dynamic state estimation and control of power systems · dynamic state estimation and control of...

Dynamic State Estimation and Control of

Power Systems

Kianoush EmamiB.S., M.S.

This thesis is presented for the degree of

Doctor of Philosophy

of The University of Western Australia

School of Electrical, Electronic and Computer Engineering.

2015

a

a c© Copyright 2015 by Kianoush Emami

a

a Dedicated to my wife Maryam.a

Abstract

This thesis reports on an observer based approach, i.e., deterministic functional

observer approach and probabilistic unscented Kalman filter and particle filter ap-

proaches to the problem of dynamic state estimation (DSE) of generators in power

systems. The proposed DSE algorithms presented in this thesis are all decentralized

in which only the measurement of the local phasor measurement units (PMUs) in-

stalled on the generator busbars are used. Consequently, any change in the power

system topology does not affect the DSE process. The performance of the above

mentioned probabilistic approaches have been tested on different IEEE test systems

and are compared when the covariances of the noise in the PMU measurements are

altered.

Furthermore, in this thesis, application of estimation techniques in load frequency

control (LFC) of highly interconnected power systems is investigated. The proposed

techniques for LFC are quasi-decentralised since the DSE based controllers need the

measurement of tie-line powers (power in the transmission line that connects two

different power system areas). The analysis and design of LFC controllers presented

in this thesis is different to traditional methods previously reported in the litera-

ture, the method proposed in this thesis considers the entire network topology. The

proposed schemes have been implemented on the IEEE 39-bus 10-generator 3-area

test system considering noise in the PMU measurements and also generator parame-

ter deviations from nominal values. The proposed probabilistic unscented transform

based method, proposed in this thesis for LFC, takes into consideration noise in PMU

measurements. Moreover, the advantages of functional observer based method in

comparison to traditional state observer based method in LFC are demonstrated in

this thesis.

The application of DSE to detect faults in dynamical systems is also investigated

in this thesis. A functional observer based fault detection technique for dynamical

systems with application to wind turbines is proposed. The proposed scheme has the

ability to detect faults independent of chosen observer parameters. The theoretical

development of the fault detection algorithm and its application to wind turbines is

presented in this thesis.

The theoretical development presented in this thesis on DSE algorithms and DSE

based LFC controllers hinges on the state of the art latest technology available in

PMUs to provide synchronized measurements using global positioning system data.

The thesis presents a contribution in decentralized estimation of generator states

and its utilization in LFC and fault detection.

Acknowledgements

The doctoral study at The University of Western Australia is a long-distance

adventure. I would like to express my heartfelt gratitude to many people who

with their inspiration, encouragement and help, this journey could be successfully

accomplished.

First of all, I would like to extend my sincerest thanks and appreciation to my su-

pervisors Professor Tyrone Fernando and Professor Brett Nener who always guided

and helped me through this journey and motivated me in times of difficulty. Profes-

sor Fernando was not only my coordinate supervisor, he is one of my best friends. I

could not have hoped for a better supervision, his patience, motivation, inspiration,

substantial support and immense knowledge was valuable in completing my thesis.

He always provided me with critical and motivating comments, and feedback on my

work. I learnt from him how to be a good researcher, how to compose a fantastic

presentation of my work and how to have patience to solve problems. I will forever

be thankful to him. I hope that I can be as lively, enthusiastic, and energetic as him.

I would like to express my heartfelt appreciation to Professor Herbert Iu, without

his illuminating instruction, consistent encouragement and substantial support, this

thesis work could not be completed. I would also like to express my sincerest thanks

to Professor Kit Po Wong and Professor Hieu Trinh for their helpful feedback and

solid support. I also like to express my sincere thanks to Head of School Profes-

sor Farid Boussaid for valuable advice and support, Professor Victor Sreeram and

my friends in the Renewable Energy laboratory for all the encouragement during a

challenging period.

Most of all, the words cannot express my heartfelt gratitude to my lovely wife

Maryam and my daughters Parnian and Ava for their patience and their valuable

support. I owe my loving thanks to my family. This work would not be possible

without the peace of mind and motivation they provided for me. I should also

extend my gratitude to my father and mother who have always been there for me.

I would like to thank the anonymous reviewers whose feedback and criticisms

made my papers and this thesis better. This research has been financially supported

by the following awards and scholarships which made this PhD possible: “Australian

Postgraduate Award”, “UWA Student Travel Award” and “UWA PhD Completion

Scholarship”.

Contents

List of Tables xiii

List of Figures xiv

List of Publications and Statement of Candidate’s Contribution xviii

1 Introduction 1

1.1 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Modern Power System Overview 9

2.1 Energy management systems . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Data acquisition and processing subsystem . . . . . . . . . . . 9

2.1.2 Energy control subsystem . . . . . . . . . . . . . . . . . . . . 12

2.1.3 Security analysis and monitoring . . . . . . . . . . . . . . . . 13

2.2 Automatic Generation Control . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Automatic generation control types . . . . . . . . . . . . . . . 15

2.2.1.1 Turbine-governor control . . . . . . . . . . . . . . . . 15

2.2.1.2 Load frequency control . . . . . . . . . . . . . . . . . 16

2.2.1.3 Economic dispatch . . . . . . . . . . . . . . . . . . . 17

2.2.2 Automatic generation unit schemes . . . . . . . . . . . . . . . 18

2.2.2.1 Power system models used in AGC . . . . . . . . . . 18

2.2.2.2 Control approaches . . . . . . . . . . . . . . . . . . . 18

2.2.2.3 AGC design strategies . . . . . . . . . . . . . . . . . 19

2.2.2.4 Impacts of excitation system and load on AGC . . . 20

2.2.2.5 Digital AGC schemes . . . . . . . . . . . . . . . . . . 21

2.2.2.6 Sensitivity of AGC to parameters variation . . . . . 21

2.2.2.7 Adaptive AGC schemes . . . . . . . . . . . . . . . . 22

2.2.2.8 Artificial intelligence based AGC schemes . . . . . . 23

2.2.2.9 Economic dispatch schemes . . . . . . . . . . . . . . 24

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Power System DSE and LFC Using Functional Observers 36

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Functional Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Power System Dynamics and Functional Observer Based Controller

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Power System DSE and LFC Using Unscented Transform 80

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2 Power System Dynamic and Problem Statement . . . . . . . . . . . . 82

4.3 State Estimation Based on The Unscented Transform . . . . . . . . . 85

4.4 Load Frequency Control Case Study . . . . . . . . . . . . . . . . . . 87

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5 Power System DSE Using Particle Filter 117

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.3 Power System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.4.1 Single-machine-infinite bus system . . . . . . . . . . . . . . . . 127

5.4.2 Multi-machine IEEE 9-bus 3-generator test system . . . . . . 131

5.4.3 Multi-machine IEEE 39-bus 10-generator test system . . . . . 140

5.5 Dealing with Bad Data . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6 DSE Approach to Fault Detection with Application to Wind Tur-

bines 159

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.2 Fault Residual In Actual System . . . . . . . . . . . . . . . . . . . . 161

6.2.1 Actual system . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.2.2 Nominal System . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.2.3 Fault Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.2.4 Asymptotic Value of Fault Residuals . . . . . . . . . . . . . . 165

6.2.5 Modified Fault Residual . . . . . . . . . . . . . . . . . . . . . 167

6.3 Higher Order Functional Observers For Fault Detection . . . . . . . . 169

6.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7 Conclusion and Future Directions 181

7.1 Summary of the Results . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

List of Tables

3.1 Generator, governor, turbine, exciter and PSS parameters. . . . . . . 54

3.2 New value of the loads at specified busbars in Case 1 and Case 2. . . 55

4.1 Generator, governor, turbine, exciter and PSS parameters. . . . . . . 95

4.2 New value of the loads at specified busbars in Case 1 and Case 2. . . 96

5.1 Noise variances for all PMU voltage and current measurements in

9-bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.2 Generator, PSS and exciter parameters in 39-bus system. . . . . . . . 142

5.3 Noise variances for all PMU voltage and current measurements in

39-bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.4 Comparison of PF and UKF computational time in 39-bus system. . . 150

List of Figures

2.1 Overview of a modern power system. . . . . . . . . . . . . . . . . . . 10

2.2 Functional diagram of EMS. . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 A simplified block diagram of a generator with turbine-governor control. 16

2.4 Steady-state TGC frequency-power regulation. . . . . . . . . . . . . . 16

2.5 Block digram of a two-area interconnected power system. . . . . . . . 17

3.1 Functional observer based control scheme. . . . . . . . . . . . . . . . 47

3.2 Sub-transient equivalent circuit of the synchronous generator l. . . . . 47

3.3 IEEE 39-bus, 10-generator (New England) test system. . . . . . . . . 49

3.4 Steam speed-governing system with GDB and GRC. . . . . . . . . . 52

3.5 Comparison of FO and SO based methods in terms of frequency re-

sponse, and tie-line power deviations in Case 1. . . . . . . . . . . . . 63

3.6 Comparison of FO and SO based methods in terms of frequency re-

sponse, and tie-line power deviations in Case 2. . . . . . . . . . . . . 64

3.7 Difference in Generator 1 frequency response and tie-line power de-

viations of all areas in Case 1 due to generator parameter variations

of 10% from nominal values. . . . . . . . . . . . . . . . . . . . . . . . 65

3.8 Difference in Generator 1 frequency response and tie-line power de-

viations of all areas in Case 2 due to generator parameter variations

of 10% from nominal values. . . . . . . . . . . . . . . . . . . . . . . . 65

3.9 Comparison between states of the nonlinear model and states of the

small-signal model of the 1st generator in Case 1. . . . . . . . . . . . 66


small-signal model of the 1st generator in Case 1 (continued). . . . . . 67


small-signal model of the 1st generator in Case 1 (continued). . . . . . 68


small-signal model of the 3rd generator in Case 1. . . . . . . . . . . . 69


small-signal model of the 3rd generator in Case 1 (continued). . . . . 70






small-signal model of the 6th generator in Case 1. . . . . . . . . . . . 73


small-signal model of the 6th generator in Case 1 (continued). . . . . 74


small-signal model of the 6th generator in Case 1 (continued). . . . . 75

4.1 Typical power system with PMUs. . . . . . . . . . . . . . . . . . . . 83

4.2 Sub-transient equivalent circuit of the synchronous generator i. . . . . 84

4.3 IEEE 39-bus, 10-generator, 3-area test system. . . . . . . . . . . . . . 90

4.4 Output and pseudo input measurements of the 1st generator in Case 1. 97

4.5 Output and pseudo input measurements of the 3rd generator in Case 1. 98

4.6 Output and pseudo input measurements of the 6th generator in Case 1. 99

4.7 State estimation of the 1st generator in Case 1. . . . . . . . . . . . . . 100

4.7 State estimation of the 1st generator in Case 1 (continued). . . . . . . 101

4.7 State estimation of the 1st generator in Case 1 (continued). . . . . . . 102

4.8 State estimation of the 3rd generator in Case 1. . . . . . . . . . . . . 103

4.8 State estimation of the 3rd generator in Case 1 (continued). . . . . . . 104



4.9 State estimation of the 6th generator in Case 1. . . . . . . . . . . . . 107

4.9 State estimation of the 6th generator in Case 1 (continued). . . . . . . 108

4.9 State estimation of the 6th generator in Case 1 (continued). . . . . . . 109

4.10 Estimation of augmented states in all generators in Case 1. . . . . . . 110

4.11 Frequency and tie-line power deviations in Case 1. . . . . . . . . . . . 111

4.12 Frequency and tie-line power deviations in Case 2. . . . . . . . . . . . 112

5.1 Particle filter state estimation scheme. . . . . . . . . . . . . . . . . . 125

5.2 Two axis equivalent circuit of the synchronous generator l. . . . . . . 125

5.3 Electrical circuit of a synchronous machine connected to an infinite

bus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.4 Comparison of PF and UKF dynamic state estimation methods in

single-machine-infinite bus system. . . . . . . . . . . . . . . . . . . . 130

5.5 Variation of dynamic state estimation error with time in single-machine-

infinite bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.6 Variation of particle filter parameters with time in single-machine-

infinite bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.7 A portion of the WSCC 3-Machine 9-Bus 3-load system. . . . . . . . 132

5.8 Estimation of rotor angles of G2 and G3 with respect to rotor angle

of G1 in 9-bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.9 Error in estimation of rotor angles of G2 and G3 with respect to rotor

angle of G1 in 9-bus system. . . . . . . . . . . . . . . . . . . . . . . . 135

5.10 Variation of estimated states of generator G1 with time in 9-bus system.136



5.13 Variation of particle filter weights with time in 9-bus system. . . . . . 139

5.14 IEEE 39-bus, 10-generator (New England) test system. . . . . . . . . 140

5.15 State estimation of generator 10 with manual excitation in 39-bus

system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.16 State estimation of generator 6 with IEEE ST1A AVR and PSS in

39-bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.16 State estimation of generator 6 with IEEE ST1A AVR and PSS in

39-bus system (continued). . . . . . . . . . . . . . . . . . . . . . . . . 147

5.17 State estimation of generator 3 with IEEE DC1A AVR in 39-bus system.148

5.17 State estimation of generator 3 with IEEE DC1A AVR in 39-bus

system (continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.18 Probability of bad data in measurement 1 and measurement 2 of

generator G3 in 39-bus system. . . . . . . . . . . . . . . . . . . . . . 153

5.19 Estimation of w3 in generator G3 in the presence of bad data and

variable noise covariance in 39-bus system. . . . . . . . . . . . . . . . 154

6.1 Fault detection process . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.2 Three fault residuals of the first example. . . . . . . . . . . . . . . . . 173

6.3 State observer and functional observer based scheme comparison. . . 174

6.4 Three fault residuals of the second example. . . . . . . . . . . . . . . 176

List of Publications and

Statement of Candidate’s Contribution

This thesis largely is based on a series of published and under review research

articles. The bibliographical details are presented below where the order is based

on the chapters the articles are included. Journal articles [1], [2], [3] and [4] are

the basis of Chapters 3, 4, 5 and 6 respectively. Although publications [5]-[8] were

completed during my PhD programme, they have not been included in this thesis.

Conference articles contain preliminary results.

The contribution I have made in the articles that I am the first author, i.e., [2],

[3], [4], [6] and [7] is 80% or above. In article [1] my contribution as the second

author is 45%. My contribution in the articles [5] and [8] is about 15%. I developed

and implemented the algorithms, performed the experiments, analysed the results

and wrote the papers. My supervisors helped me with the writing, reviewed the

papers and suggested many useful feedback for improvement. Each author has

given permission for the work to be included in the thesis.

Journal Articles

[1] Tyrone Fernando, Kianoush Emami, Shenglong Yu, Herbert H.C. Iu and Kit

Po Wong, “A novel quasi-decentralized functional observer approach to LFC of

interconnected power systems,” IEEE Transactions on Power Systems, DOI:

10.1109/TPWRS.2015.2478968, September 2015. (Chapter 3)

[2] Kianoush Emami, Tyrone Fernando, Herbert H.C. Iu, Brett Nener and Kit

Po Wong, “Application of unscented transform in frequency control of a com-

plex power system using noisy PMU data,” IEEE Transactions on Industrial

Informatics, DOI: 10.1109/TII.2015.2491222, September 2015. (Chapter 4)

[3] Kianoush Emami, Tyrone Fernando, Herbert H.C. Iu, Hieu. Trinh, and Kit

Po Wong, “Particle filter approach to dynamic state estimation of generators

in power systems,” IEEE Transactions on Power Systems, vol. 30, no. 5, pp.

2665–2675, 2015. (Chapter 5)

[4] Kianoush Emami, Tyrone Fernando, Brett Nener, Hieu Trinh, and Yang

Zhang, “Functional observer based fault detection technique for dynamical

systems,” Journal of the Franklin Institute, vol. 352, no. 5, pp. 2113–2128,

2015. (Chapter 6)

[5] Yang Zhang, Herbert H.C. Iu, Tyrone Fernando andKianoush Emami, “Co-

operative dispatch of BESS and wind power generation considering carbon

emission limitation in Australia,” IEEE Transactions on Industrial Informat-

ics, DOI:10.1109/TII.2015.2479577, August 2015.

Conference Articles

[6] Kianoush Emami, Tyrone Fernando and Brett Nener, “Power system dy-

namic state estimation using particle filter,” Proceedings of the 40th annual

conference of the IEEE Industrial Electronics Society (IECON 2014), Dallas,

TX, USA, pages 248–253, IEEE, Oct. 2014.

[7] Kianoush Emami, Brett Nener, Victor Sreeram, Hieu Trinh and Tyrone

Fernando, “A fault detection technique for dynamical systems,” Proceedings

of the 8th IEEE Industrial and Information Systems (ICIIS), Peradeniya, Sri

Lanka, pages 201–206, IEEE, Dec. 2013.

[8] Hieu Trinh, Tyrone Fernando, Kianoush Emami and D.C. Huong, “Fault de-

tection of dynamical systems using first-order functional observers,” Proceed-

ings of the 8th IEEE Industrial and Information Systems (ICIIS), Peradeniya,

Sri Lanka, pages 201–206, IEEE, Dec. 2013.

Candidate:

Coordinating Supervisor:

1CHAPTER 1

Introduction

State estimation is one of engineering fundamental tasks involving in extraction of

a system internal information from the measured data. The term state or the state

of a system stands for internal variables of a system that represent a complete in-

ternal condition or a status of the system at a given instant of time. Dynamic state

estimation, i.e., DSE, is estimation of the internal states which stimulate change

or progress within a system or process [1]. DSE is applicable to all areas of engi-

neering and science, e.g., electrical, mechanical, chemical and aerospace engineering,

robotics, economics and many more. DSE attracts interests in engineering because

of at least two main reasons:

• In order to implement a feedback controller, dynamic states should be esti-

mated due to unavailability of any direct measurement.

• Dynamic states are available by direct measurements but received measure-

ments should be validated by a supervisory system.

In control theory, a typical computer-implemented system that provides an estimate

of internal states of a real system is called an observer. Observers use measurement

of system inputs/outputs to estimate states. The methodology of an observer, first

was introduced in 1966 by D. Luenberger [2]. Following is the list of deterministic

observers commonly reported in literature:

• State or Luenberger observers.

• Functional observers (considered in this thesis, see Chapter 3 and Chapter 6).

• Sliding mode observers [3].

• Bounding or interval observers [4, 5].

A full order state observer can be designed if feedback of all system states can be

observed from measured outputs, i.e, system is observable. Functional observers are

in reduced order for class of linear time invariant systems and can be designed even

if the observed system is not observable [6, 7].

If the measurements are affected by noise, state estimation deals with estima-

tion of states based on measured data that has a random component. Following

approaches are generally considered in such cases [8]:

Chapter 1. Introduction 2

• The probabilistic approach (considered in this thesis).

• The set-membership approach.

In such estimation problems the following estimation schemes commonly reported

[9]:

• Maximum likelihood estimation (MLE) [10].

• Bayes estimation or Bayes action [11].

• Method of moments estimation [12].

• Cramer-Rao bound [13].

• Minimum mean squared error (MMSE) or Bayes least squared error (BLSE)

[14].

• Maximum a posteriori (MAP) [15].

• Minimum variance unbiased estimator (MVUE).

• Nonlinear system identification [16].

• Best linear unbiased estimator (BLUE) [17].

• Markov chain Monte Carlo (MCMC) [18].

• Wiener filter [19].

• Kalman filter, e.g., extended Kalman filter and unscented Kalman filter [20–22]

(considered in this thesis, see Chapter 4 and Chapter 5).

• Particle filter or Sequential Monte Carlo [23] (considered in this thesis, see

Chapter 5).

Particle filter (PF) and unscented Kalman filter (UKF) are used to estimate dynamic

of highly interconnected power systems in this thesis. UKF uses unscented trans-

form to estimate the states. Unscented transform provides Mean and Covariance

of the propagated Gaussian random variables (sigma points) through a nonlinear

function. Mean and Covariance can be accurately captured to the second order for

any nonlinearity. The UKF is superior to the extended Kalman filter (EKF) when

system nonlinearities increase. Particle filters have some similarities to the UKF in

that they transform a set of points via known nonlinear functions and give the results

1.1. Thesis Objectives 3

to estimate the Mean and Covariance of the states. Particle filters can deal with

nonlinear systems that have states with non-Gaussian distribution. Approximation

of the posterior probability function of the states can be obtained by generating a

large number of weighted random samples (particles). Although implementing PF

can be computationally expensive due to evaluation of a large number of samples

but it may perform better than UKF. See Chapter 5, Section 5.1.

1.1 Thesis Objectives

A power system is made of several sub-systems to deliver electricity form gener-

ation to consumption. To make a reliable network, power systems are divided into

several areas, in which each two areas are connected together with a transmission

line called tie-line. Such power systems are called interconnected power systems.

It is crucial that frequency of all areas in steady state remains identical. Load fre-

quency control, i.e., LFC is a control method that ensures all areas in steady state

have zero frequency and tie-line power deviations.

The focus in this thesis is on linear and nonlinear dynamic state estimation tech-

niques. A comprehensive study on the application of such schemes in LFC and fault

detection using latest technology in measuring devices, i.e., phasor measurement

units (PMU), has been studied.

Here is the list of objectives of this thesis:

1. Review the literature in regards to study deterministic and probabilistic ap-

proaches of dynamic state estimation.

2. Finding how dynamic state estimation can be used in load frequency control

of highly interconnected power systems.

3. Analyse the advantage and disadvantage of the proposed methods in real-time

dynamic state estimation.

4. Investigation of using dynamic state estimation in fault detection of dynamical

systems including the wind turbines.

1.2 Thesis Overview

This thesis is largely organised as a series of published or currently under re-

view articles in international refereed journals (as allowed by the regulations of The

University of Western Australia). The articles are listed as follows:

1.2. Thesis Overview 4

• Tyrone Fernando, Kianoush Emami, Shenglong Yu, Herbert H.C. Iu and Kit

Po Wong, “A novel quasi-decentralized functional observer approach to LFC

of interconnected power systems,” IEEE Transactions on Power Systems, 1st

revision submitted on 17th June 2015. (Chapter 3)

• Kianoush Emami, Tyrone Fernando, Herbert H.C. Iu, Brett Nener and Kit

Po Wong, “Application of unscented transform in frequency control of a com-

plex power system using noisy PMU data,” IEEE Transactions on Industrial

Informatics, 1st revision ongoing, July 2015. (Chapter 4)

• Kianoush Emami, Tyrone Fernando, Herbert H.C. Iu, Hieu. Trinh, and Kit

Po Wong, “Particle filter approach to dynamic state estimation of generators

in power systems,” IEEE Transactions on Power Systems, doi:10.1109/TP-

WRS.2014.2366196, 2014. (Chapter 5)

• Kianoush Emami, Tyrone Fernando, Brett Nener, Hieu Trinh, and Yang Zhang,

“Functional observer based fault detection technique for dynamical systems,”

Journal of the Franklin Institute, doi:10.1016/j.jfranklin.2015.02.006, 2015.

(Chapter 6)

Each paper constitutes an independent set of work. However, Chapter 2 as

well as the series of articles present a complete and coherent thesis. Chapters 1-2

deals with definition of the dynamic state estimation and overview of modern power

systems. The types of automatic generation control and relevant control schemes

have also been discussed in detail where potential areas of dynamic state estimation

and application of such methods in frequency control of highly interconnected power

systems has been investigated. Main chapters present novel techniques of dynamic

state estimation with investigation of using such techniques in load frequency control

of IEEE power test systems and fault detection of the wind turbines. An overview

of the chapters is as follows:

Chapter 2 provides a general discussion about modern power systems and en-

ergy management systems. Moreover, the automatic generation control (AGC), its

types and relevant control methodologies are discussed in detail.

Chapter 3 presents a novel functional observer based approach to power system

load frequency control. This chapter presents a new scheme of functional observer

design, based on a method proposed in [7], for the purpose of dynamic states es-

timation. This chapter also shows how functional observers can be designed for

a particular generator to control frequency of a multi-area interconnected power

1.3. Contribution 5

system. Since the proposed technique uses PMU measurements only from tie-lines

and generator busbars where the LFC controllers are installed, the proposed scheme

is quasi-decentralized. To show effectiveness of the method, the LFC controllers

have been implemented on a highly interconnected power system (IEEE 39-bus

10-generator 3-area test system) in presence of load fluctuations, governor dead

band (GDB) and generator rate constraint (GRC) in two different scenarios.

Chapter 4 shows application of unscented transform in load frequency control

when discrete-time PMU measurements are affected by noise. Similar to Chapter 3,

the presented technique in this chapter is quasi-decentralized. Furthermore, the

unscented transform based filter is nonlinear and can handle noise in PMU mea-

surements and generator dynamic nonlinearities. The presented scheme has been

implemented on IEEE 39-bus 10-generator 3-area test system. The results demon-

strate a solid dynamic state estimation and effective load frequency control.

Chapter 5 investigates a particle filter approach to dynamic state estimation

of power systems. This chapter demonstrates a different method to the previous

dynamic state estimation techniques presented in Chapters 3 and 4. The proposed

scheme has been implemented on different standard power test systems, i.e., single-

machine connected to infinite bus, IEEE 9-bus 3-generator system, and IEEE 39-bus

10-generator system. The proposed scheme has been compared with unscented

transform based technique which has been presented in Chapter 4.

Chapter 6 presents a fault detection technique in dynamical systems using func-

tional observers. Functional observer equations used in this chapter are a modified

version of the work proposed in [6]. Fault residual which is a function used for fault

detection is also modified in order to be independent of the observer parameters.

Therefore, the proposed scheme is in reduced order. The speed of fault detection

also can be altered by choosing appropriate observer parameters. Since the observed

system is not necessarily needed to be observable the method proposed in this chap-

ter is applicable to the systems where full order observers cannot be designed. At

the end of this chapter, applicability of the proposed fault detection technique on

wind turbines has been tested and continued with two more numerical examples.

