()

3 13.96 216.7 13.45 213.9

5 4.68 282.3 4.26 278.0

7 1.70 81.0 1.49 78.5

5.6.2 Multiple output delays

5.6.2.1 First subsystem

For the first subsystem, the delay profile is given by

d = [0,0.1,0.2,0.2,0.3,0.2] (5.46)

This delay profile implies that the first measurement is undelayed, and the rest of the

measurements are delayed as shown in (5.46). The initial condition for the observer

is set at zero. Figure 5.7 shows the delayed measurements. Similar to the case of a

single output delay, the harmonic source is taken to be at node 46. Figure 5.8 shows

the MSE of all node combinations for the first subsystem when all the measurements

are delayed by the delay profile in (5.46) for all six measurements.

Figure 5.9 shows the residual for all measurements. The results show that ∀t ≥

max(d j), where max(d j) = 0.3s, the residual approaches zero. Thus, the proposed

observer accurately estimates the system states.


Time(s)0 0.1 0.2 0.3 0.4

V31

(V)

-5000

0

5000

Time(s)0 0.1 0.2 0.3 0.4

V33

(V)

-5000

0

5000

Time(s)0 0.1 0.2 0.3 0.4

V45

(V)

-5000

0

5000

Time(s)0 0.1 0.2 0.3 0.4

I 31-3

2(A

)

-500

0

500

Time(s)0 0.1 0.2 0.3 0.4

I 33-3

4(A

)

-50

0

50

Time(s)0 0.1 0.2 0.3 0.4

I 45-4

6(A

)

-500

0

500

Figure 5.7: Delayed measurements for first sub-system.

5.6.2.2 Second subsystem

For the second sub-system, the delay profile is given by

d = [0,0.1,0.1,0.2,0.2,0.3] (5.47)

The measurements for this sub-system are shown in Figure 5.10. The initial condi-

tion for the observer is set at zero and the threshold is set at 0.05 for identifying the

harmonic source. Figure 5.11 shows the MSE of the residuals for all possible node

combinations. From the figure it is apparent that the MSE for combination 3 is below

the threshold. Combination 3 represents node 75. Therefore node 75 is identified as the

harmonic source. The residual from harmonic source identification for the combination

representing node 75 is shown in Figure 5.12. From the figure, it is apparent that the



MS

E

10-6

10-4

10-2

100

102

V31

V33

V45

I31-31

I33-34


Figure 5.8: MSE of all node combinations with multiple output delays for first sub-system

Time(s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Err

or

-1500

-1000

-500

0

500

1000

1500V31

V33

V45

I31-32

I33-34

I45-46

Figure 5.9: Residual for multiple time delays in the first sub-system

residual approaches zero for t ≥ max(d j), j = 1 . . .m. In this instance, max(d j) = 0.3.


Time(s)0 0.1 0.2 0.3 0.4

V71

(V)

-5000

0

5000

Time(s)0 0.2 0.4

V92

(V)

-5000

0

5000

Time(s)0 0.1 0.2 0.3 0.4

V84

(V)

-5000

0

5000

Time(s)0 0.2 0.4

I 92-7

5(A

)

-200

0

200

Time(s)0 0.1 0.2 0.3 0.4

I 71-8

4(A

)

-200

0

200

Time(s)0 0.2 0.4

I 32-7

1(A

)-200

0

200

Figure 5.10: Delayed measurements for second sub-system.


MS

E

10-4

10-2

100

102

104

106

V71

V92

V84

I92-75

I71-84


Figure 5.11: MSE of all node combinations with multiple output delays for secondsub-system

5.7 Summary

This chapter has provided a method of identifying harmonic sources in a power

system irrespective of the delay in the measurements. The time delay in the measure-

ments may be caused by unsynchronised measurements or delay in transmission of the


Time(s)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

e

-1000

-500

0

500

1000Error

V71

V92

V84

I92-75

I71-84

I32-71

Figure 5.12: Residual for multiple time delays in the second sub-system

measured signals to the control centre for processing. Two scenarios were investigated,

the case of a single delay on all the measurements and multiple measurement delays.

The case of single delay may be more common with delay in transmission of the signal

to the control centre, while the case of multiple delays may be more common with the

measurements being unsynchronised. The results obtained show that the proposed ob-

server design identifies the harmonic sources in steady state irrespective of the delayed

measurements. The residuals for both scenarios approach zero asymptotically, which

indicates accurate state estimation for the combination which represents the harmonic

sources.