Chapter 7 concludes the thesis by summarizing the findings of this research.

1.3 Contribution

The following is a summary of the major contributions in this thesis.

• A comprehensive review of methods for dealing with real-time dynamic state

estimation.

References 6

• Review of load frequency control methodology and related problem formula-

tion.

• A new approach to functional observer based quasi-decentralized load fre-

quency control of interconnected power systems using the state of the art

latest technology available in PMU measurements.

• A novel unscented transform based approach to quasi-decentralized load fre-

quency control of interconnected power systems considering noise in PMU

measurements.

• A novel particle filter based decentralized dynamic state estimation of inter-

connected power systems using noisy PMU measurements.

• A novel functional observer based fault detection technique for dynamical sys-

tems with application to wind turbines.

References

[1] M. S. Grewal and A. P. Andrews, Kalman Filtering: Theory and Practice with

MATLAB. New Jersey: John Wiley and Sons, 2014.

[2] D. G. Luenberger, “Observers for multivariable systems,” IEEE Transactions

on Automatic Control, vol. 11, no. 2, pp. 190–197, 1966.

[3] V. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electro-Mechanical

Systems. Boca Raton: CRC Press, 2009.

[4] M. A. Rami, C. H. Cheng, and C. de Prada, “Tight robust interval observers:

an LP approach,” in 47th IEEE Conference on Decision and Control (CDC),

2008, pp. 2967–2972.

[5] C. Combastel, “A state bounding observer for uncertain non-linear continuous-

time systems based on zonotopes,” in 44th IEEE Conference on Decision and

Control and European Control Conference, Seville, Spain, 12-15 Dec. 2005, pp.

7228–7234.

[6] M. Darouach, “Existence and design of functional observers for linear systems,”

IEEE Transactions on Automatic Control, vol. 45, pp. 940–943, 2000.

References 7

[7] T. Fernando and H. Trinh, “A system decomposition approach to the design

of functional observers,” International Journal of Control, vol. 87, no. 9, pp.

1846–1860, 2014.

[8] E. Walter and L. Pronzato, “Identification of parametric models,” Communi-

cations and Control Engineering, vol. 8, 1997.

[9] “Estimation theory,” http://en.wikipedia.org/wiki/Estimation theory.

[10] F. Scholz, “Maximum likelihood estimation,” Encyclopedia of Statistical Sci-

ences, 1985.

[11] E. L. Lehmann and G. Casella, Theory of Point Estimation. New York:

Springer Science and Business Media, 1998.

[12] L. P. Hansen, “Large sample properties of generalized method of moments es-

timators,” Journal of the Econometric Society (Econometrica), pp. 1029–1054,

1982.

[13] H. Cramer, Mathematical Methods of Statistics. Uppsala, Sweden: Princeton

University Press, 1999.

[14] R. W. Farebrother, “The minimum mean square error linear estimator and

ridge regression,” Technometrics, vol. 17, no. 1, pp. 127–128, 1975.

[15] K. P. Murphy, Machine Learning: A Probabilistic Perspective. Cambridge:

The MIT Press, 2012.

[16] S. A. Billings, Nonlinear System Identification: NARMAX Methods in the

Time, Frequency, and Spatio-Temporal Domains. Chichester, UK: John Wiley

and Sons, 2013.

[17] C. R. Henderson, “Best linear unbiased estimation and prediction under a se-

lection model,” Biometrics, pp. 423–447, 1975.

[18] C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan, “An introduction to

MCMC for machine learning,” Machine learning, vol. 50, no. 1-2, pp. 5–43,

2003.

[19] N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time

Series. Cambridge: The MIT Press, 1949.

http://en.wikipedia.org/wiki/Estimation_theory

References 8

[20] R. E. Kalman, “A new approach to linear filtering and prediction problems,”

Journal of Fluids Engineering, vol. 82, no. 1, pp. 35–45, 1960.

[21] S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,”

Proceedings of the IEEE, vol. 92, no. 3, pp. 401–422, 2004.

[22] S. J. Julier and J. K. Uhlmann, “New extension of the Kalman filter to nonlin-

ear systems,” in AeroSense 97 Conference on Photonic Quantum Computing,

Orlando, USA, Apr. 1997, pp. 182–193.

[23] F. Gustafsson, “Particle filter theory and practice with positioning applica-

tions,” IEEE Aerospace and Electronic Systems Magazine, vol. 25, no. 7, pp.

53–82, 2010.

9CHAPTER 2

Modern Power System Overview

Major components of a modern power system are various type of generators, loads,

transmission networks, measurement instruments, distributed generations (renew-

able energy sources), and monitoring and control centres [1–4]. A typical diagram of

a modern power system is shown in Figure 2.1. Such power systems are divided into

several areas to have reliable networks. At least one transmission line (tie-line) con-

nects two different areas. Power sharing between two areas take place through these

tie-lines. These type of power systems are called interconnected power systems.

Power systems are subject to disturbance continuously. Sudden changes in loads,

losses of one or more transmission lines, system configuration modification, equip-

ment outages, generator failures, and renewable energy sources connection/discon-

nection are examples of typical disturbances in power systems. Besides, nowadays

electricity utility industry looks for new approaches to deal with growing electricity

demand. They are under pressure to accommodate distributed generation into the

classic power system topologies. As distributed generations from rooftop solar to

small diesel and wind generators are more involved in the power system, the differ-

ences between distribution and transmission grids are more faded. With everything

interconnected, it is very important that such complicated networks have a safe and

reliable operation.

2.1 Energy management systems

To have a safe and reliable operation, energy management systems (EMS) are

used to monitor and control such large and complex systems. As shown in Figure 2.2,

any EMS can be categorized into three main subsystems as listed below [5]:

• Data Acquisition and Processing Subsystem.

• Energy Control Subsystem.

• Security Monitoring and Control Subsystem.

2.1.1 Data acquisition and processing subsystem

In EMS, measurements are provided by a supervisory control and data acquisi-

tion (SCADA) system. The SCADA system includes the following major compo-

nents:

2.1. Energy management systems 10

Figure 2.1: Overview of a modern power system.


Security and control subsystem

Energy control subsystem

Load forecast unit commitment

Economic interchange evaluation

Economic dispatch

Automatic generation control

Optimal power flow

Security dispatch

Environmental dispatch

Data acquisition and processing subsystem

Parameter estimation

State estimation

SCADA measurement

Network topology

Security monitoring

Restorative controls

Contingency analysis

Preventive controls

VAR dispatch

Emergency controls

Normal state

Insecurestate

Extremisstate

Emergencystate

Figure 2.2: Functional diagram of EMS.

• Remote terminal units (RTUs), to convert sensor signals to digital data. Hard-

ware of RTUs are capable of sending and receiving data from SCADA system

in order to perform boolean logic operations.

• Highly automated communication links, i.e., telemetry system and communi-

cation infrastructure.

• Programmable logic controllers (PLCs). They are used in the field to convert

sensor signals to digital data and have more sophisticated embedded control

capabilities. PLCs sometimes are used instead of RTUs because they are more

economical. However, PLCs do not have telemetry hardware.

• SCADA front end computers including software servers, Historian servers and

supervisory system to send field commands.

• Various instruments.

The main role of SCADA is receiving and processing information, managing the

real-time databases/archives, displaying the information, documenting the data,

and finally solving the dispatching tasks [6]. As it is shown in Figure 2.2, state


estimation and automatic generation control are two main components of any EMS.

Based on estimation of states, automatic generation control (AGC) is regulated and

security monitoring of the power system is possible. Therefore, state estimation is

one of the essential cores in management of the power systems. However, since EMS

uses SCADA to monitor power systems, it cannot capture power system dynamics

accurately because of the following technical issues:

• EMS is based on a steady state model of the generation units.

• SCADA system has a slow data update rate.

By introducing synchrophasor or phasor measurement unit (PMU) in 1988, real time

phasor measurements became possible. Each collected PMU measurement data is

synchronized by the global positioning system (GPS). Wide area measurement sys-

tem (WAMS) is a standalone infrastructure, complementary to the conventional

SCADA system, for a safe and reliable grid operation. WAMS uses widely dispersed

PMUs in the power system network to elevate the operator’s real-time situational

awareness. In order to overcome the stated weakness of the EMS/SCADA to esti-

mate power system dynamics, many research efforts have focused on incorporating

dynamic state estimation (DSE) in WAMS using PMUs.

2.1.2 Energy control subsystem

Utilising a control centre in order to manage power flow were suggested before

1965. However, the topic gained more attentions in 1965 after the Northwest United

States blackout. Thanks to advances in computer technology, developing intelligent

software and sophisticated computer applications and utilize them in control centres

gradually became possible. These technological advances lead to introduce EMS in

power systems. The conventional control centres had some or all of the following

functions:

• Analogue metering.

• Analogue power generation control.

• Some semi-automated substations.

• Supervisory control centre.

• Security analysis every season.


Modern control centres are more centralised and take advantage of modern digital

equipment to perform security analysis more regularly, either on-line or off-line.

Modern control centres offer much more functionality than conventional control

centres such as:

• Load forecasting.

• System planning.

• Unit commitment.

• Maintenance scheduling.

• Security monitoring.

• State estimation.

• Economic dispatch.

• Load frequency control.

2.1.3 Security analysis and monitoring

It is very important to ensure any power system is capable of withstanding

contingencies without interruption in power supply or compromising the quality.

Hence, security assessment is required to be done in order to asses vulnerability of

any power system against possible contingencies and relevant study shall be done in

real-time for the following reasons:

• Generation patterns and load demands are continuously changing, as a conse-

quence, a typical power system is always in transient states.

• Due to the size of power systems, they are always subject to disturbances

including sudden large load demand changes, contingencies in overhead trans-

mission lines or loss of any other existing power or control equipment.

System monitoring, contingency analysis and preventive, and corrective actions are

the three main components of power system security analysis.

The knowledge of system states is the most important part of any security as-

sessment and for this reason three different state operating point for power systems

are defined:

• Normal state.

2.2. Automatic Generation Control 14

• Emergency state.

• Restorative state.

One of the main reasons of blackout is cascading effect when an equipment may

be switched out due to violation of the tolerance, consequently putting more load

and pressure on other equipment and causing them to be switched out. Apparently,

the economic aspect of operation point has no priority in emergency state while it

is crucial to control and return the operating point within normal limits of power

equipment to prevent cascading effect and eliminating possible blackouts. On other

hand, the goal in the restorative state is to restore the power in the sections which

have lost the power in minimum possible time.

Security assessment can be done by employing SCADA systems, in which all

information such as voltages, currents, power flows, status of circuit breakers, are

acquired from different measurement devices and sent to central control systems.

SCADA systems generate alarm in event of overloading equipment. SCADA sys-

tems also provide operators with possibility of controlling circuit breakers, switches

and transformer taps. However, contingencies occurs in a fraction of a second, thus

operators cannot react on time, consequently using contingency analysis programs

are useful. In fact, contingency analysis is the second most important security func-

tion. To design contingency analysis programs model of the power system is used.

Contingency analysis programs monitor conditions in event of any outage and inform

the operators to start corrective actions. Maintenance of power equipment usually

requires to disconnect equipments and causes outages as well, hence contingency

analysis is also utilized for preparation of scheduled maintenances.

2.2 Automatic Generation Control

Automatic generation control unit, i.e. AGC, is a control system that adjusts bal-

ance between power production and power demand, in response to bearable faults,

disturbance and load changes in power systems. The balance in power can easily be

detected by observing frequency of the power system. Any increase in frequency in-

dicates that all generators in power system are accelerating and the generated power

is more than power consumption. Conversely, any reduction in system frequency

points out that power demand in system exceeds the instantaneous generated power

and generators are decelerating. In addition to the frequency, scheduled power ex-

change to the other areas in interconnected power systems, i.e., tie-line power, is

another variable of interest.


AGC Techniques have been developed to a significantly wide and deep extent in

recent decades. There are some valuable contributions in the design of AGC in order

to tackle load characteristics, excitation control, parallel ac/dc transmission links,

uncertainties and parameter variations in power systems. Also, some efforts have

been made to develop microprocessor based self-tuning adaptive AGC in a number of

articles. Most recently, soft computing based schemes such as neural networks, fuzzy

logic and genetic algorithms, have been incorporated into AGC to deal with system

nonlinearities and insufficient knowledge of system modelling. Apart from control

concepts, power system needs to be more resilient and smarter than before because

of the introduction of renewable energy sources like wind, solar (photovoltaic, solar

thermal), superconducting magnetic energy storage (SMES), combined heat and

energy (fuel cell, biomass and microturbines), hydroelectric, wave, geothermal and

tidal energy into the power systems. Accordingly, control philosophies related to

AGC have been adapted to deal with the new power system dynamic performance.

There are three types of automatic generation control as follows:

• Turbine-governor control (TGC).

• Load-frequency control (LFC).

• Economic dispatch.

2.2.1 Automatic generation control types

2.2.1.1 Turbine-governor control

TGC is a control action that can bring a deviation of the frequency back to

zero by adjusting mechanical power of the turbine. Figure 2.3 shows a simplified

block diagram of a generator with TGC where ∆Pm(t), ∆Pl(t), ∆Pcl(t), and ∆f(t)

are deviations of, mechanical power, load active power, AGC signal (or reference

mechanical power), and frequency of the system from the nominal values at time t

respectively. In steady-state speed regulation or droop can be adjusted by choosing

a proper value for R, typically within range of 2% to 6% of rated speed. If AGC

signal stays unchanged, i.e., ∆Pcl(t) = 0, the feedback control 1Rcan only regulate

the frequency over a small change around the operating point, i.e., ±0.5% of the

desired system frequency. For bigger value of the frequency deviation due to the

bigger load variation, AGC action is required. Frequency settles down on the desired

value when AGC assigns a new value for Pcl(t). When the number of generators in

an area is more than one, choosing a control action is a challenging task. Mechanical


Governor Turbine Generator

AGC

∆Pl(t)

+-

∆f(t)

1R

∆Pcl(t)

+

- ∆Pm(t)

Figure 2.3: A simplified block diagram of a generator with turbine-governor control.

0 0.2 0.4 0.6 0.8 10

2

4

6×10−2

∆Pm (p.u.)

∆f(p.u.)

∆Pcl = 0.5

∆Pcl = 1

Figure 2.4: Steady-state TGC frequency-power regulation.

power of the turbine in steady state can be calculated according to the following

equation,

∆Pm = ∆Pcl −1

R∆f. (2.1)

According to (2.1), Figure 2.4 demonstrates how frequency can be regulated by

choosing different values of reference mechanical power provided by AGC for a sys-

tem with speed-regulation of R = 5%, and with rated speed and desired frequency

of 1 per unit.

2.2.1.2 Load frequency control

Load frequency control is a control action that can restore both frequency and

tie-line power of a multi-area interconnected power system back to the operating

points. LFC acts with a response time of a few seconds in order to retain a stable

system frequency. In general, the product of the frequency deviation with a fre-

quency bias constant (b) is linearly combined with the weighted sum of the power

deviation of the tie-lines into a new variable called Area Control Error, i.e., ACE.

Control strategies are then required to regulate ACE in order to bring frequency and


tie-line power deviations back to zero. To provide a better understanding of LFC,

consider the system shown in Figure 2.5. The presented system is a simplified two-

Governor 1 Turbine 1 Generator 1

∆Pl1(t)

+

-∆f1(t)

1R1

∆Pcl1(t)+

-

-

∆Pm1(t)

b1

+

Governor 2 Turbine 2 Generator 2

∆Pl2(t)

+ -∆f2(t)

-

∫a1

a2

-

+

+

1R2

∆Pcl2(t)

+

b2

+ ∆Pm2(t)+

-

∫ACE1

ACE2

∫

∆Ptie1

∆Ptie2

Controller 1 ∆Pcl1(t)

Controller 2 ∆Pcl2(t)

Figure 2.5: Block digram of a two-area interconnected power system.

area interconnected power system [7], in which ∆Ptiei, i ∈ 1, 2 is the measurement

of the tie-line power at each area and ai, i ∈ 1, 2 is a constant. As it can be seen,

area control error signal in each area can be generated according to the following

equation,

ACEi(t) = ∆Ptiei(t) + bi∆fi(t), i ∈ 1, 2 (2.2)

Control signal ∆Ptiei, i ∈ 1, 2 in each area can be generated using ACE signal of

the area. The system shown in Figure 2.5 has two autonomous controllers which are

used to control and regulate the frequency and the tie-line power deviations. Since

each controller requires only the local measurement of the frequency and the power

measurement at the tie-line, the presented LFC scheme is quasi-decentralized.

2.2.1.3 Economic dispatch

Economic dispatch is another type of AGC which controls generators of an area

in order to achieve optimal power output to tackle load variations in the area, with

the least possible cost, and considering operational and system constraints.

According to the US Energy Policy Act of 2005, the term economic dispatch is

defined as [8], “the operation of generation facilities to produce energy at the lowest


cost to reliably serve consumers, recognising any operational limits of generation

and transmission facilities”.

Network system losses, generator costs such as fuel, maintenance and start-up

prices are the main factors contributing to operating costs. Furthermore, spare

generator capacity shall be available in order to deal with load demand fluctuations,

load predication errors and the inadvertent loss of scheduled generating plants, all

of which are included in an objective function. The minimum operating cost then

can be obtained by optimizing the objective function. Based on the optimization

results, at a specified time-interval, AGC calculates the amount of power that each

generator should produce to deal with the load demand fluctuations. Accordingly, a

control signal is sent by AGC to each individual generator in the area at the specified

time-interval.

2.2.2 Automatic generation unit schemes

2.2.2.1 Power system models used in AGC

To demonstrate control approaches associated with AGC, it is required to con-

sider different power system models with distinct characteristics. The majority of

research in AGC has been implemented on small-signal models of two-area/multi-

area interconnected power systems, see [9–14]. The generator rate constraint, i.e.,

GRC, for both continuous and discrete-time power systems has been incorporated

into linear models in [14, 15]. An optimized tracking method with incorporating

of energy source dynamics into AGC design, has been proposed in [15], in which

the dynamic system output is a connected load. Nonlinearities of power systems

have been considered to a fairly deep extent in [16–19]. Particularly in [19], it has

been shown that governor dead-band (GDB) nonlinearity has destabilizing effect on

conventional AGC systems. Moreover, GDB can cause continuous oscillation in fre-

quency and tie-line power transient response. In this thesis, we have considered the

aforementioned weaknesses in AGC modelling by implementing a complex highly

interconnected nonlinear power system dynamics and by considering the effect of

GRC and GDB on performance of the proposed LFC controllers, see Chapter 3,

Section 3.4.

2.2.2.2 Control approaches

Conventional optimal AGC schemes that have been extensively proposed in the

last three decades are based on the accessibility of all dynamic states. The link

between system frequency and the time domain closed-loop transient performance

has been reported in a number of research articles. However, the drawback of such


techniques is mainly a relatively large frequency settling time, see [12,20,21]. Opti-

mal modern control based AGC has also been considered in [22,23]. Such techniques

are subjected to availability of all system dynamic states by measurements which is

not a feasible practice. Therefore, many AGC research efforts have been made to

incorporate observers to estimate dynamic states, see [24–29]. For instance, in [24]

an observer based load frequency control has been proposed for a two-area inter-

connected power system. This technique uses differential approximation as well as

an adaptive state observer for identification of unknown states and unknown de-

terministic demands given tie-line power is measurable. Optimal Observer based

AGC schemes have been mostly developed using nonlinear transformations [25] and

using reduced-order models with local observers [26]. Considering the limitations

and infeasibility of implementing AGC using all state variables, suboptimal AGC

regulators have been suggested in [30–32]. The application of suboptimal AGC in

load frequency control has been considered and illustrated in [30]. In addition to

optimal and suboptimal AGC design, modal control theory has also been applied

to AGC design algorithms where modal and singular perturbation techniques are

employed to achieve a preferable decoupling effect [33]. In such techniques, local

controllers for each decoupled subsystem are devised individually to satisfy closed-

loop pole placement requirements of the subsystems. Complexity of the proposed

method is simplified by using only local information. The work proposed in [34]

uses Lyapunov second method and the minimum settling time theory to propose an

AGC regulator with minimized frequency settling time.

2.2.2.3 AGC design strategies

Assorted control strategies for AGC in early years, are based on centralized

strategies and different classes of disturbances, see [11, 12, 22, 32]. For instance, a

feedback and a loop gain based scheme has been suggested in [12] to deal with dis-

turbances. Further improvement has been made in [22] where a different feedback

has been presented to develop an optimal control based scheme. In both research

reports, the disturbances have been assumed to be deterministic, which implies

the proposed propositional controllers have been designed without taking into ac-

count the steady-state requirements and the compensation of the load disturbances.

However, although such postulations make the mathematical realization easier, fea-

sibility of such control philosophies are practically limited. The main weakness of

centralized AGC schemes are the fact that centralization requires data communica-

tion over a wide geographical distance. Moreover, such AGC strategy adds more

complexity to the AGC computation. To tackle such disadvantages of centralized


AGC schemes, many efforts have been made to design and implement decentralized

AGC techniques. Comparing to centralized methods, same results can be achieved

by decentralized control schemes. Moreover, the decentralized control schemes have

significantly less realization difficulties in comparison with the centralized control

schemes and are more feasible and reliable, see [35–42]. For example, a complete de-

centralized control technique has been proposed in [38] using a global state feedback

control where control feedback loops for each area are completely decoupled.

Two-level control strategies have also been designed in the literature in the event

of failure in one control level [43]. However, two-level control strategy does not ensure

zero steady-state error. In order to deal with such issue, multi-level optimal control

strategies have been suggested in [44–46].

Slow-fast subsystem decomposition design technique has also been reported in

the literature for a large interconnected power systems in order to reduce the control

efforts, see [47,48]. In such design technique, combination of autonomous controllers

for each individual subsystems is considered in a way that the slow subsystem con-

troller interacts with only one of the fast subsystems at any instant of time.

2.2.2.4 Impacts of excitation system and load on AGC

The assumption used by most AGC designer is the absence of interaction be-

tween the active power-frequency and reactive power-voltage control loops, which

happens only when excitation system operates much faster than governor-turbine

system. The work reported in [19] shows that there exists an interaction between

the aforementioned control loops during dynamic perturbation. Accordingly, the

damping effect of the voltage control on LFC of a two-area interconnected power

system has been addressed in [49]. Two assumptions have been considered [49],

first, reactive power-voltage control loop is much faster than active power-frequency

control loop, hence, area voltage perturbation is available as a control variable and

second, area load is not affected by area voltage perturbation. In order to develop

an approach without considering these unrealistic assumptions, the work presented

in [50] has considered the excitation control in one area and voltage perturbation as

an input variable in the other area. Moreover, the effect of voltage perturbation on

load demand has also been considered in both areas. In this thesis we do not neglect

interaction of low-speed and fast-speed control loops on the considered power sys-

tem. Furthermore, the load demands have been considered to be affected by voltage

perturbations, see Chapter 3, Section 3.4 and Chapter 4, Section 4.4.

Different load characteristics have also been incorporated into AGC design schemes

in [41,51–54]. The response of large power systems to cyclic load variations have been


studied in [51] in which the power system has been modelled as a set of first-order

linear differential equations with a Fourier series pattern load variation. LFC for un-

known deterministic loads and/or randomly varying system disturbances has been

designed in [41]. An adaptive observer has also been designed to tackle the obscu-

rity in loads. A disturbance accommodation control and its effect on inter-system

oscillations has been studied in [55], in which speed-governor and exciter control

loops with voltage dependent loads have been considered. Additional disturbance

accommodation control has demonstrated a significantly better performance than

conventional controllers. Although the complete cancellation of disturbance impacts

cannot be achieved, frequency deviation caused by disturbance effect can be fully

eliminated.

2.2.2.5 Digital AGC schemes

As digital control is more accurate, flexible, reliable, compact, less sensitive

to noise and drift, many research efforts have been made on digital AGC control

schemes in recent years [56–63]. A digital LFC based on field test results has been

presented in [57]. Since ACE signal composed of tie-line power and frequency devi-

ation cannot be measured continuously, the control signal generated by the discrete

time controller remains unchanged between the sampling time. Therefore, refer-

ence [24] has investigated the effect of sampling time on the system dynamic of a

single area discrete time power system. The work reported in [58], provides the

criterion of digital AGC system dynamic performance evaluation and the indices

that shows the effectiveness of digital controllers. More realistic works have been

studied in [64] and [62], in which the power system works in continuous time and

the controllers work in discrete mode. In this thesis, we also have implemented a

digital LFC scheme based on the continuous time interconnected power system and

the discrete time controllers, see Chapter 4, Section 4.4.

2.2.2.6 Sensitivity of AGC to parameters variation

System sensitivity study is also necessary for AGC regulators as the optimal

AGC techniques with rated values may not actually be the best control strategy

for parametric variations and uncertainties caused by both system operation and

environmental conditions. Optimal parameters need to be determined for conven-

tional AGC systems, which has been addressed in [65]. Variable structure system

(VSS) based controllers have their advantages over the controllers based on linear

control theory as VSS reduces the insensitivities to parametric variations. Hydraulic

power systems with VSS technique has been first investigated in [66]. Transient re-


sponse of the load disturbances has been improved by the use of VSS controllers

by selection of proper controller parameters where frequency deviation and tie-line

powers can be controlled effectively. LFC controllers for interconnected power sys-

tems with parametric obscurities have been investigated in [42–46,67]. Particularly,

in [67], a robust controller to parameter variations has been suggested in order to

control frequency deviation caused by load variations in a fast timely manner. The

idea of using adaptive control techniques was incorporated into the design of robust

controllers, which introduced a new methodology for robust adaptive controller de-

sign for the power systems with parametric uncertainties. Other research has been

made a contribution to decentralized robust load frequency control based on Riccati

equation [46] and later on, structured singular values are applied to the design of

decentralized LFC controllers [42]. A tentative conclusion for design methodology

can be drawn from such research works, in which structured singular values can

be used for design of decentralized load frequency controllers when frequency based

diagonal dominance is not achievable. Such empirical selection is able to achieve

desired system dynamic performance [42]. In Chapter 3, Section 3.4 we show that

the proposed functional observer based LFC controllers are insensitive to 10% vari-

ation of generator parameters as the frequency and tie-line power deviations reach

zero in a short duration.

2.2.2.7 Adaptive AGC schemes

Most conventional AGC schemes cannot perform effectively with obscure dy-

namics or sudden fluctuations in dynamic parameters, thereby new type of AGC

schemes were required to address such issues. The outcome of nearly a quarter of

a century of scientific endeavour has been focused on adaptive AGC schemes which

are usually classified into two broad classifications as follows,

• Self-tuning regulators.

• Model reference control systems.

Different adaptive AGC schemes have been investigated by a number of papers in

detail, for instance, see [68] for practical issues to satisfy control criteria in LFC,

and [47] for execution and study of an adaptive LFC system on the Hungarian power

system. A study of parameter variations in hyper-stability condition by utilising pro-

portional integral adaptation and adaptive AGC scheme, has been proposed by [49].

A multi-area adaptive LFC scheme for AGC of power systems and a reduced-order

adaptive LFC approach for interconnected hydrothermal power system have been

investigated in [50] and [69] respectively.


2.2.2.8 Artificial intelligence based AGC schemes

One of the main issues of using reduced-order model of nonlinear systems is

that they may only be valid within a certain operating point. Hence, conventional

control systems must be able to cope with the new changes in the model of the

power system due to the parameter variations. This problem can be tackled by

employing soft computing based AI techniques such as neural networks and fuzzy

logic, and evolutionary computation based AI approaches such as genetic algorithm.

Neural network (NN) based control schemes offers substantial dividends of effective

controlling of nonlinear systems, particularly when a system is operating out of its

linear operating range, which is a valid assumption in power systems. Therefore,

NN based control techniques have been gained significant attention in the field of

power systems frequency control, for example see [53, 54, 70–73]. The outcome of

implementation of NN based AGC scheme has been investigated in [53], in which

two different techniques have been developed and compared according to their per-

formances in detecting the controllable signals in the presence of a noisy random

load. The first technique has employed a neural network based algorithm to iden-

tify the pattern recognition of the controllable signals, and the second method uses

a random signal probability model to detect the controllable signals. It has been

shown that the neural network based AGC scheme has a significant better perfor-

mance in comparison to the second scheme. Furthermore, the results shown in [54]

demonstrate that using generalized neural network in LFC of power system provides

a better control performance than using individual neural networks.

Another soft computing based AI technique that has been attracted much at-

tention in AGC of power system is fuzzy logic. Fuzzy logic based AGC approaches

are fundamentally designed directly from a set of if-then rules that comprehensively

describe the dynamic of the system. This characteristic makes fuzzy logic based

schemes superior to the conventional methods that are based on mathematical model

of the power system [74–81]. In [76] and [77], adaptive fuzzy based controllers have

been proposed to update conventional PI and optimal load frequency controllers

according to load variations and different operating conditions of the power system.

In [82], artificial neuro-fuzzy inference system (ANFIS) has been incorporated into

the LFC controller in order to update the control gains of a PI controller. Takagi-

Sugeno (T-S) fuzzy based LFC has also been developed in [80] in order to deal with

nonlinearities in valve position limits and parametric uncertainties in the power sys-

tem. Genetic algorithm (GA) and particle swarm optimization (PSO) have also

been used in [81] to design an optimized fuzzy PI controller for LFC. The aforemen-

2.3. Conclusion 24

tioned fuzzy based AGC techniques are subjected to availability of if-then rules for

the control actions or on time series (T-S) modelling of the power system. Recently,

a direct-indirect adaptive fuzzy logic control technique has been proposed in [83]

in order to design a fuzzy controller that does not rely on the availability of ifthen

rules.

Genetic algorithm widely has been used for optimising complex nonlinear func-

tions of AGC [82,84–90]. For instance in [85], GA has been used to find the optimized

parameters of a conventional AGC, or in [87], GA and reinforced GA has been used

to design feedback gains of a variable structure LFC scheme.

Performance of GA based LFC schemes has been considered in various articles.

A higher order robust LFC performance has been achieved in [89] using linear matrix

inequalities and genetic algorithm. A comparison study on GA and hybrid GA-SA

based fuzzy AGC scheme can be found in [90]. The hybrid GA-SA technique is

directly depends on transient performance characteristics, and the results show that

the hybrid GA-SA scheme provides more optimal gain values compare to the GA

based scheme.

2.2.2.9 Economic dispatch schemes

Apart from the technical aspects of AGC schemes, satisfying economic constrains

is another crucial objective which should be taken into the consideration. Thus,

economic load dispatch along with AGC have been discussed and relevant schemes

have been developed in some articles, see [91–95]. Comprehensive studies about

economical load dispatch for decentralised power systems have been done in [92]

and [93]. In the work presented in [96] has been shown that classical economic

dispatch can be effectively be replaced by 2-stage real time optimal power flow

scheme. The proposed method has been successfully tested on a 685-bus and IEEE

118-bus network power systems and a zero error convergence with a short settling

time has been achieved. Apart from AGC approaches, electricity real time pricing

is an effective practice to improve system dynamic operations [96]. In the developed

system, PI feedback control law of frequency deviations determined real-time prices,

and the prices have been utilised to assist LFC. A comprehensive report investigating

various cost aspects associated with AGC has been reported in [97].

2.3 Conclusion

In this chapter a brief overview of the modern power systems, energy man-

agement systems and wide area measurement systems has been presented. It has

2.3. Conclusion 25

been demonstrated that dynamic state estimation and automatic generation con-

trol constitute the core of any energy management system. The types of automatic

generation control and relevant control schemes have also been discussed in detail.

To the best of the authors’ knowledge, to date, previous studies on AGC have been

based on approximating all generators in a given area into one single generation

unit. Furthermore, the power distribution network including transmission lines and

various busbars are all lumped into one single entity in the analysis and design of

controllers. In this thesis, all nonlinearities associated with generators, voltage/fre-

quency control loops interaction, have been incorporated into an IEEE power test

system without lumping the generators in one single generation unit and with con-

sidering the power distribution network including transmission lines and various

busbars.

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[95] C. Vournas, E. Dialynas, N. Hatziargyriou, A. Machias, J. Souflis, and B. Papa-

dias, “A flexible AGC algorithm for the hellenic interconnected system,” IEEE


[96] A. W. Berger and F. C. Schweppe, “Real time pricing to assist in load frequency

control,” IEEE Transactions on Power Systems, vol. 4, no. 3, pp. 920–926, 1989.

References 35

[97] T. Snodgrass, “Cost aspects of AGC, inadvertent energy and time error,” IEEE


36CHAPTER 3

Power System DSE and LFC Using Functional

Observers

This chapter is largely based on an article with the following details:

Tyrone Fernando, Kianoush Emami, Shenglong Yu, Her-

bert H.C. Iu, and Kit Po Wong, “A Novel Quasi-Decentralized

Functional Observer Approach to LFC of Interconnected

Power Systems,” IEEE Transactions on Power Systems, DOI:

10.1109/TPWRS.2015.2478968, September 2015.

and the following abstract:

This chapter presents a novel functional observer based quasi-

decentralized load frequency control scheme for power systems.

Based on functional observers theory, quasi-decentralized functional

observers are designed to implement any given state feedback con-

troller. The designed functional observers are decoupled from each

other and have a simpler structure in comparison to the state ob-

server based schemes. The proposed functional observer scheme is

applied to a complex nonlinear power system and the proposed de-

sign method is based on the entire network topology.

3.1 Introduction

With growing complexity of interconnected power systems, load frequency con-

trol (LFC) problem has renewed interest in recent years [1–4]. LFC is a scheme that

keeps frequency of a power network within acceptable limits by balancing power

production and consumption regardless of load variations. Furthermore, it has a ca-

pability of bringing any deviations of total power exchange amongst interconnected

areas (i.e., tie-line power deviations) back to zero. In general, LFC is implemented

on selected generation units, and each of those controllers generate a control signal

to a prime-mover to match the supply and demand of power when the loads fluctu-

ate. Tie-line power deviations are also brought to zero through an integral control

action of the area control error (ACE). A number of solutions and schemes have

3.1. Introduction 37

been proposed and developed in the last decade. For instance, linear control based

LFC schemes are reported in [5] and [6], for LFC considering time delays in power

systems see [1, 4] and [7], soft computing based LFC schemes which have attracted

considerable attention are reported in [8–12], a sliding mode technique is presented

in [3] and LFC based on a model reduction technique is presented in [2]. For other

methods of LFC, see [13–15], and in [16] a survey of various control schemes can be

found.

Previous studies on LFC have been based on approximating all generators in

a given area into one single generation unit. Furthermore, the power distribution

network including transmission lines and various busbars are all lumped into one

single entity in the analysis and design of controllers. With the growing complexity

of power distribution network, such simplified assumptions which loose the network

topology may not be representative of a complex power network. The analysis pre-

sented in this chapter makes no such assumptions and presents a quasi-decentralized

functional observer scheme to control frequency and tie-line power of a multi-area

interconnected power system. In [17], two-area linear systems connected with a

single tie-line model was considered and the application of quasi-decentralized func-

tional observer for the generation of control signals was demonstrated. Based on

the preliminary work in [17], this chapter further develops the functional observer

(FO) approach for LFC of highly interconnected power networks. In the process

of LFC signal generation only the estimate of the control signal is required, and it

is more logical to estimate the desired signal directly using a functional observer

than estimating all the individual states and then linearly combining those indi-

vidual estimates of the states to construct the control signal. Here we consider a

quasi-decentralized functional observer to generate the control signal. Functional ob-

server estimation algorithm makes use of PMU measurements of voltage and current

magnitudes and phase angles and also tie-line power measurements. The proposed

functional observer based controllers have simpler structures and their performance

on par with that of the full order observers. Furthermore, the functional observabil-

ity requirement is less stringent than the state observability requirement, see [18]

and [19]. The analysis and design considers the entire network topology.

The rest of chapter is organized as follows, in Section 3.2 the functional observer

algorithm is presented. Section 3.3 focuses on power system dynamic modeling and

functional observer based LFC. A case study of a complex power system including a

numerical study is given in Section 3.4 with the relevant functional observer design

followed by a conclusion in Section 3.5.

3.2. Functional Observers 38

3.2 Functional Observers

Consider a linear time-invariant system described by,

x(t)=Ax(t) +Bu(t), (3.1a)

y(t)=Cx(t), (3.1b)

z(t)=Lx(t), (3.1c)

where x(t) ∈ Rn, u(t) ∈ Rm and y(t) ∈ Rp are the state, input and the output

vectors, respectively. z(t) ∈ Rr is a vector to be estimated. A ∈ Rn×n, B ∈ Rn×m,

C ∈ Rp×n and L ∈ Rr×n are known constant matrices. A functional observer is a

dynamical system that can track z(t) asymptotically and has the following structure,

w(t)=Nw(t) + Jy(t) +Hu(t), (3.2a)

z(t)=Gw(t) + Ey(t). (3.2b)

The order of the observer is q ≤ n − p, and the observer parameters N ∈ Rq×q,

J ∈ Rq×p, H ∈ Rq×m and E ∈ Rr×p are such that z(t) → z(t) as t → ∞ for any

w(0) and any u(t). Without loss of generality, it is assumed that

[C

L

]is full row

rank, i.e., rank

[C

L

]= (p + r) which implies that the linear functions z(t) to be

estimated are linearly independent from the output y(t).

Remark 3.1. If the output of a system is not only a linear combination of the states

as in (3.1b) but also a linear combination of the input as given below,

ym(t) = Cx(t) +Du(t), (3.3)

then y(t) can be taken as,

y(t) = ym(t)−Du(t) = Cx(t), (3.4)

which is in the same form as in (3.1b). Since both ym(t) and u(t) are measured, the

value of y(t) is known for all t.

Definition 3.2. The triple (A,C, L) is functional observable if and only if z(t) is

a function of observable states only. Let

F(X) =

X

XA...

XAn−1

. (3.5)


Lemma 3.3. The triple (A,C, L) is functional observable if and only if

rank(F (C)

)= rank

(F

([C

L

])), (3.6)

where rank(·) is the rank of (·).

For other ways to compute functional observability, see [19]. In the subsequent

discussion we assume that the triple (A,C, L) is functional observable. Without loss

of generality we can assume that the pair (A,C) is also observable, if not, then those

unobservable states can be removed in the system description (3.1a)-(3.1c) to still

have a triple (A,C, L) which is functional observable, and also a pair (A,C) which

is observable. Clearly, for any full rank random matrix Φ of appropriate dimensions,

every output (i.e., every row) in Φy(t) is a function of all the states that constitute

entire y(t). It now follows from Lemma 3.3 that the minimum number of outputs

k from either y(t) or Φy(t) required to make a functional observer triple with the

system matrix A and the linear functional matrix L, is given by,

k = mini∈1,...,p

i, subject to :

rank(F(ΦiC

) )= rank

(F

([ΦiC

L

])), (3.7)

where Φi ∈ Ri×p is any full row rank random matrix. For any full rank Φ ∈ Rp×p, we

denote ν1, . . . , νp+r as the observability indices of the pair

(A,

[ΦC

L

]), see [20]

and [21] for how to compute observability indices. The (p+ r) observability indices

correspond to the (p+ r) rows of the matrix

[ΦC

L

]with νi corresponding to row i

of

[ΦC

L

]for i ∈ 1, . . . , p+ r. Let S be a similarity transformation matrix that

transforms the system

(A,B,

[ΦC

L

])into observable canonical form. We now

define A, B, C and L as follows,

x(t) = Sx(t), Φy(t) = Cx(t), z(t) = Lx(t), (3.8)(A, B, C, L

)=(S−1AS,S−1B,ΦCS, LS

). (3.9)


where A ∈ Rn×n can be decomposed as follows,

A =

A1 Γ1,2 . . . Γ1,p+r

Γ2,1

. . .. . .

...

.... . .

. . . Γp+r−1,p+r

Γp+r,1 . . . Γp+r,p+r−1 Ap+r

, (3.10)

and

Ai ∈ Rνi×νj , i ∈ 1, ..., p+ r . (3.11)

Now let

[C

L

]=

q1 0 . . . . . . . . . 0

0. . .

. . ....

.... . . qp

. . ....

.... . . qp+1

. . ....

.... . .

. . . 0

0 . . . . . . . . . 0 qp+r

, (3.12)

where qi ∈ R1×νi is

qi =[

0 . . . . . . 0 1], (3.13)

for i ∈ 1, . . . , p + r. Based on (3.10)-(3.13) the system(A, B, C, L

)can be

described by (p+ r) subsystems with the first p subsystems as follows,

˙xi(t) = Aixi(t) + Γi

[y(t)

z(t)

]+ Biu(t), (3.14)

where

yi(t) = ΘiΦy(t) = qixi(t), i ∈ 1, . . . , p, (3.15)

and Γi and Θi are some known matrices of appropriate dimensions. The last r

subsystems can be written as,

˙xi(t)=Aixi(t) + Γi

[y(t)

z(t)

]+ Biu(t),

zi=qixi(t), i ∈ p+ 1, . . . , p+ r, (3.16)

and also z(t) can be written as,

z(t) = Ξy(t) + Λz(t), (3.17)


where Ξ and Λ are some known matrices and z(t) =[zp+1, . . . , zp+r

]T. The r

subsystems in (3.16) only need to be combined with k subsystems in (3.14) in order

to estimate z(t) according to (3.17) since functional observability can be achieved

with a minimum of k outputs. Now consider a set of numbers W consisting of

numbers 1 to p. The elements in W are reordered so that the first k elements

correspond to subsystem numbers in (3.14) that we want to retain, and the last

(p− k) elements correspond to subsystem numbers in (3.14) that we do not want to

retain. Let

W = w1, . . . , wp . (3.18)

Now consider the similarity transformation matrix P ,

P =

Q1,1 . . . Q1,p 0 . . . 0...

. . ....

.... . .

...

Qp,1 . . . Qp,p 0 . . . 0

0 . . . 0 Iνp+1 . . . 0...

. . ....

.... . .

...

0 . . . 0 0 . . . Iνp+r

, (3.19)

where for i ≤ p, j ≤ p

Qi,j =

Iνi if i = wj

0 if i 6= wj

, (3.20)

and Iνi ∈ Rνi×νi is the identity matrix, 0 is a zero matrix of appropriate dimensions.

Using the similarity transformation P , we define A,B,C,L as follows,

x(t) = Px(t), Φy(t) = Cx(t), z(t) = Lx(t),(A, B, C, L

)=(P−1AP ,P−1B, CP , LP

)(3.21)

=((SP)−1A(SP), (SP)−1B,ΦCSP , LSP

).

There exists an invertible Υ satisfying the following conditions,

Υ =

[C

L

][C

L

]T [ C

L

][C

L

]T−1

, (3.22)

[C

L

]= Υ

[C

L

]. (3.23)


Υ−1 can be decomposed as,

Υ−1 =

Υ11 Υ12 0

Υ21 Υ22 0

Υ31 Υ32 Υ33

, (3.24)

where Υ11 ∈ Rk×k, Υ12 ∈ Rk×(p−k), Υ21 ∈ R(p−k)×k, Υ22 ∈ R(p−k)×(p−k), Υ31 ∈ Rr×k,

Υ32 ∈ Rr×(p−k), Υ33 ∈ Rr×r, and 0 is a zero matrix of appropriate dimensions. Now

define Ω ∈ Rp×p, Ω1 ∈ Rk×p and Ω2 ∈ R(p−k)×p as follows,

Ω =

[Ω1

Ω2

]= CCT

(CCT

)−1

=

[Υ11 Υ12

Υ21 Υ22

]−1

, (3.25)

and

C = ΩC. (3.26)

Let

y(t) = Cx(t) = ΩCx(t) = ΩΦy(t), (3.27)

and

z(t) = Lx(t). (3.28)

Obviously,

Υ

[y(t)

z(t)

]=

[y(t)

z(t)

]. (3.29)

z(t) =[Υ31 Υ32

]y(t) + Υ33 z(t). (3.30)

Also consider the similarity transformation matrix T as follows,

T =

Iα1 0 0

0 0 Iα2

0 Iα3 0

, (3.31)

where

α1=ν1 + . . .+ νk, (3.32a)

α2=

νk+1 + . . .+ νp if k < p

0 if k = p, (3.32b)

α3=νp+1 + . . .+ νp+r. (3.32c)

We can now define A, B, C and L as follows,(A, B, C, L

)=(T −1AT , T −1B, CT , LT

), (3.33)


and then partition A and B as follows,

(A, B

)=

([A11 A12

A21 A22

],

[B1

B2

]), (3.34)

where A11 ∈ R(α1+α3)×(α1+α3), A12 ∈ R(α1+α3)×α2 , A21 ∈ Rα2×(α1+α3), A22 ∈ Rα2×α2 ,

B1 ∈ R(α1+α3)×m and B2 ∈ Rα2×m.

From (3.12), let C1 ∈ Rk×(α1+α3), L1 ∈ Rr×(α1+α3) and C2 ∈ R(p−k)×α2 as follows,

[C1

L1

]=

q1 0 . . . . . . . . . 0

0. . .

. . ....

.... . . qk

. . ....

.... . . qp+1

. . ....

.... . .

. . . 0

0 . . . . . . . . . 0 qp+r

, (3.35)

C2=

qk+1 0 . . . 0

0. . .

. . ....

.... . .

. . . 0

0 . . . 0 qp

. (3.36)

With C1 given in (3.35), let us consider a similarity transformation matrix M as

follows,

M=[C+

1 N (C1)], (3.37)

where C+1 is the pseudo inverse of C1 and N (·) is the nullspace of (·). Now let A, B

and C as follows, (A, B, C

)=(M−1A11M,M−1Ψ, C1M

). (3.38)

where

Ψ =[B1 A12C

T2 Ω2Φ

]. (3.39)

We now partition A and B as follows,

(A, B,M

)=

([A11 A12

A21 A22

],

[B1

B2

],[M1 M2

]),

(3.40)

3.3. Power System Dynamics and Functional Observer Based Controller Design 44

where

A11 ∈ Rk×k, A12 ∈ Rk×(α1+α3−k), A21 ∈ R(α1+α3−k)×k, A22 ∈ R(α1+α3−k)×(α1+α3−k),

B1 ∈ Rk×(m+p), B2 ∈ Rα1+α3−k×(m+p),M1 ∈ R(α1+α3)×k andM2 ∈ R(α1+α3)×(α1+α3−k).

The functional observer parameters N , J , H, G and E can be obtained as follows,

F = Υ33L1M−1, (3.41a)

N = A22 − ZA12, (3.41b)[H J

]= B2 − ZB1, (3.41c)

G = FM2, (3.41d)

J = (NZ + A21 − ZA11)Ω1Φ + J , (3.41e)

E =(Υ31Ω1 + Υ32Ω2 + FM1Ω1 + FM2ZΩ1

)Φ, (3.41f)

H ∈ R(α1+α3−k)×m, J ∈ R(α1+α3−k)×p where matrix Z is obtained by any pole-

placement method so that N according to (3.41b) has eigenvalues at desired loca-

tions. The order of the functional observer q ≤ n− p is,

q =

wk∑i=w1

νi +

p+r∑i=p+1

νi − k. (3.42)

In the subsequent sections we will show how to utilize functional observers to im-

plement control signals to bring frequency deviations in a power system to within

acceptable limits.

3.3 Power System Dynamics and Functional Observer Based

Controller Design

Let us consider a power system consisting of M generators, M PMUs placed at

the terminals of each generator, B busbars and L loads as shown in Figure 3.1. The

equivalent circuit of generator l that connects to busbar l is shown in Figure 3.2.

The voltage of busbar l at time t is denoted vl(t)∠θl(t), and the current that flows

through generator l into busbar l at time t is denoted by il(t)∠γl(t). The current

il(t)∠γl(t), idl(t) and iql(t) for l ∈ 1, . . . ,M are related according to,

il(t)∠γl(t) =(idl(t) + jiql(t)

)ej(δl(t)−π/2), (3.43)

where δl(t) is the rotor angle, idl(t) is the d-axis current and iql(t) is q-axis current at

time t of generator l. In Figure 3.2, X ′dl and X ′

ql, X′′dl and X ′′

ql are d-axis and q-axis

transient and sub-transient reactances of generator l in time t, respectively. Also

in Figure 3.2, Xlsl is armature leakage reactance, Rsl is stator winding resistance,


Algorithm 3.1 Functional observer design algorithm.Step 1: Perform functional observability test

Test functional observability of triple (A,C, L) according to lemma 3.3 (or see [21]

for other ways to test functional observability), if (A,C, L) is functional observable

then continue, otherwise stop.

Step 2: Determine the value of k

Determine the value of k according to (3.7).

Step 3: Determine observability indices Find any full rank random matrix

Φ ∈ Rp×p. Now determine observability indices of the pair

(A,

[ΦC

L

]), see [20]

and [21] for how to determine observability indices.

Step 4: Find the observable canonical form

Find the observable canonical form of the pair

(A,

[ΦC

L

]), i.e., find similarity

transformation matrix S, see [20] for how to find observable canonical form. Now

find(A, B, C, L

)according to (3.9).

Step 5: Find matrices C, L, W, PFind C, L according to (3.12) and W and P according to (3.18) and (3.19) respec-

tively.

Step 6: Find similarity transformed systems

Find the similarity transformed system(A, B, C, L


Step 7: Find matrices Υ, Υ31, Υ32, Υ33, Ω

Find Υ and Ω according to (3.22) and (3.25) respectively. Also find Υ31, Υ32, Υ33

according to (3.24).

Step 8: Find similarity transformed systems

Find similarity transformation matrix T according to (3.31) and find the similarity

transformed system(A, B, C, L


Step 9: Find C1, L1, C2, MFind C1, L1, according to (3.35), C2 according to (3.36) and M according to (3.37).

Step 10: Find reduced system

Find the reduced system according to (3.38).

Step 11: Find functional observer parameters

Find functional observer parameters according to (3.41).