Chapter 6

Conclusions and recommendations

6.1 Conclusions

To address the problems of harmonic disturbances in power distribution systems,

knowledge of the harmonic sources and the amplitude and phase of each harmonic

order injected is required. This information can assist utility companies in the design

of mitigating techniques such as filters or penalise the responsible parties. Due to

the constantly varying load parameters in distribution systems, a dynamic method is

required. To achieve this, a state-space model representing the dynamics of the power

system is required. In this thesis, a method of obtaining the dynamic model of a given

power system is presented in chapter 2. Detailed applications of observer designs

to harmonic estimation and harmonic source identification have been introduced in

Chapters 3, 4 and 5.

In Chapter 2, a method of obtaining the state-space model of power systems is

presented. Kirchhoff’s current and voltage laws were applied to the system nodes and

the set of differential equations describing the power system is obtained. In addition,

binary integer linear programming combined with sequential elimination is used to

163

CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS 164

determine the optimum locations to place measurement devices in the distribution sys-

tem. Binary integer linear programming was formulated as a minimisation problem

whose solution provided the minimum number of measurements required and the op-

timum locations to place the measurements in the system. The node voltages and line

currents were measured for the nodes identified using this approach. This method guar-

antees topological observability of the power system, however, it does not ensure that

the system is numerically observable. To test for numerical observability, the Popov-

Belevitch-Hautus test was used. In the event that the rank of the observability matrix

is not full, sequential elimination was applied for identifying the additional measure-

ment set which ensures numerical observability. The results obtained from this chapter

show that the proposed method of optimum measurement placement guarantees both

topological and numerical observability.

In Chapter 3, an iterative observer algorithm was presented for carrying out har-

monic estimation in measured voltage or current signals. This algorithm was based

on the assumption that an observer may have been designed to estimate the states of

the system without the consideration of the harmonic disturbance. The harmonic in-

jection was modelled as an exogenous disturbance to the power system. The iterative

observer algorithm is designed to carry out harmonic estimation on distorted measured

signals. The THD is calculated and measurements with THD below a set threshold

are not estimated. The threshold is determined to be in line with regulatory standards.

For distorted signals, the iterative observer is applied and harmonic estimation is car-

ried out one harmonic order at a time until all the harmonic frequencies present in

the signal have been estimated. This is indicated by the residual from estimation ap-

proaching zero asymptotically. The results from this chapter show that the iterative

observer algorithm adapts to variations in the system loads. These variations can re-

sult in changes in the amplitude of harmonic injections with time and the iterative


observer algorithm adapts to these changes. In addition, the algorithm was applied

to the measurements in the presence of fundamental frequency fluctuations, decaying

DC component and measurement noise. A comparison between the iterative observer

algorithm and Kalman-Bucy filter estimates show that the algorithm terminates itera-

tion when all harmonic disturbances have been estimated, whereas, the Kalman-bucy

filter continues estimation for all harmonic orders it is designed to estimate irrespec-

tive of their presence or absence in the measured signals. Thus, the iterative observer

algorithm reduces the computational burden by only estimating distorted signals and

terminating iteration when all harmonic frequencies in the measured signal have been

estimated.

In Chapter 4, an observer-based algorithm was developed for identifying harmonic

sources in power distribution systems. The system was divided into suspicious and

non-suspicious nodes. Suspicious nodes are nodes which contain loads. Each suspi-

cious node is assumed to be the harmonic source and a combinational approach was

presented for harmonic source identification. The harmonic disturbance matrix, which

represents the relationship between harmonic injections and state variables is unknown.

As a result, node combinations were employed to identify multiple harmonic sources.

Each combination was taken to be the harmonic source and the residual from harmonic

estimation was analysed to verify if the specified combination represents the harmonic

sources. Subsequent to identifying the harmonic sources, the iterative observer algo-

rithm was then applied on these harmonic current injections to estimate the amplitude

and phase of the harmonic currents injected at the identified node combinations. For

large distribution systems the combinational approach to harmonic source identifica-

tion may result in a large number of combinations. Hence, the system was divided into

sub-systems to reduce the number of combinations. To accurately identify the har-

monic sources, the MSE of the residual after estimation was determined. A threshold


was set as a cut-off for identifying the harmonic sources. Node combinations which

yield MSE of the residual below the threshold were identified as harmonic sources,

while combinations with MSE above the threshold were not identified as harmonic

sources. Simulations were carried out on the IEEE 13-node feeder for both single and

multiple harmonic sources. The response of the algorithm to time varying harmonic

injections, changes in harmonic sources with time and measurement noise were in-

vestigated. The results showed that the proposed algorithm was able to identify the

harmonic sources irrespective of these operating conditions.