E ′dl(t) is the transient emf due to the flux in q-axis damper coil, E

′

ql(t) is the transient

emf due to field flux linkages; Ψ1d and Ψ2q are sub-transient emfs due to d and

q-axis damper coils of generator l at time t in p.u., respectively. Assuming Rsl = 0

and applying Kirchhoff’s voltage law in the circuit shown in Figure 3.2, the d-axis

current idl(t) and q-axis current iql(t) at time t for generator l, l ∈ 1, . . . ,M takes

the following form,

idl(t)=Λl

(vl(t), θl(t), δl(t), E

′

ql(t),Ψ1dl(t)), (3.44)

iql(t)=Φl

(vl(t), θl(t), δl(t), E

′

dl(t),Ψ2ql(t)), (3.45)

where Λl(·) and Φl(·) are some known nonlinear functions. The current il(t)∠γl(t)

and voltage vl(t)∠θl(t) satisfy the following power balance equations at each busbar

l ∈ 1, . . . , B in the power network,

PLl(t) + jQLl(t) + vl(t)ejθl(t)il(t)e

−jγl(t) −B∑

r=1

vl(t)vr(t)Ylrej(θl(t)−θr(t)−αlr) = 0,

(3.46)

where Ylr∠αlr is the admittance of the line connecting buses l and r, PLl(t) and

QLl(t) are active power and reactive power consumed by the loads connected to

busbar l at time t. Obviously, at a generator busbar we have il(t)e−jγl(t) 6= 0,

and at a load busbar we have il(t)e−jγl(t) = 0 in (3.46). The dynamics of the

transmission network is much faster than the dynamics of rotating machines, so

the voltages vl(t)∠θl(t) and currents il(t)∠γl(t) on each busbar on the network can

change instantaneously, and those voltages and currents can be regarded as inputs

or outputs of the generators connected to it. Let us now denote xl(t), l ∈ 1, . . . ,Mto be the states of the generator connected to busbar l at time t, and its dynamics

can be written as,

xl(t) = fl(xl(t), vl(t), θl(t), Vrefl, Pmlop , Pcl(t)

), (3.47)

where Vrefl is the reference voltage, Pmlop is the mechanical output power of the

generator turbine at the operating point, Pcl(t) is the governor control signal and

fl(·) are some known nonlinear functions, all of which are applicable to generator

l, l ∈ 1, . . . ,M. The PMU measurements vl(t)∠θl(t) can be considered as the

pseudo inputs in (3.47), thus each generator’s dynamics can be decoupled from the

other generators and the power network. The PMU current measurement il(t)∠γl(t)

can be considered as the following output function,[il(t)

γl(t)

]= hl

(xl(t), vl(t), θl(t)

), (3.48)


!"#$%

!"#$L

!"#$%!&!'()

*'+(),

-+!./+'!)012!%

31'(%1/)4+5'"/

31'(%1/)4+5'"/

6$'7(+1'"/)

89#!%:!%)9"#!;)

31'(%1//!%),

<%!"),

<%!")0

-+!./+'!)012!%

012!%)=!(21%>)2+(?)))))@

$#9"%#

-+!./+'!#

6$'7(+1'"/)

89#!%:!%)9"#!;)

31'(%1//!%)M

1G

MG

!"#$%!&!'()

*'+(),

Figure 3.1: Functional observer based control scheme.

Figure 3.2: Sub-transient equivalent circuit of the synchronous generator l.

3.4. Case Study 48

where hl(·) are some known nonlinear functions that can be derived from (3.43)-(3.45).

The small-signal model of (3.47)-(3.48) for lth generator can be written in the form

of (3.1a) and (3.3). The governor control signal zl(t) which can be designed using

well known techniques such as pole placement, optimal control, etc., can be approx-

imated using the functional observer output Pcl(t) = zl(t) according to (3.2a)-(3.2b)

where the vectors ul(t) and yl(t) are the measurements from the nonlinear system

and can be obtained from (3.46)-(3.48) as follows,

ul(t)=

θl(t)− θlop

vl(t)− vlop

Pcl(t)

, yl(t)=[ il(t)− ilop

γl(t)− γlop

]−Dlul(t), (3.49)

where ”op” is the operating point. To maintain tie-line power deviations to zero the

integral of the area control error in the control signal Pcl(t) [17] can be incorporated

as,

Pcl(t)=Glwl(t) + Elyl(t) +Ktiel

∫∆Ptie(total)(t)dt, (3.50)

where ∆Ptie(total)(t) is the total measurement of all the tie-line powers in the area.

Since the control signal requires only the local busbar current and voltage measure-

ments from PMUs and the power measurements at the tie-lines, the control signal

Pcl(t) is quasi-decentralized. For voltage and current measurements, PMUs are only

required on generator basbars where LFC controllers are implemented.

3.4 Case Study

Here we consider the IEEE 39-bus, 10-generator test system shown in Figure

3.3. In each area, one generator has a type I, IEEE ST1A AVR with PSS, excitation

system and all other generators are equipped with excitation systems of type II,

IEEE DC1A AVR without PSS. Speed-governing system is categorized into two

main types: (1) mechanical-hydraulic and (2) electro-hydraulic with/without steam

feedback. Hydro and steam turbines are also considered in the generation units of

this case study. The steam turbines used in the case study are tandem-compound,

double or single reheat. All generation units, i.e., l ∈ 1, . . . , 10 have the following

3.4. Case Study 49

! "

#$

$

!%

#

#

#%

!#

"

%

#

& &

'

()*+,

!'

!

#-

%.#"

% -

%/

%-

%'

!$

#'

.

/

!/

#.%%

%

%&

#&

!.

#/

!-

%#

##

!&

%"

$

Figure 3.3: IEEE 39-bus, 10-generator (New England) test system.

dynamics [22],

E ′ql(t)=

1

T ′dol

(− E ′

ql(t)− (Xdl −X ′dl)

(idl(t)− E ′

ql(t)−X ′

dl −X ′′dl

(X ′dl −Xlsl)2

(Ψ1dl(t)

+(X ′dl −Xlsl)idl(t)

))+ Efdl(t)

),

Ψ1dl(t)=1

T ′′dol

(−Ψ1dl(t) + E ′

ql(t)− (X ′dl −Xlsl)idl(t)

),

E ′dl(t)=

1

T ′qol

(− E ′

dl(t) + (Xql −X ′ql)

(iql(t) + E ′

dl(t)−X ′

ql −X ′′ql

(X ′ql −Xlsl)2

(Ψ2ql(t)

+(X ′ql −Xlsl)iql(t)

))),

Ψ2ql(t)=1

T ′′qol

(−Ψ2ql(t)− E ′

dl(t)− (X ′ql −Xlsl)iql(t)

),

δl(t)=ωbl

(ωl(t)− ωsl

),

3.4. Case Study 50

ωl(t)=ωsl

2Hl

(Pml(t)−

(X ′′

ql −X ′′dl

)idl(t)iql(t)−Dlωl(t) +Dlωsl −

X ′′dl −Xlsl

X ′dl −Xlsl

E ′ql(t)iql(t)

−X ′dl −X ′′

dl

X ′dl −Xlsl

Ψ1dl(t)iql(t)−X ′′

ql −Xlsl

X ′ql −Xlsl

E ′dl(t)idl(t) +

X ′ql −X ′′

ql

X ′ql −Xlsl

Ψ2ql(t)idl(t)

).

(3.51)

The dynamics of the IEEE DC1A AVR without PSS applicable to generation units

1, 2, 4, 5, 6, 9 and 10, i.e., l ∈ 1, 2, 4, 5, 6, 9, 10 are given below [22],

Efdl(t)=1

TEl

(−(KEl + Axle

BxlEfdl(t))Efdl(t) + VRl(t)

),

Rfl(t) =1

TFl

(−Rfl(t) +

KFl

TFl

Efdl(t)

),

VRl(t) =1

TAl

(−KAlKFl

TFl

Efdl(t)− VRl(t) +KAlRfl(t) +KAl

(Vrefl − vl(t)

)).(3.52)

The dynamics of the IEEE ST1A AVR with PSS applicable to generation units 3,

7 and 8 are given below [22],

Efdl(t)=1

TRl

(KAl

(Vrefl +

KplT1plT3pl

T2plT4pl

(ωl(t)− ωsl

)+ y1pl(t) + y2pl(t) + y3pl(t)

−vl(t))− Efdl(t)

),

y1pl(t)=1

Twl

(T ′pl

(ωl(t)− ωsl

)− y1pl(t)

),

y2pl(t)=1

T2pl

(T ′′pl

(ωl(t)− ωsl

)− y2pl(t)

),

y3pl(t)=1

T4pl

(T ′′′pl

(ωl(t)− ωsl

)− y3pl(t)

), (3.53)

where

T ′pl =

−KplT2wl+KplTwlT1pl+KplTwlT3pl−KplT1plT3pl

(Twl−T2pl)(Twl−T4pl),

T ′′pl =

−KplTwlT1plT2pl+KplTwlT1plT3pl+KplTwlT22pl−KplTwlT2plT3pl

T2pl(Twl−T2pl)(T2pl−T4pl),

T ′′′pl =

KplTwlT1plT3pl−KplTwlT1plT4pl−KplTwlT3plT4pl+KplTwlT24pl

T4pl(Twl−T4pl)(T4pl−T2pl).

The dynamics of the mechanical-hydraulic speed governing system and hydro-turbine

including Generator Rate Constraint (GRC) is considered in generation unit 1, i.e.,

l = 1, is given below [23],

P1gl(t)=1

T1gl

(K1gl(ωl(t)− ωsl)− P1gl(t)

),

P2gl(t)=1

T2gl

(K2gl(ωl(t)− ωsl)− P2gl(t)

),

P1T l(t)=1

T2T l

(satmax

min

(Pcl(t)− P1gl(t)− P2gl(t)

)− P1T l(t)

), (3.54)

3.4. Case Study 51

where satαβ(·) is the saturation function, α and β are the upper and lower saturation

levels of (·) respectively, P1gl(t) and P2gl(t) are speed governing system dynamic

states and P1T l(t) is a state variable of the hydro-turbine. The mechanical power

Pml in (3.51) can be obtained from the following equation,

Pml(t)=Pmlop +K1T l

(T2T l − T1T l

)P1T l(t)

+K1T lT1T l

(satmax

min

(Pcl(t)− P1gl(t)− P2gl(t)

)), (3.55)

where Pmlop is the mechanical power of the generator l at the operating point.

The dynamics of Westinghouse electro-hydraulic speed governing system with steam

feedback including Governor Dead-Band (GDB) and GRC nonlinearities applicable

to generation units 3, 5, 7, 8, 9 and 10, i.e., l ∈ 3, 5, 7, 8, 9, 10 are given below [23],

P1gl(t)=1

T1gl

(K1gl

(1− T2gl

T1gl

)(ωl(t)− ωsl)− P1gl(t)

),

Pdgl(t)=1

Tdgl

(N1dglTdgl −N2dgl

Tdgl

(Pcl(t)− Pdgl(t)

)−K1gl

T2gl

T1gl

(ωl(t)− ωs

)),

P2gl(t)=satupdown

(1

T3gl

(N2dgl

Tdgl

(Pcl(t)−K1gl

T2gl

T1gl

(ωl(t)− ωsl)− P1gl(t))

+Pdgl(t)− satmaxmin

(P2gl(t)

))), (3.56)

The dynamics of General Electric and Westinghouse electro-hydraulic governors

including GDB and GRC nonlinearities without steam feedback used in generation

units 2, 4 and 6, i.e., l ∈ 2, 4, 6 are given below, [23],

Pdgl(t)=1

Tdgl

(N1dglTdgl −N2dgl

Tdgl

(Pcl(t)− Pdgl(t)

)−K1gl

(ωl(t)− ωs

)),

P2gl(t)=satupdown

(1

T1gl

(N2dgl

Tdgl

(Pcl(t)−K1gl(ωl(t)− ωsl)

)+ Pdgl(t)

−satmaxmin

(P2gl(t)

))). (3.57)

The dynamics of tandem-compound double and single reheat steam turbines appli-

cable to generation unit 2-10, i.e., l ∈ 2, . . . , 10 are given below [23],

P1T l(t)=1

T1T l

(satmax

min

(P2gl(t)

)− P1T l(t)

),

P2T l(t)=1

T2T l

(P1T l(t)− P2T l(t)

),

P3T l(t)=1

T3T l


),

P4T l(t)=1

T4T l


). (3.58)

3.4. Case Study 52

Mechanical output power of the steam turbines can be obtained as follows,

Pml(t)=Pmlop +(K1T lP1T l(t) +K2T lP2T l(t) +K3T lP3T l(t) +K4T lP4T l(t)

). (3.59)

All parameters of the generators, exciters, PSS, governors, GDB, GRC and turbines

are shown in table 3.1. A block diagram representation of the steam speed governing

system is shown in Figure 3.4. Assume stator winding resistance Rsl of all the

K1g(1+sT2g)1+sT1g

N1dgl+sN2dgl

1+sTdgl

1T3gl

1s

∆ωl(t) −

Pcl(t)

+ +

Pup

Pdown

Pmax

Pmin

P2gl(t)

−

Figure 3.4: Steam speed-governing system with GDB and GRC.

generators is 0. By applying Kirchhoff’s voltage law to the circuit in Figure 3.2,

terminal currents of the lth generator according to (3.44) and (3.45) can be written

as,

idl(t)=1

X ′′dl

(X ′′

dl −Xlsl

X ′dl −Xlsl

E ′ql(t)− vl(t)cos

(δl(t)− θl(t)

)+

X ′d −X ′′

dl

X ′dl −Xlsl

Ψ1dl(t)

),

iql(t)=1

X ′′ql

(−

X ′′ql −Xlsl

X ′ql −Xlsl

E ′dl(t) + vl(t)sin

(δl(t)− θl(t)

)+

X ′q −X ′′

ql

X ′ql −Xlsl

Ψ2ql(t)

).

(3.60)

By substituting (3.60) into relevant generator dynamic equations (3.51)-(3.59), the

dynamics of all generation units can be rewritten in the form of (3.47). From (3.43),

it follows for l ∈ 1, 2, . . . , 10,[il(t)

γl(t)

]=

√i2dl(t) + i2ql(t)

tan−1(

−idl (t)

iql (t)

)+ δl(t)

. (3.61)

Using (3.60) in (3.61) we can obtain the dynamic output equation of the form (3.48).

Implementation: The initial conditions for the states of all generators are found by

performing a load flow calculation considering the active and reactive power data for

all the buses in the IEEE 39-bus, 10-generator test system data given in Matpower

toolbox [24]. The implemented system considers GDB and GRC nonlinearities in

the governor system as shown in Figure 3.4. Only one generator in each area has

3.4. Case Study 53

been assumed to have a functional observer based controller. These generation

units are G1 (12th order), G3 (16th order) and G6 (13th order). We consider two

cases in the implementation, Case 1 when active and reactive power of the loads on

busbar numbers 1, 12 and 24 are increased and Case 2 when the loads on busbar

numbers 7, 21 and 26 are decreased. Table 3.2 lists the change of the loads in both

cases. Here we show only the functional observer design for the 1st generator, the

same procedure can be utilized for generators G3 and G6. State feedback control

laws which are linear combinations of generator states can be designed to maintain

system frequency at the desired level and also maintain tie-line power deviations at

zero. Here we implement two state feedback controllers: (i) a pole-placement based

controller and (ii) a PI based controller. Both of those state feedback controllers were

implemented in two ways, first using a functional observer (FO) based approach and

then using a state observer (SO) based approach. Controllers were tuned to obtain a

satisfactory overshoot/undershoot in the frequency deviation and zero steady-state

error in tie-line power deviations. In both cases the respective observers, FO and

SO, estimate the state feedback control law. The small-signal model of G1 can be

obtained as follows,[A B

C D

]=

−1.72 0 0 −0.17 0.92 0 0 0 0 0 0 0 0.12 −0.92 0

0 −24.8 10 0 −2.19 0 0 0 0 0 0 0 14.65 2.19 0

0 0.06 −0.34 0 −0.03 0 0.1 0 0 0 0 0 0.18 0.03 0

−14.29 0 0 −32 −18.41 0 0 0 0 0 0 0 −2.49 18.41 0

0 0 0 0 0 377 0 0 0 0 0 0 0 0 0

0.4 −0.02−0.05−0.1 −0.54−0.05 0 0 0 0.24 0.24 0.36−0.03 0.54 −0.24

0 0 0 0 0 0 −1.85 0 1.27 0 0 0 0 0 0

0 0 0 0 0 0 0.51 −2.86 0 0 0 0 0 0 0

0 0 0 0 0 0 −36 200 −5 0 0 0 −200 0 0

0 0 0 0 0 0.45 0 0 0 −0.026 0 0 0 0 0

0 0 0 0 0 4.54 0 0 0 0 −1.92 0 0 0 0

0 0 0 0 0 0 0 0 0 −1 −1 −1 0 0 1

−24.35 8.51 17.73 5.84 35.26 0 0 0 0 0 0 0 −21.73−35.26 0

−7.46−3.45−7.19 1.79 9.04 0 0 0 0 0 0 0 11.84 −8.04 0

The proportional and integral constants for the traditional PI control law Pc(t) =

Lx(t) +Ktie

∫∆Ptie(total)(t)dt, are defined according to the following L matrix,

L =[0 0 0 0 −0.0158 −1 0 0 0 0 0 0

],

3.4. Case Study 54

Table 3.1: Generator, governor, turbine, exciter and PSS parameters.

Generator Parameters

l Hl Dl Xdl X ′dl X ′′

dl Xql X ′ql X ′′

ql Xlsl T ′dol T ′′

dol T ′qol T ′′

qol

1 42 4 0.1 0.031 0.025 0.069 0.028 0.025 0.0125 10.2 0.05 1.5 0.035

2 30.3 9.8 0.295 0.070 0.005 0.282 0.170 0.05 0.035 6.56 0.05 1.5 0.035

3 35.8 10 0.250 0.053 0.045 0.237 0.088 0.045 0.0304 5.7 0.05 1.5 0.035

4 28.6 10 0.262 0.044 0.035 0.258 0.166 0.035 0.0295 5.69 0.05 1.5 0.035

5 26 3 0.67 0.132 0.05 0.62 0.166 0.05 0.054 5.4 0.05 0.44 0.035

6 34.8 14 0.254 0.05 0.04 0.241 0.081 0.04 0.0224 7.3 0.05 0.4 0.035

7 26.4 8 0.295 0.049 0.04 0.292 0.186 0.04 0.0322 5.66 0.05 1.5 0.035

8 24.3 9 0.29 0.057 0.057 0.280 0.091 0.045 0.028 6.7 0.05 0.41 0.035

9 34.5 14 0.211 0.057 0.057 0.205 0.059 0.045 0.0298 4.79 0.05 1.96 0.035

10 248 33 0.296 0.006 0.006 0.029 0.005 0.004 0.003 5.9 0.05 1.5 0.035

Governor and Turbine Parameters

l K1gl K2gl T1gl T2gl T3gl K1T l K2T l K3T l K4T l T1T l T2T l T3T l T4T l

1 17.44 2.56 38.5 0.52 - 10 - - - -2 1 - -

2 20 - 0.1 - - 0.22 0.22 0.3 - 0.3 7 0.4 -

3 20 - 2.8 1 0.15 0.3 0.4 0.3 0.26 0.3 7 5 0.4

4 20 - 0.02 - - 0.22 0.22 0.3 - 0.3 7 0.4 -

5 20 - 3.8 2 0.15 0.3 0.4 0.3 0.26 0.3 7 5 0.4

6 20 - 0.1 - - 0.22 0.22 0.3 - 0.3 7 0.4 -

7 20 - 2.8 1.5 0.1 0.3 0.4 0.3 0.26 0.3 7 5 0.4

8 20 - 1.8 1 0.15 0.22 0.22 0.3 - 0.3 7 0.4 -

9 20 - 2.8 1 0.1 0.3 0.4 0.3 0.26 0.3 7 5 0.4

10 20 - 5.8 2.5 0.4 0.22 0.22 0.3 - 0.3 7 0.4 -

GDB and GRC Parameters

l Tgdl N1gdl N2gdl Pup Pdown Pmax Pmin

1 - - - - - 2Pop -Pop

2,3,4,5,6,7,8,9,10 0.1 0.8 -0.2 0.1 -0.1 2Pop -Pop

IEEE DC1A AVR without PSS Parameters

l KAl TAl KEl TEl KFl TFl Axl Bxl

1,2,4,5,6,9,10 40 0.2 1 0.785 0.063 0.35 0.07 0.91

IEEE ST1A AVR with PSS Parameters

l KAl Kpl T1pl Tp2l T3pl T4pl Twl TRl

3 20 0.032 0.1 0.2 0.1 0.25 10 0.01

7,8 20 10 0.1 0.2 0.1 0.25 40 0.1

3.4. Case Study 55

Table 3.2: New value of the loads at specified busbars in Case 1 and Case 2.

Case 1 Case 2PPPPPPPPPPPPower

Bus No. 1 12 24 7 21 26

Active (MW) 290.34 208.52 486.42 34.55 157.11 25.46

Reactive (MVAR) 40.91 -87.86 185.57 -184.27 -107.91 -15.34

and the pole-placement controller is defined according to the following L matrix,

L =[0 0 0 0 −0.0158 −1 0 0 0 0.25 0.125 −0.5

].

In the following we show how to design the functional observer based pole-placement

controller according to Algorithm 3.1,

Step 1: For the triple (A,C, L) from (3.6) we have,

rank(F (C)

)= rank

(F

([C

L

]))= 12.

Hence according to Lemma 3.3 the triple (A,C, L) is functional observable.

Step 2: According to (3.7) we have,

k = 1.

Step 3: The full rank random matrix Φ ∈ R2×2 is chosen as,

Φ =

[0.8535 0.4859

0.4687 0.5158

].

The numeric values of observability indices of the pair

(A,

[ΦC

L

])can be found

as,

ν1 = 5, ν2 = 4, ν3 = 3.

Step 4: Observable canonical form according to (3.9) is,A 103B

10−3C

L

=

3.4. Case Study 56

0 0 0 0 -992.5 0 0 0 -366.52 0 0 -0.44 -1.32 5.1 0.02

1 0 0 0 -1809.27 0 0 0 -142.64 0 0 -5.13 -0.24 4.12 -13.91

0 1 0 0 -3607.9 0 0 0 -522.39 0 0 -7.8 -0.22 10.58 -4.33

0 0 1 0 -379.64 0 0 0 11.19 0 0 0 0.02 0.4 -0.15

0 0 0 1 -34.85 0 0 0 0 0 0 0 0 0.01 0

0 0 0 0 -13537.21 0 0 0 -6847.74 0 0 -4.78 -4.59 87.15 7.33

0 0 0 0 -22660.19 1 0 0 -10929.53 0 0 20.08 -7.16 142.02 35.33

0 0 0 0 -304.36 0 1 0 5.15 0 0 40.08 -0.09 1.88 -0.85

0 0 0 0 0 0 0 1 -23.97 0 0 0 -0.01 0.04 0

0 0 0 0 3678.31 0 0 0 1860.25 0 0 1.27 1.25 -23.68 -2.05

0 0 0 0 6548.59 0 0 0 3152.87 1 0 -6.23 2.06 -41.01 -11.98

0 0 0 0 413.84 0 0 0 145.94 0 1 -12.75 0.12 -2.54 -1.27

0 0 0 0 20834.87 0 0 0 0 0 0 0

0 0 0 0 8235.68 0 0 0 796.64 0 0 0

0 0 0 0 0 0 0 0 -600.97 0 0 205.61

.

Step 5: According to (3.12), (3.18) and (3.19), matrices C, L, W and P can be

calculated as follows, C

L

= 0 0 0 1 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 0 0 0 1

,

W=2, 1,

P=

0 Iv1 0

Iv2 0 0

0 0 Iv3

=

0 I5 0

I4 0 0

0 0 I3

.

Step 6: Similarity transformed matrices as in (3.21) can be computed as,A 103B

10−3C

L

=

3.4. Case Study 57

0 0 0 -6847.74 0 0 0 0 -13537.21 0 0 -4.78 -4.59 87.15 7.33

1 0 0 -10929.53 0 0 0 0 -22660.19 0 0 20.08 -7.16 142.02 35.33

0 1 0 5.15 0 0 0 0 -304.36 0 0 40.08 -0.09 1.88 -0.85

0 0 1 -23.97 0 0 0 0 0 0 0 0 -0.01 0.04 0

0 0 0 -366.52 0 0 0 0 -992.5 0 0 -0.44 -1.32 5.1 0.02

0 0 0 -142.64 1 0 0 0 -1809.27 0 0 -5.13 -0.24 4.12 -13.91

0 0 0 -522.39 0 1 0 0 -3607.9 0 0 -7.8 -0.22 10.58 -4.33

0 0 0 11.19 0 0 1 0 -379.64 0 0 0 0.02 0.4 -0.15

0 0 0 0 0 0 0 1 -34.85 0 0 0 0 0.01 0

0 0 0 1860.25 0 0 0 0 3678.31 0 0 1.27 1.25 -23.68 -2.05

0 0 0 3152.87 0 0 0 0 6548.59 1 0 -6.23 2.06 -41.01 -11.98

0 0 0 145.94 0 0 0 0 413.84 0 1 -12.75 0.12 -2.54 -1.27

0 0 0 0 0 0 0 0 20834.87 0 0 0

0 0 0 796.64 0 0 0 0 8235.68 0 0 0

0 0 0 -600.97 0 0 0 0 0 0 0 205.61

.

Step 7: According to (3.22), (3.24) and (3.25), matrices Υ, Υ31, Υ32, Υ33 and Ω

are obtained as follows,

Υ=

−4.96× 10−7 1.26× 10−6 0

4.8× 10−8 0 0

−1.45× 10−6 3.67× 10−6 4.86× 10−3

,

Υ31=−600.97, Υ32 = 0, Υ33 = 205.61,

Ω=

[−0.496× 10−6 1.255× 10−6

0.048× 10−6 0

].

Step 8: Similarity transformation matrix as in (3.31) and similarity transformed

matrices as in (3.33) can be computed as,

T =

I4 0 0

0 0 I5

0 I3 0

,

A 103B

10−3C

L

=

3.4. Case Study 58

0 0 0 -6847.74 0 0 -4.78 0 0 0 0 -13537.21 -4.59 87.15 7.33

1 0 0 -10929.53 0 0 20.08 0 0 0 0 -22660.19 -7.16 142.02 35.33

0 1 0 5.15 0 0 40.08 0 0 0 0 -304.36 -0.09 1.88 -0.85

0 0 1 -23.97 0 0 0 0 0 0 0 0 -0.01 0.04 0

0 0 0 1860.25 0 0 1.27 0 0 0 0 3678.31 1.25 -23.68 -2.05

0 0 0 3152.87 1 0 -6.23 0 0 0 0 6548.59 2.06 -41.01 -11.98

0 0 0 145.94 0 1 -12.75 0 0 0 0 413.84 0.12 -2.54 -1.27

0 0 0 -366.52 0 0 -0.44 0 0 0 0 -992.5 -1.32 5.1 0.02

0 0 0 -142.64 0 0 -5.13 1 0 0 0 -1809.27 -0.24 4.12 -13.91

0 0 0 -522.39 0 0 -7.8 0 1 0 0 -3607.9 -0.22 10.58 -4.33

0 0 0 11.19 0 0 0 0 0 1 0 -379.64 0.02 0.4 -0.15

0 0 0 0 0 0 0 0 0 0 1 -34.85 0 0.01 0

0 0 0 0 0 0 0 0 0 0 0 20834.87

0 0 0 796.64 0 0 0 0 0 0 0 8235.68

0 0 0 -600.97 0 0 205.61 0 0 0 0 0

.

Step 9: According to (3.35), (3.36) and (3.37), matrices C1, L1, C2 and M can be

calculated as follows, C1

L1

=[ 0 0 0 1 0 0 0

0 0 0 0 0 0 1

],

C2=[0 0 0 0 1

],

M=

0 0 0 −1 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

1 0 0 0 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

.

Step 10: According to (3.38), reduced order system can be obtained as below, A 103B

C

=

3.4. Case Study 59

-23.97 0 1 0 0 0 0 -0.01 0.04 0 0 0

-10929.53 0 0 -1 0 0 20.08 -7.16 142.02 35.33 -0.93 -0.53

5.15 1 0 0 0 0 40.08 -0.09 1.88 -0.85 -0.01 -0.01

6847.74 0 0 0 0 0 4.78 4.59 -87.15 -7.33 0.55 0.32

1860.25 0 0 0 0 0 1.27 1.25 -23.68 -2.05 0.15 0.09

3152.87 0 0 0 1 0 -6.23 2.06 -41.01 -11.98 0.27 0.15

145.94 0 0 0 0 1 -12.75 0.12 -2.54 -1.27 0.02 0.01

1 0 0 0 0 0 0

.

Step 11: Governor control signal Pc1(t) can be obtained using approximation

of z1(t) in (3.1c) by choosing Ktie1 = 0.05 in (3.50) and a functional observer

(3.2a)-(3.2b) of order 6 as follows,[N J 103H

G E

]=

0−413331.49−100 20.08 −18.1−44.61 2096.15 −15397.82 35.33

1 −289.89 0 00 40.08 −0.01 −0.03 1.38 −9.01 −0.85

0 350721.85 0 00 4.78 15.37 37.88 −1780.12 13098.78 −7.33

0 102345.48 0 00 1.27 4.49 11.06 −519.56 3824.16 −2.05

0 129766.05 0 10 −6.23 5.68 14.01 −658.27 4837.75 −11.98

0 10027.77 0 01−12.75 0.44 1.08 −50.91 374.47 −1.27

0 0 0 00 205.61 −0.34 −0.84

.

For generation unit 3, the proportional and integral constants for the traditional PI

control law Pc(t) = Lx(t)+Ktie

∫∆Ptie(total)(t)dt, and the pole-placement controllers

are defined according to the following L matrices,

L=[0 0 0 0 −0.04 −10 0 0 0 0 0 0 0 0 0 0

],

and

L=

[0 0 0 0 −0.04 −10 0 0 0 0 0 0 0.1 0.3 0.1 −0.7

],

respectively. The small-signal model of G3 can be obtained as,[A B

C D

]=

3.4. Case Study 60

−2.11 0 0 −0.88 0.38 0 0 0 0 0 0 0 0 0 0 0 0.41 −0.38 0

0 −23.6 13.51 0 −7.21 0 0 0 0 0 0 0 0 0 0 0 6.94 7.21 0

0 0.37 −1.03 0 −0.35 0 0.18 0 0 0 0 0 0 0 0 0 0.34 0.35 0

−19.3 0 0 −55.62 −24.58 0 0 0 0 0 0 0 0 0 0 0 −26.37 24.58 0

0 0 0 0 0 376.99 0 0 0 0 0 0 0 0 0 0 0 0 0

0.05 −0.08 −0.14 −0.16 −0.33 −0.14 0 0 0 0 0 0 0 0.01 0 0 −0.09 0.33 0

0 0 0 0 0 12.73 −100 2000 2000 2000 0 0 0 0 0 0 −2000 0 0

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 −0.16 0 0 −5 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0.24 0 0 0 −4 0 0 0 0 0 0 0 0 0

0 0 0 0 0 4.59 0 0 0 0 −0.36 0 0 0 0 0 0 0 0

0 0 0 0 0 −47.62 0 0 0 0 −6.67 −6.67 0 0 0 0 0 0 6.67

0 0 0 0 0 0 0 0 0 0 0 3.33 −3.33 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0.14 −0.14 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0.2 −0.2 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.5 −2.5 0 0 0

−2.47 7.14 12.87 7.2 20.84 0 0 0 0 0 0 0 0 0 0 0 −6.74 −20.84 0

−0.74 −0.5 −0.9 2.15 1.96 0 0 0 0 0 0 0 0 0 0 0 3.05 −0.96 0

.

According to Algorithm 3.1, the numeric values of observability indices, and func-

tional observer parameters of 3rd generation unit are computed as follows,

ν1 = 6, ν2 = 5, ν3 = 5,

and [N 103J 103H

G E

]=

000−1821838.58−10000 −136.47 10.15 8.54 26.25 −39.830.02

100 −463697.04 0 0000 −361.68 2.61 2.21 6.62 −9.96 0.05

010 −35686.26 0 0000 −161.63 0.2 0.18 0.52 −0.79 0

001 −61.91 0 0000 −15.87 0 0 0 0 0

000 2225400.78 0 0000 26.73 −12.12−10.06−31.59 46.25 0

000 −205996.7 0 0000 −2.75 1.12 0.93 2.93 −4.28 0

000 −301676.04 0 1000 −15.29 1.67 1.39 4.35 −6.5 0

000 −152056.18 0 0100 −43.13 0.85 0.72 2.23 −3.43 0

000 −31226.77 0 0010 −37.52 0.18 0.15 0.46 −0.71 0

000 −2167.05 0 0001 −11.43 0.01 0.01 0.03 −0.05 0

000 0 0 0000670139.06 0.11 0.09

,

For generation unit 6, the proportional and integral constants for the traditional PI

control law Pc(t) = Lx(t)+Ktie

∫∆Ptie(total)(t)dt, and the pole-placement controllers

are defined according to the following L matrices,

L=[0 0 0 0 −0.05 −10 0 0 0 0 0 0 0

],

3.4. Case Study 61

and

L=

[0 0 0 0 −0.05 −10 0 0 0 0.5 −0.8 0 −0.2

],

respectively. The small-signal model of G6 can be computed as,[A B

C D

]=

−8.13 0 0 −2.66 2.24 0 0 0 0 0 0 0 0 2.08 −2.24 0

0 −25 11.2 0 −10.1 0 0 0 0 0 0 0 0 9.89 10.1 0

0 0.21 −0.79 0 −0.33 0 0.14 0 0 0 0 0 0 0.32 0.33 0

−16 0 0 −58.14 −31.69 0 0 0 0 0 0 0 0 −29.4 31.69 0

0 0 0 0 0 376.99 0 0 0 0 0 0 0 0 0 0

0.08 −0.1 −0.17 −0.19 −0.43 −0.2 0 0 0 0 0 0 0 −0.09 0.43 0

0 0 0 0 0 0 −3.29 0 1.27 0 0 0 0 0 0 0

0 0 0 0 0 0 0.51 −2.86 0 0 0 0 0 0 0 0

0 0 0 0 0 0 −36 200 −5 0 0 0 0 −200 0 0

0 0 0 0 0 −200 0 0 0 −10 0 0 0 0 0 10

0 0 0 0 0 0 0 0 0 3.33 −3.33 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0.14 −0.14 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 2.5 −2.5 0 0 0

−3.48 8.01 14.1 8.18 24.96 0 0 0 0 0 0 0 0 −7.71 −24.96 0

−1.01 −0.65 −1.14 2.38 2.24 0 0 0 0 0 0 0 0 3.65 −1.24 0

.

The numeric values of observability indices, and functional observer parameters for

6th generation unit can be computed according to Algorithm 3.1 as below,

ν1 = 5, ν2 = 4, ν3 = 4,

and [N 103J 103H

G E

]=

0−1350.49−1000 76.81 −0.4 −3.53 18.54 −16.73 0.15

1 −64.87 0 000 2.59 0 0 0.74 −0.72 0.01

0 5140.14 0 000 −316.64 1.81 17.62 −71.27 66.14 0.26

0 24646.33 0 000 −1650.3 7.55 88.19 −352.89 369.06 0

0 18818.52 0 100−1346.84 5.98 68.62 −268.86 278.13 −1.09

0 5431.75 0 010 −432.38 1.78 20.49 −77.54 80.1 0.07

0 364.82 0 001 −40.43 0.12 1.36 −5.24 5.52 0.19

0 0 0 000 26089.79 −0.12−1.49

.

As for generator G1, following the functional observer design Algorithm 3.1, we can

design functional observers of order 10 and 7 to implement the control signals for

generators G3 and G6 respectively. The implementation results are presented in

3.4. Case Study 62

Figures 3.5 and 3.6 where the frequency of the 1st generator and tie-line powers

of the three areas are shown. Five control scenarios were implemented for Case 1

and Case 2: (1) droop control only, i.e., Pc(t) = 0, (2) a functional observer based

feedback controllers on G1, G3 and G6 to implement both PI and pole-placement

controllers, (3) a state observer based feedback controllers on G1, G3 and G6 to

implement both PI and pole-placement controllers. Functional observer of order 6,

10 and 7 achieved for the generation units 1, 3 and 6, respectively are in comparison

to 12th, 16th and 13th order required for full order state observers, and 10th, 14th and

11th order required for reduced Luenberger observers. Comparison results of tie-line

power deviation and frequency response for both FO and SO based controllers shown

in Figures 3.5 and 3.6 demonstrate that the performance of FO based controllers are

comparable to SO based controllers even though FO based controllers have much

simpler structure in comparison to SO based controllers. The simpler structures pos-

sible with FO based controllers are consistent for both types of feedback controllers

(i.e., traditional PI controllers and pole-placement based controllers) implemented.

As it can be seen, by utilizing functional observer based controller, total tie-line

power of each area goes to zero in a timely manner and frequency can come back to

its operating point quickly. The performance of the full order state observer (SO) and

functional observer is comparable, yet decoupled functional observer based method

having much simpler structures. To test the sensitivity of controllers to generator

parameter variations, at the time of fault, all generator parameters for generators

G1, G3 and G6 were changed by 10% from its nominal values. Figures 3.7 and 3.8

shows the difference in frequency for G1 and tie-line power when parameters devi-

ated 10% from its nominal values. The controllers are insensitive to those parameter

deviations as the responses reach zero in a short duration as evident from Figures 3.7

and 3.8. Dynamic states of the nonlinear model and the small-signal model of the

1st, 3rd and 6th generation units in Case 1 are compared in Figures 3.9-3.11 when

FO based pole-placement controllers are used in the generation units.

3.4. Case Study 63

0 25 50 75 100 125 1500.984

0.988

0.992

0.996

1

1.004

ω1(t) FO based feedback control SO based feedback control

FO based PI control SO based PI control

Droop control only

0 25 50 75 100 125 150−2.5

−1.25

0

1.25

Ptie(M

W)(A

rea1)

∆Ptie(total) ∆Ptie(39→1)

∆Ptie(4→3) ∆Ptie(14→15)

0 25 50 75 100 125 150−0.6

−0.3

0

0.3

0.6

1

Ptie(M

W)(A

rea2)


∆Ptie(16→17)

0 25 50 75 100 125 150−1.5

−0.7

0

0.8

1.6

2.5

Ptie(M

W)(A

rea3)


∆Ptie(3→4) ∆Ptie(17→16)

Figure 3.5: Comparison of FO and SO based methods in terms of frequency response,

and tie-line power deviations in Case 1.

3.4. Case Study 64

0 25 50 75 100 125 1500.996

1

1.004

1.008

1.012ω1(t)

0 25 50 75 100 125 150−2

−1.4

−0.7

0

0.7

1.5

Ptie(M

W)(A

rea1)


∆Ptie(4→3) ∆Ptie(14→15)

0 25 50 75 100 125 150−0.5

0

0.5

1

Ptie(M

W)(A

rea2)


∆Ptie(16→17)

0 25 50 75 100 125 150−1.5

−0.7

0

0.7

1.5

Time (s)

Ptie(M

W)(A

rea3)


∆Ptie(3→4) ∆Ptie(17→16)

Figure 3.6: Comparison of FO and SO based methods in terms of frequency response,

and tie-line power deviations in Case 2.

3.4. Case Study 65

0 20 40 60 80 100−1.2

−0.8

−0.4

0

0.4

0.8×10−3

Difference

inω1(t)

0 20 40 60 80 100−0.6

−0.4

−0.2

0

0.2

0.4

Time (s)

Differen

cein

∆Ptie(total)

Area 1

Area 2

Area 3

Figure 3.7: Difference in Generator 1 frequency response and tie-line power devi-

ations of all areas in Case 1 due to generator parameter variations of 10% from

nominal values.

0 20 40 60 80 100−4

−3

−2

−1

0

1×10−4

Difference

inω1(t)

0 20 40 60 80 100−0.3

−0.15

0

0.15

0.3

Time (s)

Difference

in∆Ptie(total)

Area 1

Area 2

Area 3

Figure 3.8: Difference in Generator 1 frequency response and tie-line power devi-

ations of all areas in Case 2 due to generator parameter variations of 10% from

nominal values.

3.4. Case Study 66

0 25 50 750.06

0.11

0.15

0.2E

′ d1(t)

Nonlinear state

Linearized state

0 25 50 751.04

1.05

1.06

1.07

Ψ1d1(t)

Nonlinear state

Linearized state

0 25 50 751.08

1.1

1.13

1.15

E′ q1(t)

Nonlinear state

Linearized state

0 25 50 75−0.28

−0.22

−0.16

−0.1

Time (s)

Ψ2q1(t)

Nonlinear state

Linearized state

Figure 3.9: Comparison between states of the nonlinear model and states of the

small-signal model of the 1st generator in Case 1.

3.4. Case Study 67

0 25 50 75−70

−50

−30

−10

10δ 1(t)

Nonlinear state

Linearized state

0 25 50 750.986

0.99

0.994

0.998

1.002

ω1(t)

Nonlinear state

Linearized state

0 25 50 751.2

1.3

1.4

1.5

Efd1(t)

Nonlinear state

Linearized state

0 25 50 750.22

0.23

0.24

0.25

0.26

0.27

Time (s)

Rf1(t)

Nonlinear state

Linearized state


small-signal model of the 1st generator in Case 1 (continued).

3.4. Case Study 68

0 25 50 751.45

1.55

1.65

1.75

1.85

1.95VR1(t)

Nonlinear state

Linearized state

0 25 50 75−6

−4

−2

1×10−2

P1g1(t)

Nonlinear state

Linearized state

0 25 50 75−3

−1

×10−2

P2g1(t)

Nonlinear state

Linearized state

0 25 50 750

0.09

0.18

0.27

0.35

Time (s)

P1T1(t)

Nonlinear state

Linearized state


small-signal model of the 1st generator in Case 1 (continued).

3.4. Case Study 69

0 25 50 750.44

0.48

0.52

0.56

0.6E

′ d3(t)

Nonlinear state

Linearized state

0 25 50 750.65

0.73

0.81

0.9

Ψ1d3(t)

Nonlinear state

Linearized state

0 25 50 750.9

0.95

1

1.04

E′ q3(t)

Nonlinear state

Linearized state

0 25 50 75−0.85

−0.79

−0.73

−0.67

−0.6

Time (s)

Ψ2q3(t)

Nonlinear state

Linearized state


small-signal model of the 3rd generator in Case 1.

3.4. Case Study 70

0 25 50 75−70

−50

−30

−10

10δ 3(t)

Nonlinear state

Linearized state

0 25 50 750.986

0.99

0.994

0.998

1.002

ω3(t)

Nonlinear state

Linearized state

0 25 50 752.2

2.4

2.6

2.8

Efd3(t)

Nonlinear state

Linearized state

0 25 50 75−0.5

0.5

1.5

2.5×10−4

Time (s)

P1s3(t)

Nonlinear state

Linearized state


small-signal model of the 3rd generator in Case 1 (continued).

3.4. Case Study 71

0 25 50 75−0.5

1.2

2.9

4.5×10−4

P2s3(t)

Nonlinear state

Linearized state

0 25 50 75−8

−5

−2

1×10−4

P3s3(t)

Nonlinear state

Linearized state

0 25 50 75−0.14

−0.1

−0.06

−0.02

0.02

P1g3(t)

Nonlinear state

Linearized state

0 25 50 75−0.5

0.5

1.5

2.5

3.5

Time (s)

P2g3(t)

Nonlinear state

Linearized state



3.4. Case Study 72

0 25 50 75−0.5

0.5

1.5

2.5

3.5

P1T3(t)

Nonlinear state

Linearized state

0 25 50 75−0.5

0.4

1.3

2.2

3

P2T3(t)

Nonlinear state

Linearized state

0 25 50 75−0.5

0.4

1.3

2.2

3

P3T3(t)

Nonlinear state

Linearized state

0 25 50 75−0.5

0.4

1.3

2.2

3

Time (s)

P4T3(t)

Nonlinear state

Linearized state



3.4. Case Study 73

0 25 50 750.44

0.48

0.52

0.56

0.6E

′ d6(t)

Nonlinear state

Linearized state

0 25 50 750.7

0.76

0.82

0.88

0.95

Ψ1d6(t)

Nonlinear state

Linearized state

0 25 50 750.94

0.985

1.03

1.075

1.12

E′ q6(t)

Nonlinear state

Linearized state

0 25 50 75−0.82

−0.77

−0.72

−0.67

−0.62

Time (s)

Ψ2q6(t)

Nonlinear state

Linearized state


small-signal model of the 6th generator in Case 1.

3.4. Case Study 74

0 25 50 75−70

−50

−30

−10

10δ 6(t)

Nonlinear state

Linearized state

0 25 50 750.986

0.99

0.994

0.998

1.002

ω6(t)

Nonlinear state

Linearized state

0 25 50 752.2

2.29

2.38

2.47

2.56

2.65

Efd6(t)

Nonlinear state

Linearized state

0 25 50 750.39

0.42

0.45

0.48

Time (s)

Rf6(t)

Nonlinear state

Linearized state


small-signal model of the 6th generator in Case 1 (continued).

3.4. Case Study 75

0 25 50 753.2

3.5

3.8

4.1

4.4

4.6VR6(t)

Nonlinear state

Linearized state

0 25 50 75−0.5

0.5

1.5

2.5

3.5

Time (s)

P1g6(t)

Nonlinear state

Linearized state

0 25 50 75−0.5

0.5

1.5

2.5

3.5

P1T6(t)

Nonlinear state

Linearized state

0 25 50 75−0.5

0.5

1.5

2.5

3.5

P2T6(t)

Nonlinear state

Linearized state

0 25 50 75−0.5

0.5

1.5

2.5

3.5

Time (s)

P3T6(t)

Nonlinear state

Linearized state


small-signal model of the 6th generator in Case 1 (continued).

3.5. Conclusion 76

3.5 Conclusion

In this chapter we have presented a quasi-decentralized functional observer ap-

proach to load frequency control. The designed functional observers are decoupled

from each other and have a simpler structure in comparison to the state observer

based schemes. The proposed design method is based on preserving the entire net-

work topology, and the control system is analyzed on the 39-bus 10-generator IEEE

test system. The simulation results show that the proposed functional observer

based scheme which produces simpler decoupled structures can effectively control

the frequency and tie-line power deviations.

References 77

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80CHAPTER 4

Power System DSE and LFC Using Unscented

Transform

This chapter is largely based on an article with the following details:

Kianoush Emami, Tyrone Fernando, Herbert H.C. Iu,

Brett Nener, and Kit Po Wong, “Application of Unscented Trans-

form in Frequency Control of a Complex Power System Using Noisy

PMU Data,” IEEE Transactions on Industrial Informatics, DOI:

10.1109/TII.2015.2491222, September 2015.


This chapter presents a novel unscented transform based quasi-

decentralized load frequency control scheme for power systems. The

designed load frequency controllers are decoupled from each other

and can cope with noisy and discrete PMU data. The proposed un-

scented transform based scheme is applied to a complex nonlinear

power system. Furthermore, the design and analysis of the proposed

controllers are based on considering the entire network topology.

4.1 Introduction

The availability of phasor measurement units (PMUs) in power networks has pre-

sented an opportunity to develop algorithms to obtain estimates of dynamic states of

all generating units, and to develop complementary algorithms for control functions

of the power grid. Complete real time knowledge of the state of a power grid pro-

vides a complete and a reliable database, based on which, control functions can be

reliably deployed to control frequency deviation due to sudden load variations. Due

to significant recent investment in power system infrastructure, PMUs are widely

available in power distribution networks [1–5]. PMUs can provide synchronized mea-

surements of the state of a power system at a rate of up to 120 samples per second.

Processing these PMU measurements through innovative monitoring algorithms can

provide an accurate picture of the state of the power system including the dynamic


states of generators, which can be utilized in control algorithms to device methods

to improve reliable power distribution.

The frequency of the power network is tightly linked to the supply and demand

of active power. To maintain the frequency constant at a desired level the supply

and the demand of active power has to be matched. Introduction of distributed gen-

erations in the power network has increased the intermittency of power supply. For

instance the output of renewable energy generators are linked to weather variations:

wind turbine output fluctuates with wind speed or hydro power generation capacity

is also linked to rain fall. The intermittency in supply and demand of active power

presents a challenge in load frequency control (LFC). LFC has renewed interest in

recent years with greater penetration of renewable sources and increased complexity

of the power system. LFC is a scheme that keeps frequency of a power network

within acceptable limits by balancing power production and consumption regardless

of load variations [6]. Furthermore, it has a capability of bringing any deviations of

total power exchange amongst interconnected areas (i.e., tie-line power deviations)

back to zero. Generally, LFC is achieved by a primary and a secondary control

mechanism. The initial readjustment of the frequency is handled by the primary

control mechanism and the secondary control mechanism takes over the fine adjust-

ment of the frequency by resetting the frequency error to zero through an integral

action. Tie-line power deviations are also brought to zero through an integral con-

trol action of the area control error (ACE). A number of solutions and schemes have

been proposed and developed for LFC. A large number of approaches are based on

tunning the gain of a fixed parameter PI controller [7,8]. Fuzzy logic based schemes

and other soft computing based LFC schemes have attracted considerable attention

and are reported in [9–13]. For other methods of LFC see [14–16] and [17] for a

survey of various control schemes.

Previous studies on LFC have not considered the availability of PMUs on the

power network. Furthermore, those studies are also based on approximating all gen-

erators and transmission lines into one simplified linearized block which may not

be valid for highly interconnected multi-node power system with many generation

units in a single area. With the availability of PMUs such simplifying assumptions

is no longer necessary, the analysis and design of LFC can be carried out consid-

ering the entire network topology. The analysis presented in this chapter makes

no such assumptions and present a quasi-decentralized unscented transform (UT)

based scheme to control frequency and tie-line power of a multi-area interconnected

power system. The proposed method takes into account the noise in the PMU mea-

4.2. Power System Dynamic and Problem Statement 82

surements in estimating the dynamic states of the generators. The rest of chapter is

organized as follows, in Section 4.2, power system dynamic is presented. Section 4.3

focuses on unscented transform based filtering algorithm. Load frequency control

and a case study of a complex power system including a numerical study is given in

Section 4.4 with the relevant UT based control algorithm followed by a conclusion

in Section 4.5.