In Chapter 5, the observer-based algorithm for harmonic source identification was

extended to the case of unsynchronised measurements and output delay. This is useful

in wide area measurement systems when measurements from remote areas take more

time to arrive at the control centre for processing. It is also instrumental for cases of

PMU outage. Both cases of single and multiple output delay systems were considered

in the observer design for time delay systems. For single output delay systems, a new

state variable was created and the system was transformed to become a linear system

with input delay. For multiple output delay systems, this method is not accurate, due

to the variation in time delay for each measurement. The observers designed for both

cases were incorporated into the harmonic source identification algorithm. Results

showed that for the case of single output delay, the residual approaches zero ∀t > 0 for

the node combinations which represent harmonic sources. However, for multiple out-

put delay systems, the residual approaches zero ∀t ≥max(d) for the harmonic source

node combination. Therefore harmonic sources were accurately identified irrespective

of the measurement delay.

In summary, this research has achieved the following:

• Determined the optimum locations to place measurement devices in power dis-

tribution systems to ensure complete observability.


• Designed an iterative observer algorithm to carry out harmonic estimation in

measured signals.

• Developed an observer-based algorithm to identify harmonic sources in power

distribution systems.

• Extended the harmonic source identification algorithm to cases where the mea-

surements are unsynchronised or delayed.

• The effects of resonance on harmonic source identification

6.2 Recommendations for further study

This thesis has provided a method of estimating the harmonic content in a measured

signal as well as identify the harmonic sources in a given power distribution system.

Also, a method of identifying harmonic sources with delay in the measurements taken

from the distribution system has been proposed. In addition to the contributions of this

thesis, further studies need to be carried out to take into account:

• Direct measurement of the harmonic current injections,

• A more robust method of determining the threshold for termination of iteration

in the proposed iterative observer algorithm,

• A more robust method of determining the threshold used to identify harmonic

sources in the proposed fault observer algorithm,

• Application of the algorithms proposed in this thesis to an experimental model

of a power distribution system,


The proposed iterative observer approach to harmonic estimation in a measured

signal is carried out with the assumption that the harmonic current injection is not di-

rectly measured. This is because measuring the harmonic disturbance introduces an

additional term in the measurement equation. The matrix describing the relationship

between the harmonic injections and the system measurements is unknown and needs

to be determined. Therefore, further research needs to be carried out on harmonic

estimation while taking into consideration the harmonic injection which has been di-

rectly measured and determining a method of estimating the additional term. Another

area of research which may be explored in the area of harmonic estimation is accu-

rate determination of the threshold for termination of iteration. Although this thesis

provides a threshold which is below the recommended limit of harmonic injections in

power distribution systems, a more suitable technique of selecting this threshold needs

to be researched. Also, a more robust approach to the determination of the observer

gain for harmonic source identification using the fault observer algorithm needs to be

addressed. This will provide utility companies with an effective method of determin-

ing the threshold for identifying the harmonic sources in a given power distribution

system.

Another research area which needs to be addressed is the application of the methods

proposed in this theses to actual power systems. For harmonic estimation, measure-

ments need to be taken from an experimental model of a power distribution system over

a given time. This measurement needs to incorporate the combined effects of changes

in harmonic current injections with time, variations in fundamental frequency, sudden

load changes and the presence of a decaying DC component. The proposed iterative

observer algorithm should be applied to the measurements taken from the distribution

system model to determine the amplitudes and phase angles of the injected harmonic

frequencies. With regards to the identification of harmonic sources, the proposed fault


observer algorithm needs to be applied to an experimental model of a power distribu-

tion system to identify the exact nodes where the harmonic injections occur.

Furthermore, the effects of resonance on harmonic injections and identifying har-

monic sources in the presence of resonance should be investigated. Resonance in

power distribution systems may result from the presence of power factor correction

shunt capacitors. The distinction between voltage distortions which are as a result of

resonant conditions and voltage distortion resulting from nonlinear loads needs to be

established. A method of identifying harmonic sources as well as diminish the effects

of resonance in the system also needs to be researched.