4.2 Power System Dynamic and Problem Statement

Let us consider a power system consisting of N areas, M generators, M PMUs

placed at the terminals of each generator, B busbars and L loads as shown in Figure

4.1. Furthermore, it is assumed that dynamics of the transmission network is much

faster than the dynamics of rotating machines, so the voltages Vl(k)∠θl(k) and

currents Il(k)∠γl(k), l ∈ 1, 2, · · · , B on each busbar on the network can change

instantaneously. Thus, these variable of the network can be regarded as algebraic

variables. A sub-transient equivalent circuit that connects generator i to busbar i is

shown in Figure 4.2 in which E ′qi(k) is the transient emf due to field flux linkages in

p.u., Ψ1di(k) and Ψ2qi(k) are subtransient emfs due to flux in d−q axes damper coils

in p.u., E ′di(k) is the transient emf due to flux in q-axis damper coil in p.u. and δi(k)

is the rotor angle in radian. These variables constitute part of the ith generator state

vector xi(k). Parameters of the ith generator which are shown in Figure 4.2 are d−q

axes synchronous reactances in p.u. (i.e., Xdi, Xqi), d− q axes transient reactances

in p.u. (i.e., X ′di, X

′qi), d − q axes subtransient reactances in p.u. (i.e., X ′′

di, X′′qi),

armature leakage reactance in p.u. (i.e., Xlsi), and armature resistor Rsi. As can be

seen in Figure 4.2, the dynamic of generator i is coupled with the algebraic variables,

Vi(k)∠θi(k) and Ii(k)∠γi(k). These variables are also coupled with other voltages

and currents in the network. The dynamic of the ith generator can be represented as

a difference algebraic equation (DAE), refer to equations (4.25)-(4.37) for a complete

description. Assuming armature resistor Rsi = 0 and applying Kirchhoff’s voltage

law in the circuit shown in Figure 4.2, d−q axes current of the ith generator terminal,

Idi(k) and Iqi(k) can take the following form,

Idi(k)=Λi

(xi(k), Vi(k), θi(k)

), (4.1a)

Iqi(k)=Φi

(xi(k), Vi(k), θi(k)

), (4.1b)


N Area Power Network withM Generators, B Busbars and L Loads

N

M B L

PMU

`

UT Based Controller 1

Con

trol S

igna

l

Tie-line Pow

er

c1P k

1G

1

1I

k°

k

1

1V

kµ

k

1

1Ik°k

1

1V

kµ

k

Satellite Synchronisation

PMU

PMUM

`

UT Based Controller M

Con

trol S

igna

l

Tie-line Pow

er

cMP k

MG

M

MI

k°

k

M

MV

kµ

k

M

M

I

k

°

k

M

M

V

k

µ

k

PMU

Figure 4.1: Typical power system with PMUs.

where Λi(·) and Φi(·) are some known nonlinear functions, and the magnitude and

phase angle of the ith generator can be written as,[Ii(k)

γi(k)

]=

√I2di(k) + I2qi(k)

tan−1(

−Idi(k)Iqi(k)

)+ δi(k)

. (4.2)

The generator dynamic given in (4.27)-(4.37) can be written in the following compact

form,

xi(k + 1)=f(xi(k), zi(k), ui(k)

), (4.3a)

yi(k)=h(xi(k), zi(k), ui(k)

), (4.3b)

where f(·), h(·) are some known nonlinear functions, xi(k) ∈ Rni×1 is the state

vector, input signal ui(k) = Pci(k) is the governor control signal at time k, yi(k)

and zi(k) are algebraic variables such that zi(k) =[Vi(k) θi(k)

]Tand yi(k) =[

Ii(k) γi(k)]T

. Furthermore, the algebraic variables including the current Il(k)∠γl(k)

and voltage Vl(k)∠θl(k), l ∈ 1, 2, · · · , B at each busbar satisfy the following power


Figure 4.2: Sub-transient equivalent circuit of the synchronous generator i.

balance equations in the power network,

PLl(k) + jQLl(k) + vl(k)ejθl(k)il(k)e

−jγl(k) −B∑

r=1

vl(k)vr(k)Ylrej(θl(k)−θr(k)−αlr) = 0,

(4.4)

where Ylr and αlr are magnitude and phase angle of the admittance between busbar

l and r. PLl(k) and QLl(k) are also the total active and reactive power of the

loads connected to busbar l, respectively. Obviously, at a generator busbar we have

Ii(k)e−jγi(k) 6= 0, and at a load busbar we have Ii(k)e

−jγi(k) = 0 in (4.4). It is

clear that to calculate value of the dynamic states in (4.3a) at time k + 1, (4.3a)

and (4.4) should be solved simultaneously. However, PMUs placed at the different

selected busbars can provide the measurements of Vl(k)∠θl(k) and Il(k)∠γl(k) at

every time instant k providing a snapshot of the status of the power network. If

PMU measurement of the algebraic variable zi(k) =[Vi(k) θi(k)

]Tis regarded

a pseudo input as in (4.3a), then the dynamic of each generator can be decoupled

from the rest of the network equations. To provide more explanation, let us define,

Vi(k)=Vi(k) + η1i(k),

θi(k)=θi(k) + η2i(k), (4.5)

as the measurement of Vi(k) and θi(k) by the ith PMU and η1i(k) and η2i(k) are the

noise presented in the measurement. η1i(k) and η2i(k) are assumed to be normally

distributed with 0 mean and variances σ2η1i

and σ2η2i, respectively, i.e.,

ηi(k) =

[η1i(k)

η2i(k)

]∼N

([0]2×1

, Qi

), (4.6)

4.3. State Estimation Based on The Unscented Transform 85

where

Qi =

[σ2η1i

0

0 σ2η2i

]. (4.7)

The measured output can be written as,

yi(k) =

[Ii(k)

γi(k)

]=

[Ii(k) + ς1i(k)

γi(k) + ς2i(k)

]=

[Ii(k)

γi(k)

]+ ςi(k), (4.8)

where ς1i(k) and ς2i(k) have normal distributions with 0 mean and variances, σ2ς1i

and σ2ς2i, respectively, i.e.,

ςi(k) =

[ς1i(k)

ς2i(k)

]∼N

([0]2×1

, Ri

), (4.9)

where

Ri =

[σ2ς1i

0

0 σ2ς2i

]. (4.10)

Considering equations (4.5) and (4.8), (4.3a) and (4.3b) can be rewritten in the

following form,

xi(k + 1)=f(xi(k), ui(k), Vi(k), θi(k), ηi(k)

), (4.11a)

yi(k)=h(xi(k), ui(k), Vi(k), θi(k), ηi(k)

)+ ςi(k), (4.11b)

where f(·) and h(·) are some known nonlinear functions. As evident from (4.11),

the algebraic variable vector zi(k) has been replaced with the measured PMU data.

Consequently, the solution of the algebraic equation (4.4) is not necessary to deter-

mine the generator state vector at time k + 1 according to (4.11a). In the following

sections we show how the unscented transform can be used to estimate the state

of generators according to (4.11) and use it for frequency control in the presence of

load fluctuations.

4.3 State Estimation Based on The Unscented Transform

The unscented transformation (UT) is a method to estimate statistics of a ran-

dom variable subjected to a given nonlinear transformation [18, 19]. Let us assume

that υ is a τ dimensional random variable distributed normally with mean υ and

covariance Pυυ. If υ undergoes a nonlinear transformation,

ζ = Υ(υ), (4.12)

4.3. State Estimation Based on The Unscented Transform 86

then UT can provide the estimation of the mean ζ and covariance Pζζ of ζ. A set

of 2τ +1 points called sigma points (χ) with mean υ and covariance Pυυ are chosen

to estimate the mean (ζ) and covariance (Pζζ) of the transformed points using the

following equations,

χ0 = υ,

χr = υ +(√

(τ + λ)Pυυ

)r; r ∈ 1, 2, · · · , τ,

χr+τ = υ −(√

(τ + λ)Pυυ

)r; r ∈ 1, 2, · · · , τ, , (4.13)

where(√

(τ + λ)Pυυ

)ris the rth row or column of the matrix square root of (τ +

λ)Pυυ. Also λ = α2(τ + κ) − τ is a scaling parameter where α is a factor which

specifies the spread of the sigma points, and κ = 0 is the second scaling parameter.

Furthermore, mean and covariance of ζ are approximated based on the following

corresponding weights,

W 0m =

λ

(λ+ τ), (4.14a)

W 0c =

λ

(λ+ τ)+(1− α2 + β

), (4.14b)

W rm = W r

c =1

2(λ+ τ); r ∈ 1, 2, · · · , 2τ, (4.14c)

where β is a factor to incorporate prior knowledge of the distribution of υ, e.g.,

β = 2 for normal distributions. The mean and covariance of the random variable ζ

in (4.12) can be calculated using the following equations,

ζr = Υ(χr); r ∈ 0, 1, · · · , 2τ, (4.15)

ζ =2τ∑r=0

W rmζ

r, (4.16)

Pζζ =2τ∑r=0

W rc (ζ

r − ζ)(ζr − ζ)T . (4.17)

Now consider the decoupled dynamic of the ith generation unit (4.11a) and (4.11b).

As we discussed before, it is assumed that PMUs measurement noise has a normal

distribution with zero mean. If we assume that covariances of the measurement

noise in pseudo inputs are constant then the state vector xi(k) and measurement

noise ηi(k) can be considered as a new augmented state vector, Xi(k), as follows,

Xi(k)=

[xi(k)

ηi(k)

], (4.18)

4.4. Load Frequency Control Case Study 87

where Xi(k) ∈ R(ni+2)×1 is the augmented state vector of the ith generation unit.

Mean and covariance of the augmented state vectorXi(k) can be obtained as follows,

Xi(k)=

[xi(k)[0]2×1

], (4.19)

PXiXi(k)=

[Pxixi

(k) Pxiηi(k)

Pxiηi(k) Qi

]. (4.20)

Considering (4.8) and (4.18), (4.11) can be rewritten as follows,

Xi(k + 1)=f(Xi(k), ui(k)

), (4.21a)

yi(k)=h(Xi(k), ui(k)

)+ ςi(k), (4.21b)

where ui(k) is an input vector such that,

ui(k)=

Pci(k)

Vi(k)

θi(k)

. (4.22)

Using the UT described in the previous section, state estimation of the ith generation

unit described by equation (4.21) can be implemented using algorithm 4.1. At the

end of each filtering algorithm iteration Xi(k) provides on-line estimate of the ith

generator augmented state vector Xi(k). Estimation of the state vector xi(k) can

be extracted from the augmented state vector Xi(k) according to (4.19).

4.4 Load Frequency Control Case Study

To simplify the analysis, but without loss of generality, we assume there is

only one tie line connects different areas. We also assume that the tie-line power

measurements are available by the PMUs installed at corresponding busbars. Let

us define measurement of tie-line power between area a and area b at time k as

Ptiea→b(k); a, b ∈ 1, 2, · · · , N; a 6= b and total tie-line power PMU measurements of

area a at time k as Ptota(k); a ∈ 1, 2, · · · , N. In this chapter we propose a decen-

tralized feedback control law Pci(k) = Fi∆xi(k) at time k to calculate the control

signal Pci(k) of the ith generator where ∆xi(k) is the deviation of the states from op-

erating point. Discrete on-line estimation of the nonlinear states are available from

the unscented filter algorithm presented in Section 4.3. The feedback gain F can

be designed using well known techniques such as pole placement, optimal control,

etc. Furthermore, the designed control signal Pci(k) can ensure zero steady-state

value for total tie-line power deviation by incorporating area control error (ACE).


Algorithm 4.1 Unscented transform based filtering algorithm.Step 0:

• Find fi(·) and hi(·) in (4.21), Qi in (4.7), Ri in (4.10), set κi = 0 and let

αi = 10−3, βi = 2.

• Select initial value of the state vector xi(0) and select it as xi(0).

• Augment initial value of the state vector with PMU measurement noise mean,

i.e., Xi(0) =

[xi(0)[0]2×1

].

• Initiate covariance of the augmented state vector Xi(0) as PXiXi(0) =[

Pxixi(0) 0

0 Qi

].

• Set k = 1.

Step 1: Time Update,

• Consider υ = Xi(k − 1) and Pυυ = PXiXi(k − 1) in (4.13).

• Generate 2(ni + 2) + 1 sigma points according to (4.13), i.e., χi(k − 1) =[χ0i (k − 1) · · · χ

2(ni+2)i (k − 1)

].

• Associate weights according to (4.14), i.e.,

Wmi =[W 0

mi W 1mi · · · W

2(ni+2)mi

],

Wci =[W 0

ci W 1ci · · · W

2(ni+2)ci

].

• Calculate transferred points according to (4.15), i.e., Xi(k) = f(χi(k −

1), ui(k − 1)).

• Calculate mean X−i (k) and covariance PXiXi

of the transferred points Xi(k),

according to (4.16) and (4.17), i.e., X−i (k) =

2(ni+2)∑r=0

W rmiX

ri (k),

PXiXi=2(ni+2)∑r=0

W rci

(Xr

i (k)−X−i (k)

) (Xr

i (k)−X−i (k)

)T.

Step 2: Measurement Update,

• Calculate measurement update based on the transferred sigma points Xi(k),

obtained from Step 1, i.e., Yi(k) = h(Xi(k), ui(k)

).

• Calculate mean Y −i (k) according to (4.16), i.e., Y −

i (k) =2(ni+2)∑r=0

W rciY

ri (k).


Algorithm 4.1 Unscented transform based filtering algorithm (continued).

• Calculate covariance of Yi(k), PYiYi, according to (4.17), i.e.,

PYiYi= Ri +

2(ni+2)∑r=0

W rci

(Y ri (k)− Y −

i (k)) (

Y ri (k)− Y −

i (k))T.

• Calculate cross-covariance PXiYias,

PXiYi=2(ni+2)∑r=0

W rci

(Xr

i (k)−X−i (k)

) (Y ri (k)− Y −

i (k))T

.

Step 3: Filtering,

• Calculate filter gain Ki(k) as, Ki(k) = PXiYiP−1YiYi

.

• Update state mean based on PMU measurement as,

X(k) = X−i (k) +Ki(k)

(yi(k)− Y −

i (k)).

• Calculate covariance PXiXi(k) as,

PXiXi(k) = PXiXi

−Ki(k)PYiYiKT

i (k).

Step 4:

• Reset PXiXi(k) =

[Pxixi

(k) Pxiηi(k)

Pxiηi(k) Pηiηi(k)

]to PXiXi

(k) =

[Pxixi

(k) Pxiηi(k)

Pxiηi(k) Qi

].

• Reset Xi(k) =

[xi(k)

ηi(k)

]to Xi(k) =

[xi(k)[0]2×1

].

• Increment k

• Goto Step 1.


For instance, ACE can be computed for area a by summing deviation of the total

tie-line power measurements, i.e,

ACEa=∑k

∆Ptota(k)∆t∑k

∆Ptiea→1(k)∆t+ · · ·+∑k

∆Ptiea→b(k)∆t,

(4.23)

where a 6= b. Either only one controller in each area regulates ACE back to zero, or

on the other hand if many controllers are used, then a fraction of the ACE correction

signal is incorporated into each of those controllers. In our analysis we assume that

only one generator in each area regulates the ACE. Thus the control signal of ith

generator in area a can be obtained as follows,

Pci(k) = Fi∆xi(k) +Ktiea

∑k

∆Ptota(k)∆t. (4.24)

Control signal computed in (4.24) guarantees zero steady-state in tie-line power

deviation and also the network frequency deviation. Since the control signal requires

only the local busbar current and voltage measurements from PMUs and the power

measurements at the tie-lines, the control signal Pci(k) is quasi-decentralized.

Here we consider the IEEE 39-bus, 10-generator, 3-area test system shown in

Figure 4.3. Two tie-lines interconnect Area 1 and Area 3. Area 2 is intercon-

nected to Area 1 and Area 3 with one tie-line. In each area, one generator has a

type I, IEEE ST1A AVR with PSS, excitation system and all other generators are

equipped with excitation systems of type II, IEEE DC1A AVR without PSS. Speed-

governing system is categorized into two main types: (1) mechanical-hydraulic and

(2) electro-hydraulic with/without steam feedback. Hydro and steam turbines are

also considered in the generation units of this case study. The steam turbines used

in the case study are tandem-compound, double or single reheat. Consider dynamic

equation presented by (4.21a). Dynamic state vector xi(k) of ith generator shown

in (4.18) includes, generator general dynamic state vector x1i(k), excitation system

state vector x2i(k), governor system state vector x3i(k) and turbine system state

vector x4i(k), i.e., xi(k) =

x1i(k)

x2i(k)

x3i(k)

x4i(k)

. Dynamic of generator i can be described by


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.

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Figure 4.3: IEEE 39-bus, 10-generator, 3-area test system.

the following discritized equations,x1i(k + 1)

x2i(k + 1)

x3i(k + 1)

x4i(k + 1)

=

x1i(k)

x2i(k)

x3i(k)

x4i(k)

+∆t

f1i(xi(k), ui(k)

)f2i(xi(k), ui(k)

)f3i(xi(k), ui(k)

)f4i(xi(k), ui(k)

)

, (4.25)

where ∆t is sampling time, f1i(·) = [f 11i(·) f 2

1i(·) · · · ]T, f2i(·) = [f 1

2i(·) f 22i(·) · · · ]

T,

f3i(·) = [f 13i(·) f 2

3i(·) · · · ]Tand f4i(·) = [f 1

4i(·) f 24i(·) · · · ]

Tare known nonlinear dy-

namic functions. Generator i general dynamic state vector is,

x1i(k) =[E ′

qi(k) Ψ1di(k) E′di(k) Ψ2qi(k) δi(k) ωi(k)

]T, (4.26)

where ωbi, ωsi, ωi(k) are base, synchronous and current rotor angle speed in p.u.,

respectively. Generator general dynamic for all units, i.e., i ∈ 1, . . . , 10 have the


following nonlinear functions [20],

f 11i(·)=

1

T ′doi

(− E ′

qi(k)− (Xdi −X ′di)

(Idi(k)− E ′

qi(k)−X ′

di −X ′′di

(X ′di −Xlsi)2

(Ψ1di(k)

+(X ′di −Xlsi)Idi(k)

))+ Efdi(k)

), (4.27a)

f 21i(·)=

1

T ′′doi

(−Ψ1di(k) + E ′

qi(k)− (X ′di −Xlsi)Idi(k)

), (4.27b)

f 31i(·)=

1

T ′qoi

(− E ′

di(k) + (Xqi −X ′qi)

(Iqi(k) + E ′

di(k)−X ′

qi −X ′′qi

(X ′qi −Xlsi)2

(Ψ2qi(k)

+(X ′qi −Xlsi)Iqi(k)

))), (4.27c)

f 41i(·)=

1

T ′′qoi

(−Ψ2qi(k)− E ′

di(k)− (X ′qi −Xlsi)Iqi(k)

), (4.27d)

f 51i(·)=ωbi

(ωi(k)− ωsi

), (4.27e)

f 61i(·)=

ωsi

2Hi

(Pmi(k)−

(X ′′

qi −X ′′di

)Idi(k)Iqi(k)−Di

(ωi(k)− ωsi

)−X ′′

di −Xlsi

X ′di −Xlsi

E ′qi(k)Iqi(k)−

X ′di −X ′′

di

X ′di −Xlsi

Ψ1di(k)Iqi(k)

−X ′′

qi −Xlsi

X ′qi −Xlsi

E ′di(k)Idi(k) +

X ′qi −X ′′

qi

X ′qi −Xlsi

Ψ2qi(k)Idi(k)

), (4.27f)

where Pmi(k) is the mechanical power generated by the turbine at time k. The

nonlinear dynamic functions of the IEEE DC1A AVR without PSS applicable to

generation units 1, 2, 4, 5, 6, 9 and 10, i.e., x2i(k) =[Efdi(k) Rfi(k) VRi(k)

]T, are

given below [20],

f 12i(·)=

1

TEi

(−(KEi + Axie

BxiEfdi(k))Efdi(k) + VRi(k)

), (4.28a)

f 22i(·)=

1

TFi

(−Rfi(k) +

KFi

TFi

Efdi(k)

), (4.28b)

f 32i(·)=

1

TAi

(−KAiKFi

TFi

Efdi(k)− VRi(k) +KAiRfi(k) +KAi

(Vrefi − Vi(k)

)),

(4.28c)

where Efdi(k) is excitation system field voltage in p.u., VR is AVR regulator voltage

in p.u., Rf is AVR regulator voltage rate feedback in p.u. and Vrefi is AVR regulator

reference voltage. The dynamic functions of the IEEE ST1A AVR with PSS appli-

cable to generation units 3, 7 and 8, i.e., x2i(k) =[Efdi(k) y1pi(k) y2pi(k) y3pi(k)

]T,


are as follows [20],

f 12i(·)=

1

TRi

(KAi

(Vrefi +

KpiT1piT3pi

T2piT4pi

(ωi(k)− ωsi

)+ y1pi(k) + y2pi(k)

+y3pi(k)− Vi(k))− Efdi(k)

), (4.29a)

f 22i(·)=

1

Twi

(T ′pi

(ωi(k)− ωsi

)− y1pi(k)

), (4.29b)

f 32i(·)=

1

T2pi

(T ′′pi

(ωi(k)− ωsi

)− y2pi(k)

), (4.29c)

f 42i(·)=

1

T4pi

(T ′′′pi

(ωi(k)− ωsi

)− y3pi(k)

), (4.29d)

where

T ′pi =

−KpiT2wi+KpiTwiT1pi+KpiTwiT3pi−KpiT1piT3pi

(Twi−T2pi)(Twi−T4pi),

T ′′pi =

−KpiTwiT1piT2pi+KpiTwiT1piT3pi+KpiTwiT22pi−KpiTwiT2piT3pi

T2pi(Twi−T2pi)(T2pi−T4pi),

T ′′′pi =

KpiTwiT1piT3pi−KpiTwiT1piT4pi−KpiTwiT3piT4pi+KpiTwiT24pi

T4pi(Twi−T4pi)(T4pi−T2pi),

y1pi(k), y2pi(k) and y3pi(k) are PSS state variables. The dynamic functions of the

mechanical-hydraulic speed governing system and hydro-turbine is considered in

generation unit 1, i.e., x3i(k) =[P1gi(k) P2gi(k)

]Tand x4i(k) =

[P1Ti(k)

]Tare

given below [21],

f 13i(·)=

1

T1gi

(K1gi(ωl(k)− ωsi)− P1gi(k)

), (4.30a)

f 23i(·)=

1

T2gi

(K2gi(ωl(k)− ωsi)− P2gi(k)

)(4.30b)

f 14i(·)=

1

T2Ti

(Pci(k)− P1gi(k)− P2gi(k)− P1Ti(k)

), (4.30c)

where P1gi(k) and P2gi(k) are speed governing system dynamic states and P1Ti(k) is

state variable of the hydro-turbine. The mechanical power Pmi(k) in (4.27) can be

obtained from the following equation,

Pmi(k)=Pmiop +K1TiT1Ti

(Pci(k)− P1gi(k)− P2gi(k)

)+K1Ti

(T2Ti − T1Ti

)P1Ti(k),

(4.31)

where Pmiop is the mechanical power of the generator i at the operating point. The

dynamic functions of Westinghouse electro-hydraulic speed governing system with

steam feedback applicable to generation units 3, 5, 7, 8, 9 and 10, i.e., x3i(k)=[P1gi(k) P2gi(k)

]T, are given below [21],

f 13i(·)=

1

T1gi

(K1gi

(1− T2gi

T1gi

)(ωl(k)− ωsi)− P1gi(k)

), (4.32a)


f 23i(·)=

1

T3gi

(Pci(k)−K1gi

T2gi

T1gi

(ωl(k)− ωsi)− P1gi(k)− P2gi(k)

). (4.32b)

The dynamic function of General Electric and Westinghouse electro-hydraulic gov-

ernors without steam feedback used in generation units 2, 4 and 6, i.e., x3i(k)=[P2gi(k)

]T, is given below [21],

f 13i(·)=

1

T1gi

(Pci(k)−K1gi(ωl(k)− ωsi)− P2gi(k)

). (4.33)

The dynamic functions of tandem-compound double and single reheat steam tur-

bines applicable to generation unit 2-10, i.e., x4i(k)=[P1Ti(k) P2Ti(k) P3Ti(k) P4Ti(k)

]T,

are given below [21],

f 14i(·)=

1

T1Ti

(P2gi(k)− P1Ti(k)

), (4.34a)

f 24i(·)=

1

T2Ti

(P1Ti(k)− P2Ti(k)

), (4.34b)

f 34i(·)=

1

T3Ti

(P2Ti(k)− P3Ti(k)

), (4.34c)

f 44i(·)=

1

T4Ti

(P3Ti(k)− P4Ti(k)

). (4.34d)

Mechanical output power of the steam turbines can be obtained as follows,

Pmi(k) = Pmiop +(K1TiP1Ti(k)+K2TiP2Ti(k)+K3TiP3Ti(k)+K4TiP4Ti(k)

). (4.35)

All parameters of the ten generators, exciters, PSS, governors and turbines are shown

in table 4.1.

Assume stator winding resistance Rsi of all the generators is 0. By applying Kirch-

hoff’s voltage law to the circuit in Figure 4.2, terminal currents of the ith generator

can be written of the form (4.1) as,

Idi(k)=1

X ′′di

(X ′′

di −Xlsi

X ′di −Xlsi

E ′qi(k)− Vi(k)cos

(δi(k)− θi(k)

)+

X ′d −X ′′

di

X ′di −Xlsi

Ψ1di(k)

),

(4.36a)

Iqi(k)=1

X ′′qi

(−

X ′′qi −Xlsi

X ′qi −Xlsi

E ′di(k) + Vi(k)sin

(δi(k)− θi(k)

)+

X ′q −X ′′

qi

X ′qi −Xlsi

Ψ2qi(k)

).

(4.36b)

By substituting (4.37) into relevant generator dynamic equations (4.27)-(4.35), con-

sidering PMU measurements Vi(k), θi(k) as pseudo inputs, the dynamics of all gen-

eration units can be rewritten in the form of (4.11a). Using (4.37) in (4.2) and


Table 4.1: Generator, governor, turbine, exciter and PSS parameters.


l Hl Dl Xdi X′di X′′

di Xqi X′qi X′′

qi Xlsi T ′doi T ′′

doi T ′qoi T ′′

qoi

1 42 4 0.1 0.031 0.025 0.069 0.028 0.025 0.0125 10.2 0.05 1.5 0.035

2 30.3 9.8 0.295 0.070 0.005 0.282 0.170 0.05 0.035 6.56 0.05 1.5 0.035

3 35.8 10 0.250 0.053 0.045 0.237 0.088 0.045 0.0304 5.7 0.05 1.5 0.035

4 28.6 10 0.262 0.044 0.035 0.258 0.166 0.035 0.0295 5.69 0.05 1.5 0.035

5 26 3 0.67 0.132 0.05 0.62 0.166 0.05 0.054 5.4 0.05 0.44 0.035

6 34.8 14 0.254 0.05 0.04 0.241 0.081 0.04 0.0224 7.3 0.05 0.4 0.035

7 26.4 8 0.295 0.049 0.04 0.292 0.186 0.04 0.0322 5.66 0.05 1.5 0.035

8 24.3 9 0.29 0.057 0.057 0.280 0.091 0.045 0.028 6.7 0.05 0.41 0.035

9 34.5 14 0.211 0.057 0.057 0.205 0.059 0.045 0.0298 4.79 0.05 1.96 0.035

10 248 33 0.296 0.006 0.006 0.029 0.005 0.004 0.003 5.9 0.05 1.5 0.035

Governor and Turbine Parameters

l K1gi K2gi T1gi T2gi T3gi K1Ti K2Ti K3Ti K4Ti T1Ti T2Ti T3Ti T4Ti

1 17.44 2.56 38.5 0.52 - 10 - - - -2 1 - -

2 10 - 0.1 - - 0.22 0.22 0.3 - 0.3 7 0.4 -

3 1 - 2.8 1 0.15 0.3 0.4 0.3 0.26 0.3 7 5 0.4

4 10 - 0.02 - - 0.22 0.22 0.3 - 0.3 7 0.4 -

5 1 - 3.8 2 0.15 0.3 0.4 0.3 0.26 0.3 7 5 0.4

6 10 - 0.1 - - 0.22 0.22 0.3 - 0.3 7 0.4 -

7 1 - 2.8 1.5 0.1 0.3 0.4 0.3 0.26 0.3 7 5 0.4

8 1 - 1.8 1 0.15 0.22 0.22 0.3 - 0.3 7 0.4 -

9 1 - 2.8 1 0.1 0.3 0.4 0.3 0.26 0.3 7 5 0.4

10 1 - 5.8 2.5 0.4 0.22 0.22 0.3 - 0.3 7 0.4 -

IEEE DC1A AVR without PSS Parameters

l KAi TAi KEi TEi KFi TFi Axi Bxi

1,2,4,5,6,9,10 40 0.2 1 0.785 0.063 0.35 0.07 0.91

IEEE ST1A AVR with PSS Parameters

l KAi Kpi T1pi Tp2i T3pi T4pi Twi TRi

3,7,8 20 10 0.1 0.2 0.1 0.25 40 0.1

considering PMU measurements Ii(k), γi(k), we can obtain the dynamic output

equation of the form (4.11b).

Implementation: The initial conditions for the states of all generators are found by

performing a load flow calculation considering the active and reactive power data

for all the buses in the IEEE 39-bus, 10-generator test system data given in Mat-

power toolbox [22]. Only one generator in each area has been assumed to have

a UT based controller. These generation units are G1 (n1 = 12), G3 (n3 = 16)

and G6 (n6 = 13). It is assumed power system and filter are in steady-state at

the beginning of the study. We consider two cases in the implementation, Case

1 when active and reactive power of the loads on busbars number 8, 18 and 21

are increased, and case 2 when the loads on busbars 4, 20 and 27 are decreased.

Loads are changed after 1s of the implementation. Table 4.2 lists the change of

the loads in both cases. It is assumed that PMU installed on the ith generation


Table 4.2: New value of the loads at specified busbars in Case 1 and Case 2.

Case 1 Case 2PPPPPPPPPPPPower

Bus No. 8 18 21 4 20 27

Active (MW) 674.2 348.5 457.1 395.56 492.3 60.6

Reactive (MVAR) -100.3 8.2 -67.9 -282.4 -154.9 -170

unit is affected by the noise with the following distributions, η1i(k) ∼ N (0, 10−8),

η2i(k) ∼ N (0, 10−7), ς1i(k) ∼ N (0, 10−8), ς2i(k) ∼ N (0, 10−7). Consequently,

Qi =

[10−8 0

0 10−7

], Ri =

[10−8 0

0 10−7

], and we have chosen αi = 10−3, β = 2

in the UT based filtering algorithm. Furthermore, we have chosen the initial covari-

ance as PXiXi(0) =

[diag

(10−10, 10−10, · · · , 10−10︸︷︷︸

ni

, 10−8, 10−7)]. Sampling time ∆t

is chosen as 10ms and feedback matrices F1, F3 and F6 are given below,

F3=

[0 0 0 0 −0.2 −1 0 0 0 −2 −1 4

],

F3=

[0 0 0 0 −0.04 −150 0 0 0 0 0 0.2 0 0.1 0.01 0

],

F6=

[0 0 0 0 −0.05 −150 0 0 0 0.5 −0.8 0 0.5

].

Controller gains Ktie1 = Ktie2 = Ktie3 = 0.05 are chosen for all three areas. The im-

plementation results for the UT based filter in Case 1 are presented in Figures 4.4–4.9

where the state estimation of 1st, 3rd, 6th genertion units and PMU measurements

(i.e., pseudo inputs and outputs) Vi(k), θi(k), Ii(k), γi(k); i ∈ 1, 3, 6 are shown.

Figure 4.10 shows the estimation of augmented states (i.e., ηi(k) =

[η1i(k)

η2i(k)

];

i ∈ 1, 3, 6). Frequency of the 1st generator and tie-line power deviations of the

three areas are also shown in Figures 4.11 and 4.12. For each case we implemented

three control scenarios: (1) the frequency of G1 without any control, (2) a UT

based control on G1 only when feedback control vectors F3 and F6 are set to 0,

and (3) UT based control on G1, G3 and G6. As can be seen in Figures 4.11 and

4.12, UT based filter can estimate states precisely. Furthermore, by utilizing UT

based controller, total tie-line power deviation of each area goes to zero in a timely

manner and frequency can come back to its operating point quickly. The method

proposed and the results reported in this chapter is based on taking into account


the entire network topology and is different to previously methods where an area of

the entire network is simplified into one linearized entity. For instance, the states

of the generators and the network frequency as shown in Figures 4.11 and 4.12,

cannot be compared with previous methods because the network topology is lost

in the simplifying assumptions of those previously reported methods. Furthermore,

previous reported methods do not consider noise in measurements as considered in

this chapter. PMU measurements of generator current and voltage magnitude and

phase angles for the 1st, 3rd and 6th generation units are shown in Figures 4.4-4.6.

Figures 4.7-4.9 show that UT based estimators can flawlessly follow the nonlinear

dynamic states in presence of noise in the measurements.


0 10 20 30 40 501.034

1.04

1.046

1.052V1(k)

0 10 20 30 40 50−18

−12

−6

0

θ 1(k)

0 10 20 30 40 502

4

6

8

9

I 1(k)

0 10 20 30 40 50−18

−12

−6

0

Time (s)

γ1(k)

Figure 4.4: Output and pseudo input measurements of the 1st generator in Case 1.


0 10 20 30 40 500.89

0.92

0.95

0.98

1.01V3(k)

0 10 20 30 40 50−18

−12

−6

0

θ 3(k)

0 10 20 30 40 506.9

7.5

8.1

8.7

I 3(k)

0 10 20 30 40 50−18

−12

−6

0

Time (s)

γ3(k)

Figure 4.5: Output and pseudo input measurements of the 3rd generator in Case 1.


0 5 10 15 20 25 301.01

1.03

1.05

1.07V6(k)

0 10 20 30 40 50−18

−13

−8

−3

2

θ 6(k)

0 10 20 30 40 506.4

6.9

7.4

7.9

8.4

I 6(k)

0 10 20 30 40 50−18

−12

−6

0

Time (s)

γ6(k)

Figure 4.6: Output and pseudo input measurements of the 6th generator in Case 1.


0 10 20 30 40 500.05

0.1

0.15

0.2

0.25

0.3

E′ d1(k)

Real state UT based est.

0 10 20 30 40 501.01

1.03

1.05

1.07

Ψ1d1(k)


0 10 20 30 40 501.08

1.1

1.12

1.14

E′ q1(k)


0 10 20 30 40 50−0.4

−0.3

−0.2

−0.05

Time (s)

Ψ2q1(k)


Figure 4.7: State estimation of the 1st generator in Case 1.


0 10 20 30 40 50−16

−10

−4

2

δ 1(k)


0 10 20 30 40 500.99

0.995

1

1.004

ω1(k)


0 10 20 30 40 501.2

1.3

1.38

1.45

Efd1(k)


0 10 20 30 40 500.22

0.235

0.25

0.265

Time (s)

Rf1(k)


Figure 4.7: State estimation of the 1st generator in Case 1 (continued).


0 10 20 30 40 501.45

1.55

1.65

1.75

1.85

VR1(k)


0 10 20 30 40 50−1.8

−1.3

−0.8

−0.3

0.2×10−2

P1g1(k)


0 10 20 30 40 50−2.1

−1.2

−0.3

0.6×10−2

P2g1(k)


0 10 20 30 40 50−0.06

0.11

0.28

0.45

0.62

Time (s)

P1T1(k)


Figure 4.7: State estimation of the 1st generator in Case 1 (continued).


0 10 20 30 40 500.45

0.47

0.49

0.51

E′ d3(k)


0 10 20 30 40 500.65

0.71

0.77

0.83

0.9

Ψ1d3(k)


0 10 20 30 40 500.8

0.9

1

1.1

E′ q3(k)


0 10 20 30 40 50−0.7

−0.68

−0.66

−0.64

−0.62

Time (s)

Ψ2q3(k)


Figure 4.8: State estimation of the 3rd generator in Case 1.


0 10 20 30 40 50−16

−10

−4

2

δ 3(k)


0 10 20 30 40 500.99

0.994

0.997

1.001

1.004

ω3(k)


0 10 20 30 40 501.8

2.1

2.4

2.7

Efd3(k)


0 10 20 30 40 500

0.25

0.5

0.75

1×10−2

Time (s)

y 1p3(k)


Figure 4.8: State estimation of the 3rd generator in Case 1 (continued).


0 10 20 30 40 50−0.04

−0.005

0.03

0.065

0.1y 2

p3(k)


0 10 20 30 40 50−0.2

−0.15

−0.1

−0.05

0

0.05

y 3p3(k)


0 10 20 30 40 50−4

−3

−2

−1

0

1×10−3

P1g3(k)


0 10 20 30 40 500

0.5

1

1.5

2

Time (s)

P2g3(k)




0 10 20 30 40 500

0.5

1

1.5

2

P1T3(k)


0 10 20 30 40 50−0.2

0.2

0.6

1

1.4

P2T3(k)


0 10 20 30 40 50−0.2

0.2

0.6

1

1.4

P3T3(k)


0 10 20 30 40 50−0.2

0.2

0.6

1

1.4

Time (s)

P4T3(k)




0 10 20 30 40 500.47

0.5

0.53

0.56

E′ d6(k)


0 10 20 30 40 500.76

0.8

0.84

0.88

0.92

Ψ1d6(k)


0 10 20 30 40 500.96

0.995

1.03

1.065

1.1

E′ q6(k)


0 10 20 30 40 50−0.76

−0.73

−0.7

−0.67

−0.64

Time (s)

Ψ2q6(k)


Figure 4.9: State estimation of the 6th generator in Case 1.


0 10 20 30 40 50−16

−10

−4

2

δ q6(k)


0 10 20 30 40 500.99

0.994

0.997

1.001

1.004

ω6(k)


0 10 20 30 40 502.1

2.2

2.3

2.4

2.5

2.6

Efd6(k)


0 10 20 30 40 500.38

0.41

0.44

0.47

Time (s)

Rf6(k)


Figure 4.9: State estimation of the 6th generator in Case 1 (continued).


0 10 20 30 40 503

3.4

3.8

4.2

4.6

VR6(k)


0 10 20 30 40 50−0.5

0.375

1.25

2.125

3

P2g6(k)


0 10 20 30 40 50−0.5

0.375

1.25

2.125

3

P1T6(k)


0 10 20 30 40 50−0.5

0.375

1.25

2.125

3

P2T6(k)


0 10 20 30 40 50−0.5

0.375

1.25

2.125

3

Time (s)

P3T6(k)


Figure 4.9: State estimation of the 6th generator in Case 1 (continued).


0 10 20 30 40 50−1.5

−0.875

−0.25

0.375

1×10−3

η 11(k)

0 10 20 30 40 50−4

−2.25

−0.5

1.25

3×10−3

η 21(k)

0 10 20 30 40 50−6

−2.5

1

4.5

8×10−4

η 13(k)

0 10 20 30 40 50−1.5

−0.625

0.25

1.125

2×10−3

η 23(k)

0 10 20 30 40 50−4

−1

2

5×10−4

η 16(k)

0 10 20 30 40 50−1

−0.375

0.25

0.875

1.5×10−3

Time (s)

η 26(k)

Figure 4.10: Estimation of augmented states in all generators in Case 1.


0 20 40 60 80 1000.984

0.988

0.992

0.996

1

1.004ω1(k) UT based control on G1,G3,G6

UT based control on G1 only

Open loop

0 20 40 60 80 100−3

−1.5

0

1.2

2.5

Area1tie-linepow

ers

∆Ptot1 (k) ∆Ptie1→3 (k)

∆Ptie1→3 (k) ∆Ptie1→2 (k)

0 20 40 60 80 100−1

0

1.2

2.5

Area2tie-linepow

ers ∆Ptot2 (k) ∆Ptie2→1 (k)

∆Ptie2→3 (k)

0 20 40 60 80 100−5

−2.5

0

3

Time (s)

Area3tie-linepow

ers



Figure 4.11: Frequency and tie-line power deviations in Case 1.


0 20 40 60 80 1000.995

1

1.005

1.01

1.015

1.02ω1(k) UT based control on G1,G3,G6

UT based control on G1 only

Open loop

0 20 40 60 80 100−2.5

−1.2

0

1.5

Area1tie-linepow

ers



0 20 40 60 80 100−1.5

−0.7

0

0.7

1.5

Area2tie-linepow


∆Ptie2→3 (k)

0 20 40 60 80 100−2

0

2

4

Time (s)

Area3tie-linepow



Figure 4.12: Frequency and tie-line power deviations in Case 2.

4.5. Conclusion 114

4.5 Conclusion

In this chapter we have presented a quasi-decentralized UT based approach to

load frequency control. The designed UT based controllers are decoupled from

each other and can handle noisy PMU data. The proposed design method is based

on preserving the entire network topology, and the control system is analyzed on

the 39-bus 10-generator IEEE test system. The simulation results show that the

proposed UT based scheme can effectively control the frequency and tie-line power

deviations using noisy PMU data.

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2011.

118CHAPTER 5

Power System DSE Using Particle Filter

This chapter is largely based on a published article with the following details (addi-

tional case studies have been included in Section 5.4):

Kianoush Emami, Tyrone Fernando, Herbert H.C. Iu, Heiu

Trinh, and Kit Po Wong, “Particle filter approach to dynamic state

estimation of generators in power systems,” IEEE Transactions on



This chapter presents a novel particle filter based dynamic state

estimation scheme for power systems where the states of all the

generators are estimated. The proposed estimation scheme is decen-

tralized in that each estimation module is independent from others

and only uses local measurements. The particle filter implementa-

tion makes the proposed scheme numerically simple to implement.

What makes this method superior to the previous methods which are

mainly based on the Kalman filtering technique is that the estima-

tion can still remain smooth and accurate in the presence of noise

with unknown changes in covariance values. Moreover, this scheme

can be applied to dynamic systems and noise with both Gaussian

and non Gaussian distributions.

5.1 Introduction

With emerging new energy resources in power grids, the necessity of having a

sustainable and a reliable power supply is the most crucial task of any energy man-

agement system (EMS). A computer aided EMS that uses supervisory control and

data acquisition (SCADA) systems, which is often referred to as EMS/SCADA, can

assist utility grids to monitor, optimize and control the generation units and/or

transmission system. However, these EMS/SCADA systems cannot capture power

system dynamics accurately because the EMS is based on a steady state model of

the generation units, and the SCADA system has a slow data update rate. In 1988,


a device called synchrophasor or phasor measurement unit (PMU) was invented to

calculate the real time phasor measurements. Each collected measurement data

is synchronized by the Global Positioning System (GPS). In typical applications,

PMUs sample widely dispersed locations in the power system grid, tag the measure-

ments with GPS time stamp and send it to Phasor Data Concentrators (PDC) and

then to a Wide Area Measurement System (WAMS). As the name specifies, WAMS

is a wide spread system that can also be a standalone infrastructure complementary

to the conventional SCADA system. It is specifically designed for a safe and reli-

able grid operation by elevating the operator’s real-time situational awareness. In

order to overcome the stated weakness of the EMS/SCADA to estimate power sys-

tem dynamics, many research efforts have focused on incorporating dynamic state

estimation (DSE) in WAMS using PMUs [1–4]. Among the techniques used for

DSE in power systems are the supercalibrator technique [5–8] and the Kalman fil-

ter based schemes [9–15]. A linearization based technique like extended Kalman

filter (EKF) [10] has its drawbacks, which has been addressed in [11] and com-

pared with the unscented Kalman filter (UKF). Power system dynamic is nonlinear

and linearization is not a feasible solution to DSE of generators in a power system.

Unscented transform (UT) based schemes are introduced in many papers [11–15]

which approximate transformations of Gaussian variables by propagating a number

of points called Sigma points through a nonlinear function (i.e., system dynam-

ics). Mean and covariance are approximated as a linear combination of the sigma

points. UKF uses the unscented transform to compute an approximate mean and

covariance in nonlinear filtering problems. Although reasonable performance has

been reported for UKF in recent publications, there are still some drawbacks to the

method as listed below:

• A truly global approximation cannot be done using UKF due to the small set

of sigma points.

• Does not work well with nearly singular covariances, i.e. with nearly deter-

ministic systems.

• Like its predecessor Kalman filter, UKF often has practical implementation

issues due to the fact that noise covariance matrices cannot be estimated well.

If good estimation of the noise covariance is not provided, Cholesky factoriza-

tions may not be completed due to uncorrelated received data (posteriori error

covariance matrix is not positive/semi-positive definite) and consequently the

estimation process can halt.


• Can only be applied to systems with unimodal distribution and Gaussian noise.

Based on the points stated above, the applicability of UKF to DSE of generators in a

power system can be limited, there can be instances where an UKF based algorithm

may not perform adequately. Measurement of PMUs are always noisy due to the

nature of the power system and noise covariance may change over time from known

initial values. Although some extensive research has been carried out to estimate

the noise covariance matrices in KF algorithm e.g. auto-covariance least-squares

(ALS) [16,17], such a technique has also not been previously applied in power system

dynamic state estimation. Using the particle filter (PF) approach, it is possible to

overcome the drawbacks of UKF. Unlike the Sigma points being a function of the

system order, in PF approach, the designer can choose the number of particles

which represent the posterior distribution of the estimated states. By choosing a

higher number of particles, it is possible to achieve better accuracy of the estimates.

Furthermore, unlike UKF, PF is not limited to systems driven by Gaussian noise and

the PF algorithm does not halt when the Cholesky factorization of error covariance

matrix cannot be completed. The application of PF technique to DSE of generators

in power systems is at its infancy, see [18–20]. The merits of using PF in DSE in

power systems and associated challenges with regard to computations are stated in

[19], and also it is stated the need for a parallel implementation of the PF algorithm.

In fact, such a parallel implementation of the PF algorithm is necessary for it to be

feasible for its real-time application. In [20], an extended PF technique is presented

for a single machine system connected to an infinite bus. The particle filter scheme

proposed in this chapter complements previous studies [11–15] on DSE, and it is

a decentralized parallel implementation making it feasible for real-time application

in a large complex power system. We present a comprehensive study of applying

the PF technique in the DSE of generators in IEEE 39-bus, 10-generator (New

England) test system. Our results show that the proposed estimation can still

remain smooth and accurate in the presence of noise with unknown changes in

covariance values. Moreover, this scheme can be applied to dynamic systems and

noise with both Gaussian and non Gaussian distributions. The chapter is organized

as follows, in Section 5.2, the particle filter algorithm is presented. Power system

dynamics are considered in Section 5.3. Case study of complex power systems are

given in Section 5.4 with the relevant particle filter design. The results are compared

with the UKF algorithm. Bad data detection is addressed in Section 5.5, followed

by a conclusion in Section 5.6.

5.2. Particle Filter 121

5.2 Particle Filter

Consider a system with states x(k) ∈ Rn at time k, k = 0, 1, .. which evolves

according to a Markov process with initial states x(0) distributed according to the

probability distribution function p (x(0)), i.e.,

x(0) ∼ p (x(0)) . (5.1)

The distribution of states x(k + 1) at time k + 1 given the states x(k) at time k is

according to the transition density function p (x(k + 1)|x(k)), i.e.,

x(k + 1) ∼ p (x(k + 1)|x(k)) . (5.2)

The observation y(k) ∈ Rm given the states x(k) at the time k is distributed ac-

cording to the likelihood density function p(y(k)|x(k)), i.e.,

y(k) ∼ p (y(k)|x(k)) . (5.3)

The Bayesian recursive solution to the posterior density p (x(k)|y(k)) is of the form,

p (x(k)|y(k))=p (y(k)|x(k)) p (x(k)|y(k − 1))

p (y(k)|y(k − 1)), (5.4a)

p (y(k)|y(k − 1))=

∫p (y(k)|x(k)) p (x(k)|y(k − 1)) dx(k), (5.4b)

p (x(k + 1)|y(k))=∫

p (x(k + 1)|x(k)) p (x(k)|y(k)) dx(k). (5.4c)

Given an observation y(k), a particle filter can approximate the posterior density

p(x(k)|y(k)) recursively over time using the incoming observations and a mathe-

matical model of the system in state space based on the Bayesian paradigm. In a

particle filter implementation of the Bayesian recursion (5.4), a finite N number of

random samples, x1(0), . . . , xN(0), called particles are drawn from the the proba-

bility density function p(x(0)), and each of those particles are assigned a weight of

1/N , i.e.,

wi(1|0) =

1

N, i = 1, . . . , N. (5.5)

The posterior density p(x(k)|y(k)) is estimated recursively by propagating those

particles x1(0), . . . , xN(0) forward in time assigning appropriate weights for each

of those particles as follows,

p (x(k)|y(k − 1)) =N∑i=1

wi(k|k−1)Dδ

(x(k)− xi(k)

), (5.6)

5.2. Particle Filter 122

where Dδ(·) denotes Dirac impulse function. The Bayesian recursion (5.4) can be

rewritten as,

p (x(k)|y(k))=N∑i=1

wi(k|k)Dδ

(x(k)− xi(k)

), (5.7)

where

wi(k|k) =

1

s(k)p(y(k)|xi(k)

)wi

(k|k−1), (5.8)

s(k) =N∑i=1

p(y(k)|xi(k)

)wi

(k|k−1). (5.9)

If y(k) ∈ Rm, i.e., the system has m outputs, then p (y(k)|xi(k)) in (5.9) can be

calculated as,

p(y(k)|xi(k)

)=

m∏d=1

p(yd(k)|xi(k)

). (5.10)

By introducing a proposal density function q(xi(k + 1)|xi(k), y(k + 1)) and using im-

portance sampling principle [21], the subsequent particles xi(k + 1) can be obtained,

and the posterior density of p (x(k + 1)|y(k)) can be estimated as [22],

p (x(k + 1)|y(k)) =N∑i=1

wi(k+1|k)Dδ

(x(k + 1)− xi(k + 1)

), (5.11)

where

wi(k+1|k) =

p (xi(k + 1)|xi(k))

q (xi(k + 1)|xi(k), y(k + 1))wi

(k|k). (5.12)

The calculated weights according to (5.12) are normalized such that,

N∑i=1

wi(k+1|k) = 1. (5.13)

The estimated state x(k) of x(k) is taken as,

x(k) =N∑i=1

wi(k|k)x

i(k). (5.14)

One of the most important steps in particle filtering is the resampling step. There

are several resampling methods proposed in the literature such as systematic, multi-

nomial, stratified and residual resampling, etc. Systematic resampling is found to

be more favourable due to its implementation simplicity, resampling quality and

computational complexity (see [23] and [24]) which is the choice for the proposed

5.3. Power System Dynamics 123

algorithm. Without resampling the weight of one of the effective particles becomes

1 while the other weights tend to 0. In this case, sample degeneracy, or sample

depletion, or sample impoverishment is affecting PF to break down to a set of in-

dependent trajectories [25]. Accordingly, resampling has been adopted to all recent

PF algorithms. Although resampling is not practically necessary in each time itera-

tion, it needs to be performed when the number of efficient particles falls under the

threshold. The threshold can be chosen as Nthr =2N3

or Nthr =N2. Resampling then

provides required feedback information from the observation as only the trajectories

that perform well will survive. Efficient number of particles Neff and its estimation

Neff can be obtained as [26],

Neff =N

1 +N2V ar(wi(k|k))

, (5.15)

and

Neff =1

N∑i=1

(wi(k|k))

2

. (5.16)

Finding a right proposal density is also an important step in particle filter design.

When the signal to noise ratio is small, often the proposal density is chosen as the

state transition density function p(x(k + 1)|x(k)), then (5.12) becomes,

wi(k+1|k) = wi

(k|k). (5.17)

The particle filter algorithm proposed in this study can be summarized as in Algo-

rithm 5.1.

5.3 Power System Dynamics

Let us consider a power system consisting of M generators, M PMUs placed at

the terminals of each generator, B busbars and L loads as shown in Figure 5.1. The

equivalent circuit of generator l that connects to busbar l is shown in Figure 5.2.

The voltage of busbar l at time k is denoted vl(k)∠θl(k), and the current that flows

through generator l into busbar l at time k is denoted by il(k)∠γl(k). The current

il(k)∠γl(k), idl(k) and iql(k) for l ∈ 1, . . . ,M are related according to,

il(k)∠γl(k) =(idl(k) + jiql(k)

)ej(δl(k)−π/2), (5.18)

where δl(k) is the rotor angle, idl(k) is the d-axis current and iql(k) is q-axis current

at time k of generator l. In Figure 5.2, the d-axis transient synchronous reactance

X ′dl, the q-axis transient synchronous reactance X ′

ql and stator winding resistance


Algorithm 5.1 Particle filter algorithm

Step 0: Initialization (k = 0)

• Let q (x(k + 1)|x(k), y(k + 1)) = p (x(k + 1)|x(k)), and also let the number of

particles be N .

• Draw particles xi(0), i = 1, 2, ..., N randomly from p(x(0)) and let wi1|0 =

1N.

Step 1: Measurement update

• For i = 1 to N assign a weight to each particle according to wi(k|k) =

wi(k|k−1)p (y(k)|xi(k)).

• Normalize the weights wi(k|k) =

wi(k|k)∑N

j=1 wj(k|k)

.

Step 2: Systematic adaptive resampling

• Calculate Neff , set Nth = N2. If Neff < Nth then continue, otherwise go to

Step 3.

• Calculate cumulative ci = ci−1 + wik:kNi=2, c1 = 0 by finding cumulative sum

of the elements of w(k|k), i.e., c1, c2, . . . , cN

• Draw a starting sample from a zero mean, 1N

variance, normal distribution N ,

i.e., s1 ∼ N (·, 0, 1N).