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Appendices

190

Appendix A

IEEE 13-node distribution test feeder

data

The IEEE 13-node distribution feeder is shown in Figure A.1 The data for this test

Figure A.1: IEEE 13-node distribution feeder

system obtained from [99] is as follows:

191

APPENDIX A. IEEE 13-NODE DISTRIBUTION TEST FEEDER DATA 192

Source system:Node 50

Short circuit MVA 1100 at 82 degrees lagging, Balanced

Substation: Node 50-31

Transformer: 5 MVA, 115 kV delta - 4.16 kV wye grounded

Impedance, z = 1 + j8% at 50Hz

Transformers: Node 33-34

500 kVA, 4.16 kV Delta - 480 V wye

Impedance, z = 1.1 + j2.0%

The upper triangle of phase domain impedance(Ω/mile) and admittance(µS/mile) are

shown below. For nonexistent phases, the matrices have been padded with zeros.

501:

Zabc =

0.3477+ j1.0141 0.1565+ j0.4777 0.1586+ j0.4361

0.3375+ j1.0478 0.1535+ j0.3849

0.3414+ j1.0348

Yabc =

j6.2450 − j1.7664 − j1.3951

j5.8271 − j0.7461

j5.6985

502:

Zabc =

0.7538+ j1.1775 0.1586+ j0.4361 0.1565+ j0.4777

0.7475+ j1.1983 0.1535+ j0.3849

0.7436+ j1.2112


Yabc =

j5.6587 − j1.1943 − j1.5024

j5.2262 − j0.6626

j5.3220

503:

Zabc =

0.0000+ j0.0000 0.0000+ j0.0000 0.0000+ j0.0000

1.3294+ j1.3471 0.2066+ j0.4591

1.3238+ j1.3569

Yabc =

0.0000 0.0000 0.0000

j4.7097 − j0.8999

j4.6658

504:

Zabc =

1.3238+ j1.3569 0.0000+ j0.0000 0.2066+ j0.4591

0.0000+ j0.0000 0.0000+ j0.0000

1.3294+ j1.3471

Yabc =

j4.6658 0.0000 − j0.8999

0.0000 0.0000

j4.7097


505:

Zabc =

0.0000+ j0.0000 0.0000+ j0.0000 0.0000+ j0.0000

0.0000+ j0.0000 0.0000+ j0.0000

1.3395+ j1.3295

Yabc =

0.0000 0.0000 0.0000

0.0000 0.0000

j4.6178

508: Underground cable

Zabc =

0.8506+ j0.4037 0.3191+ j0.0325 0.3191+ j0.0325

0.8597+ j0.4458 0.2848− j0.0145

0.8597+ j0.4458

Yabc =

j94.6212 0.0000 0.0000

j94.6212 0.0000

j94.6212

509: Underground cable

Zabc =

0.9806+ j0.5146 0.0000+ j0.0000 0.0000+ j0.0000

0.0000+ j0.0000 0.0000+ j0.0000

0.0000+ j0.0000

Yabc =

j0.3915 0.0000 0.0000

0.0000 0.0000

0.0000


Zabc is converted to Z012 which represents the zero, positive and negative sequence

impedance as follows:

Zs =Zaa +Zbb +Zcc

3(A.1)

Zm =Zab +Zac +Zbc

3(A.2)

Thus

Zabc =

Zs Zm Zm

Zm Zs Zm

Zm Zm Zs

(A.3)

Let

T =

1 1 1

1 a2 a

1 a a2

(A.4)

where a = cos(2π/3)+ j ∗ sin(2π/3). Hence,

Z012 = A−1ZabcA (A.5)

Similarly,

Y012 = A−1YabcA (A.6)

where Zabc and Yabc represent matrices of self and mutual impedances and admittances

respectively.


A.1 Line connectivity data

Table A.1: Line data

From node To node Line length(km) ID

32 45 0.1524 503

32 33 0.1524 502

45 46 0.0914 503

31 32 0.6096 501

84 52 0.2438 509

32 71 0.6096 501

71 84 0.2438 504

71 150 0.3048 501

92 75 0.1524 508

84 911 0.0914 505

71 92 0.1524 508

harmonic estimation and source identification in …

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