• Do N iterations starting from = 1 . . . N , calculate s = s1+(−1)N

and at each

th iteration find the index of the first value on the cumulative c1, c2, . . . , cNthat is greater than s and register the index of ci in a set S.

• Select those particles corresponding to the elements in S and assign identical

weight 1N

to all selected particles.

Step 3: Time update

• Estimate state mean according to x(k) =∑N

i=1 wi(k|k)x

i(k).

• Generate predicted particles xi(k + 1) ∼ p (x(k + 1)|xi(k)).

• Compute importance weight wi(k+1|k) = wi

(k|k).

• Increment k and iterate from step 1.


Rsl are equivalent circuit parameters of generator l. Considering the voltage drops

in the circuit shown in Figure 5.2 and applying Kirchhoff’s voltage law, it is easy

to see that the d-axis current idl(k) and q-axis current iql(k) at time k for generator

l, l ∈ 1, . . . ,M takes the following form,

idl(k)=Ψl

(vl(k), θl(k), δl(k), E

′

dl(k), E′

ql(k)), (5.19)

iql(k)=Φl

(vl(k), θl(k), δl(k), E

′

dl(k), E′

ql(k)), (5.20)

where E′

dl(k) is the transient emf due to the flux in q-axis damper coil and E′

ql(k) is

the transient emf due to field flux linkages of generator l at time k, and Ψl(·),Φl(·) aresome known nonlinear functions. The current il(k)∠γl(k) and voltage vl(k)∠θl(k)

satisfy the following power balance equations at each busbar l ∈ 1, . . . , B in the

power network,

PLl(k) + jQLl(k) + vl(k)ejθl(k)il(k)e

−jγl(k) −B∑

r=1

vl(k)vr(k)Ylrej(θl(k)−θr(k)−αlr) = 0,

(5.21)

where Ylr∠αlr is the admittance of the line connecting buses l and r, PLl(k) and

QLl(k) are active power and reactive power consumed by the loads connected to

busbar l at time k. Obviously, at a generator busbar we have il(k)e−jγl(k) 6= 0, and

at a load busbar we have il(k)e−jγl(k) = 0 as in (5.21). A load flow analysis can

determine voltage vl(k)∠θl(k) at every busbar l ∈ 1, . . . , B, however, the PMUs

placed at the generator busbars can provide the measurements of vl(k)∠θl(k) and

il(k)∠γl(k) at every time instant k providing a snapshot of the status of the power

network.

Remark 5.1. In this chapter we make the same assumption regarding the dynamics

of the transmission network and the dynamics of rotating machines as in [15]. The

dynamics of the transmission network is much faster than the dynamics of rotating

machines, so the voltages vl(k)∠θl(k) and currents il(k)∠γl(k) on each busbar on the

network can change instantaneously, and those voltages and currents can be regarded

as inputs or outputs of the generators connected to it [15].

Let us now denote xl(k) to be the state of the generator connected to busbar l

at time k, unlike the voltage vl(k)∠θl(k) and current il(k)∠γl(k) which can change

instantly, the states xl(k), l ∈ 1, . . . ,M of a generator cannot change instantly,

and its dynamics take the following form,

xl(k + 1) = fl (xl(k), vl(k), θl(k), idl(k), iql(k), vrefl, Tml) , (5.22)


PMU

Transmission Network

with Busbars

Load 1

Load

PMU 1

Part

icle

Fi

lter

Estim

ator

1

Part

icle

Fi

lter

Estim

ator

m

GPS Synchronization Signal

Loca

l C

ontr

olle

r 1

Loca

l C

ontr

olle

r m

Figure 5.1: Particle filter state estimation scheme.

Figure 5.2: Two axis equivalent circuit of the synchronous generator l.


where vrefl is the reference voltage, Tml is the mechanical input torque and fl(·)are some known nonlinear functions, all of which are applicable to generator l, l ∈1, . . . ,M. Typically the state vector xl(k) consist of some or all of variables,

E ′dl(k), E

′ql(k), rotor angle δl(k) (rad), angular velocity ωl(k) (p.u.) and scaled field

voltage Efdl(k) (p.u.), rate feedback Rfl(k) (p.u.) and scaled input to the main ex-

citer VRl(k) (p.u.) which are generator’s excitation system dynamic state variables.

Considering (5.19) and (5.20), we can rewrite (5.22) as follows,

xl(k + 1) = fl (xl(k), vl(k), θl(k), vrefl, Tml) . (5.23)

A PMU that is placed at the generator terminal can measure vl(k)∠θl(k) and

il(k)∠γl(k). The PMU voltage and current measurements consist of noise, con-

sequently, the measured voltages and currents at every busbar l, l ∈ 1, . . . ,M,differs from its actual values,

il(k)

γl(k)

vl(k)

θl(k)

=

il(k)

γl(k)

vl(k)

θl(k)

+

ηl,1(k)

ηl,2(k)

ηl,3(k)

ηl,4(k)

, (5.24)

where il(k)∠γl(k) and voltage vl(k)∠θl(k) are the PMU measurements, ηl,1(k),

ηl,2(k), ηl,3(k) and ηl,4(k) are noise in the measurements of PMU l at time k. Using

(5.24) in (5.23) we have,

xl(k + 1)=fl

(xl(k), vl(k), θl(k), vrefl, Tml, ηl,3(k), ηl,4(k)

), (5.25)

where fl(·) are some known nonlinear functions. From (5.25), it is clear that for

a given state xl(k), noise terms ηl,1(k), ηl,2(k), ηl,3(k), ηl,4(k), mechanical torque Tml

and reference voltage vrefl, the next state xl(k + 1) is determined by the bus volt-

age vl(k)∠θl(k) and il(k)∠γl(k). Clearly, vl(k)∠θl(k) and il(k)∠γl(k) are the driv-

ing inputs of the nonlinear dynamical system in (5.25). The current measurement

il(k)∠γl(k) can be written as,

yl(k) =

[il(k)

γl(k)

]= hl

(xl(k), vl(k), θl(k), ηl,1(k), ηl,2(k), ηl,3(k), ηl,4(k)

), (5.26)

where hl(·) are some known nonlinear functions that can be derived from (5.18)-(5.20)

and (5.24). Using vl(k) and il(k) as the input and/or the output is also reported

in [15]. The transition density function p (x(k + 1)|x(k)) according to (5.2) and the

likelihood p(y(k)|x(k)) according to (5.3) can be obtained from (5.25) and (5.26)

respectively.

5.4. Case Studies 128

1 2

PMU

GPS Synchronization Signal

PF Estimator

Figure 5.3: Electrical circuit of a synchronous machine connected to an infinite bus.

Remark 5.2. Equations (5.18)-(5.26) which are according to the KVL and power

balance equations in a power system lead to the state transition density function

and the likelihood function used in the proposed PF algorithm. The particle filter

proposed in this chapter is an alternative to the UKF algorithm proposed in [15].

We can perform particle filtering in the estimation of the dynamic states of all

generating units using Algorithm 5.1 presented in Section 5.2.

5.4 Case Studies

5.4.1 Single-machine-infinite bus system

Here we consider a power system composed of a single generator, i.e., M = 1,

l = 1, and two connected transmission lines to an infinite bus, i.e., B = 2. We

assume the synchronous machine has a constant electro-magnetic field, i.e. E ′q1(k) =

1.0566 p.u., non-salient pole, i.e., X ′d1 = X ′

q1 with transient reactance X ′d1=0.1 p.u..

Excitation system and damping winding dynamic of the generator are neglected, i.e.,

E ′d1(k) = 0. The resistance of the lines, transformers and synchronous machine (Rs1)

are 0. The mechanical torque Tm1 = 1 p.u. from the prime mover is assumed to be

constant during the transient condition. The two transmission lines are modelled

with two equal reactances 2XL = 0.4 p.u. between generator and the infinite bus.

At time t = 1s a three-phase to ground short circuit fault happens on one of the lines

close to the generator bus. The fault is cleared after 6 cycles (6× 1f

swhere f = 60

Hz) by the protection system. The infinite bus has a constant voltage, v2(k) = 1

p.u. and zero phase angle, θ2(k) = 0. The single machine-infinite bus system is

obtained from Figure 5.1 is shown in Figure 5.3. The dynamic of the generator in

the form of (5.22) is given by the following equation [27],


[δ1(k + 1)

ω1(k + 1)

]=

[δ1(k)

ω1(k)

]+

[∆t(ω1(k)− ωs1)ωbωs1∆t

2H1(Tm1 − Te1(k)−D1(ω1(k)− ωs1))

],(5.27)

where Te(k) is the electrical torque acting on the rotor in p.u., δ1 is the angle of

the rotor in radian, ω1 is the rotor angular velocity, ωs = 1 and ωb = 2πf rad/s

are synchronous and base angular velocities, respectively, H1 = 20s−1 is inertia

constant, D1 = 10 is damping coefficient and ∆t is the sampling time. Electrical

power can be obtained from Figure 5.2 as follows,

Pe1(k) = ws1Te1 = E ′d1id1(k) + E ′

q1iq1(k) + (X ′q1 −X ′

d1)id1(k)iq1(k). (5.28)

Applying Kirchhoff’s voltage law to Figure 5.3, id1(k) and iq1(k) can be obtained as,

id1(k)=1

X ′d1

(E ′

q1 − v1(k)cos (δ1(k)− θ1(k))),

iq1(k)=1

X ′d1

(v1(k)sin (δ1(k)− θ1(k))) . (5.29)

Substituting (5.29) in (5.28) and considering v1(k) and θ1(k) as pseudo inputs, elec-

trical torque can be rewritten as,

Te1(k) =E ′

q1

ωs1X ′d1

((v1(k)− η1,3

)sin(δ1(k)−

(θ1(k)− η1,4

))), (5.30)

where v1(k) and θ1(k) are PMU measurements of voltage magnitude and phase angle

of the generator terminal respectively and η1 is the normally distributed measure-

ment noise, i.e., N(·, 0, σ2

η1(k)

)with 0 mean and σ2

η1(k)variance. By substituting

(5.30) in (5.27), the single machine-infinite bus dynamic can be described as,

x1(k + 1) = f1

(x1(k), v1(k), θ1(k), Tm1, vref1, η1,3(k), η1,4(k)

). (5.31)

Observation function h1(·) for the measurement i1(k)∠γ1(k) can be obtained from

(5.29) as follows,

y1(k)=

[yl,1(k)

y1,2(k)

]= h1 (x1(k)) +

[η1,1(k)

η1,2(k)

]=

[i1(k) + η1,1(k)

γ1(k) + η1,2(k)

]

=

√i2d1(k) + i2q1(k) + η1,1(k)

tan−1(

−id1 (k)

iq1 (k)

)+ δ1(k) + η1,2(k)

. (5.32)

The power system is considered to be in steady state initially and the initial condi-

tions can be obtained as follows,

x1(k = 0) =

[δ1(0)

ω1(0)

]=

[sin−1(

Tm1(XL+X′d1)

E′q1V∞

)

ωs1

]. (5.33)


Considering (5.31), transitional prior distribution in (5.2) can be approximated as

follows,

p

(x1(k + 1)|x1(k)

)= N

(xl(k + 1), f

(x1(k), v1(k), θ1(k), Tm1, 0, 0

)+ µ1(k), σ

2µ1

),

where µ1(k) is process noise and is assumed to have normal distributionN(·, 0, σ2

µ1(k)

)with 0 mean and σ2

µ1(k)variance. Using (5.10) and (5.32) we approximate likelihood

in (5.3) as follows,

p

(y1(k)|x1(k)

)=p

(i1(k)|x1(k)

)× p

(γ1(k)|x1(k)

)=N

(h1 (x1(k)) , i1(k), σ

2i1(k)

)×N

(h1 (x1(k)) , γ1(k), σ

2γ1(k)

),

(5.34)

where i1(k) and γ1(k) are the measurement that are received from the PMU.

Implementation: Initial particles δi1(0) and ωi1(0), i ∈ 1, . . . , N are drawn from the

following distribution,[δi1(0)

ωi1(0)

]∼ N

(·, [(δ1(0), ω1(0))]2×1, [diag(10

−4)]2×2

). (5.35)

We choose the number of particles N=250, resampling threshold Nthr is chosen as

50% of the total number of particles and the sampling time is selected as ∆t=20ms.

Process and measurement noise are assumed to have zero mean Normal distribution

as, [µ1,1(k)

µ1,2(k)

]∼ N

(·, [0]2×1 ,

[diag(10−10)

]2×2

),

and [η1,1(k)

η1,2(k)

]∼ N

(·, [0]2×1 ,

[diag(10−6)

]2×2

).

Likelihood variance σ2i1(k)

= σ2γ1(k)

= 10−3. Here we compare the proposed par-

ticle filter method with the UKF method. The designed values of the UKF pa-

rameters are taken as, Q1(k) = [diag(10−10)]2×2, R1(k) = [diag(10−6)]2×2 and

P1(0) = [diag(10−3)]2×2. Initial value of states in UKF is selected as x1ukf (0) =

[δ1(0) + 0.01 ω1(0) − 0.01]T . The dynamic state estimation and relevant errors

are shown in Figures 5.4 and 5.5. As it can be seen in Figures 5.4 and 5.5, PF and

UKF have a fairly similar performance, however, during the period of fault PF can

smoothly and quickly handle the disturbance while UKF tends to diverge during the


0 1 2 3 4 5 60.18

0.51

0.84δ1(k)

Real State

PF Est.

UKF Est.

0 1 2 3 4 5 60.994

0.999

1.004

ω1(k)

Real State

PF Est.

UKF Est.

0 1 2 3 4 5 60.994

0.999

1.004

Time (s)

ω1(k)

0 1 2 3 4 5 60.18

0.51

0.84δ1(k)

Figure 5.4: Comparison of PF and UKF dynamic state estimation methods in single-

machine-infinite bus system.

0 1 2 3 4 5 6−0.01

0

0.01

δ1(k)−δ1(k)

0 1 2 3 4 5 6−2

0

2x 10

−3

ω1(k)−ω1(k)

PF

UKF

0 1 2 3 4 5 6−2

0

2x 10

−3

Time (s)ω1(k)−ω1(k)

0 1 2 3 4 5 6−0.01

0

0.01

δ1(k)−δ1(k)

Figure 5.5: Variation of dynamic state estimation error with time in single-machine-

infinite bus system.


Time (s) Time (s)

Figure 5.6: Variation of particle filter parameters with time in single-machine-infinite

bus system.

fault. The particles and their weights are also plotted in Figure 5.6. In the following

case studies, we will show that PF provides flawless estimations than UKF when the

covariances of the noise in the PMU measurements are deviate from known initial

values.

5.4.2 Multi-machine IEEE 9-bus 3-generator test system

Here we consider the IEEE-9 bus test system representing a portion of the West-

ern System Coordinating Council (WSCC) 3-Machine, 9-Bus, 3-load system with

IEEE DC1A AVR without PSS. We simulate a 3-phase-to-ground short circuit fault

at t = 1.2s along the line between buses 4 and 5 as shown in Figure 5.7. The fault

is cleared after 6 cycles, i.e. t = 1.3s, by the protection system. The dynamic state

model of the generator l, l ∈ 1, . . . , 3 presented in [28] can be discretized in the

form of (5.22) as follows,

E ′dl(k + 1)=E ′

dl(k) +∆t

T ′ql

(−E ′

dl(k) + (Xqi −X ′ql)iql(k)

)+ µl,1(k),

E ′ql(k + 1)=E ′

ql(k) +∆t

T ′dl

(−E ′

ql(k)− (Xdl −X ′dl)idl(k) + Efdl(k)

)+ µl,2(k),

δl(k + 1)=δl(k) + ∆t(ωl(k)− ωsl

)ωbl + µl,3(k),

ωl(k + 1)=ωl(k) +∆tωsl

2Hl

(Tml − E ′

dl(k)idl(k)− (X ′ql −X ′

dl)idl(k)iql(k)

−E ′ql(k)iql(k)−Dl(ωl(k)− ωsl)

)+ µl,4(k),

Efdl(k + 1)=Efdl(k) +∆t

TEl

(VRl(k)−

(KEl + SEl(Efdl(k))

)Efdl(k)

)+ µl,5(k),


Slack

2 7 8 9 3

65

4

1

163 MW67 MVAR

18 KV1.025 pu

18:230

230 KV

072.0008.078 jZ +=0745.00 jY +=

230 KV 230 KV

230 KV230 KV

230 KV

16.5 KV1.04 pu

230:16.5

0625.027 jZ =

1008.00119.089 jZ +=1045.00 jY +=

230:13.8

13.8 KV1.025 pu

85 MW-10.9 MVAR

100 MW35 MVAR

90 MW30 MVAR

125 MW50 MVAR

161.0

032.0

57j

Z+

=

153.0

0j

Y+

=

085.001.045 jZ +=

088.0

0j

Y+

=

092.0017.046 jZ +=

079.0

0j

Y+

=

0576.0

014

jZ

+=

00

jY

+=

0586.039 jZ =

71.6 MW27 MVAR

17.0

039.0

69j

Z+

=

179.0

0j

Y+

=

00 jY += 00 jY +=

3G2G

1G

Figure 5.7: A portion of the WSCC 3-Machine 9-Bus 3-load system.

Rfl(k + 1)=Rfl(k) +∆t

TFl

(−Rfl(k) +

KFl

TFl

Efdl(k))+ µl,6(k),

VRl(k + 1)=VRl(k) +∆t

TAl

(−KAlKFl

TFl

Efdl(k)− VRl(k)

+KAlRfl(k) +KAl(vrefl − vl(k)))+ µl,7(k). (5.36)

Assume stator winding resistance Rsl of all the generators is 0. By applying Kirch-

hoff’s voltage law to the circuit in Figure 5.2, terminal currents of the lth generator

according to (5.19) and (5.20) can be written as,

idl(k)=1

X ′dl

(E ′

ql(k)− vl(k)cos(δl(k)− θl(k))),

iql(k)=1

X ′ql

(−E ′

dl(k) + vl(k)sin(δl(k)− θl(k))). (5.37)

By substituting (5.24) and (5.37) into (5.36), the dynamic equation (5.36) can be

rewritten in the form of (5.25) as follows,

xl(k + 1)=fl(xl(k), vl(k), θl(k), Tml, vrefl , ηl,1(k), ηl,2(k)) , (5.38)


Using (5.38) we approximate the probability density p (x(k + 1)|x(k)) as follows,

p (xl(k + 1)|xl(k))=N(xl(k + 1), fl

(xl(k), vl(k), θl(k), Tml, vrefl, 0, 0

)+ µl(k), σ

2µl

),

(5.39)

where the process noise µl(k) is normally distributed with 0 mean and variance σ2µl,

i.e., µl(k) ∼ N (0, σ2µl). From (5.18), it follows for l ∈ 1, 2, 3,[

il(k)

γl(k)

]=

√i2dl(k) + i2ql(k)

tan−1(

−idl (k)

iql (k)

)+ δl(k)

. (5.40)

Using (5.24) and (5.37) in (5.40), we can write the observation equation in the form


yl(k)=

[yl,1(k)

yl,2(k)

]=

[il(k) + ηl,1(k)

γl(k) + ηl,2(k)

]=

[il(k)

γl(k)

]

=hl

(xl(k), vl(k), θl(k), ηl,3(k), ηl,4(k)

)+

[ηl,1(k)

ηl,2(k)

]. (5.41)

The measurement noise ηl,i(k), i ∈ 1, . . . , 4 is normally distributed with 0 mean

and variance σ2ηl,i

, i.e., ηl,i(k) ∼ N (·, 0, σ2ηl,i

).

Using (5.10) and (5.41) we approximate the probability density p (yl(k)|xl(k)) as

follows,

p (yl(k)|x1(k))=p (il(k)|xl(k))× p (γl(k)|xl(k))

= N(hl

(xl(k), vl(k), θl(k), 0, 0

), il(k), σ

2il(k)

)×N

(hl

(xl(k), vl(k), θl(k), 0, 0

), γl(k), σ

2γl(k)

). (5.42)

It is necessary to compute the initial condition of all the states and the constant

inputs (Tml) and (vrefl). The steady state operating point is found by performing

a load flow and those steady state solutions which are used as the initial condi-

tions are chosen as, x1(0) = [0 1.0560 0.0625 1.0 1.082 0.195 1.105]T , x2(0) =

[0.622 0.788 1.0664 1.0 1.7890 0.322 1.902]T and x3(0) = [0.624 0.768 0.946 1.0 1.403

0.252 1.4530]T .

Implementation: Initial particles are drawn from the following initial distribution,

xil(k = 0)∼N (·, [xl(0)]7×1, [diag(10

−4)]7×7), i ∈ 1, . . . , N. (5.43)

To implement decentralized estimators, three distributed particle filters are used to

estimate the states of the three generators. All three estimators are independent and


Table 5.1: Noise variances for all PMU voltage and current measurements in 9-bus

system.

Time period 1 Time period 2 Time period 3

(0 to 3 seconds) (3 to 6 seconds) (6 to 10 seconds)

Variance of ηl,1(k) 10−10 10−6 10−5

Variance of ηl,2(k) 10−8 10−5 10−4

Variance of ηl,3(k) 10−10 10−6 10−5

Variance of ηl,4(k) 10−8 10−5 10−4

are designed separately. We choose N = 500 particles with a resampling threshold

Nthr =50% and sampling time ∆t = 20 ms in each estimator. Process noise is as-

sumed to have the following distribution, µl(k) ∼ N(·, [0]7×1 , [diag(10

−7)]7×7

), l ∈

1, 2, 3 for the entire duration. Variances in (5.42) are chosen as, σ2i1(k)

= 2× 10−3,

σ2γ1(k)

= 7×10−3, σ2i2(k)

= σ2i3(k)

= 1×10−3 and σ2γ2(k)

= σ2γ3

= 2×10−3 for the whole

duration. UKF parameters are chosen as, Ql(k) = [diag(10−7)]7×7, l ∈ 1, 2, 3,Rl(k) = [diag(10−8, 10−10)]2×2, l ∈ 1, 2, 3, Pl(0) = [diag(10−3)]7×7, l ∈ 1, 2, 3 and

xlukf (0) = xl(0)+10−3×[rand ∼ (−0.5, 0.5))]7×1, l ∈ 1, 2, 3 for the entire duration.During the implementation noise variance is altered in three time frames as stated

in table 5.1. In order to investigate the performance of the PF and UKF algorithms

for unknown changes in noise covariance, the design parameters for PF and UKF

filters were left unchanged even though noise variances were altered in different time

frames. The rotor angle swings between generator G2 and G1, δ2(k)−δ1(k), and that

of G3 and G1, δ3(k)− δ1(k) are plotted in Figure 5.8. Estimation errors of the rotor

angles also are shown in Figure 5.9. All remaining states of the three generators

are shown in Figures 5.10-5.12. The PF weights are shown in Figure 5.13. As can

be seen in Figures 5.8-5.12, the UKF estimations deteriorate from the start of the

second time frame and progressively getting worse in the third time frame implying

an inability to cope with unknown changes in noise covariances. On the other hand

as evident from the same Figures 5.8-5.12, the PF estimates remain smooth and

continuous in all time frames implying an ability to cope with unknown changes in

noise covariances. Furthermore, the UKF algorithm may fail because error covari-

ance matrix P (k) is no longer positive definite and no solution can be found for

the Cholesky factorization. The performance of the PF algorithm is relatively much

better than UKF in presence of unknown changes in noise covariance.


0 2 4 6 8 100.2

0.6

1

1.4

1.8

δ 2(k)−δ 1(k)

Real state PF est. UKF est.

0 2 4 6 8 100.4

0.6

0.8

1

1.2

1.4

Time (s)

δ 3(k)−δ 1(k)

Real state PF est. UKF est.

Figure 5.8: Estimation of rotor angles of G2 and G3 with respect to rotor angle of

G1 in 9-bus system.

0 2 4 6 8 10−10

−5

0

Est.Error

ofδ 2(k)−δ 1(k)

PF est. error

UKF est. error

0 2 4 6 8 10−12

−8

−4

0

4

Time (s)

Est.Error

ofδ 3(k)−δ 1(k)

PF est. error

UKF est. error

Figure 5.9: Error in estimation of rotor angles of G2 and G3 with respect to rotor

angle of G1 in 9-bus system.


0 2 4 6 8 10−0.2

−0.15

−0.1

−0.05

0

0.05E

′ d1(k)

Real state

PF est.

UKF est.

0 2 4 6 8 101.03

1.05

1.07

1.09

E′ q1(k)

Real state

PF est.

UKF est.

0 2 4 6 8 100.998

1.007

1.016

1.025

ω1(k)

Real state

PF est.

UKF est.

0 2 4 6 8 100.8

1.2

1.5

1.8

2.2

Efd1(k)

Real state

PF est.

UKF est.

0 2 4 6 8 100.15

0.2

0.25

0.3

Rf1(k)

Real state

PF est.

UKF est.

0 2 4 6 8 100.5

1.4

2.3

3.2

4.1

5

Time (s)

E′ d1(k)

Real state

PF est.

UKF est.

Figure 5.10: Variation of estimated states of generator G1 with time in 9-bus system.


0 2 4 6 8 100.35

0.45

0.55

0.65

E′ d2(k)

Real state

PF est.

UKF est.

0 2 4 6 8 100.75

0.87

0.99

1.1

E′ q2(k)

Real state

PF est.

UKF est.

0 2 4 6 8 100.995

1.007

1.019

1.03

ω2(k)

Real state

PF est.

UKF est.

0 2 4 6 8 101.6

1.9

2.2

2.5

2.8

Efd2(k)

Real state

PF est.

UKF est.

0 2 4 6 8 100.32

0.35

0.38

0.41

0.44

Rf2(k)

Real state

PF est.

UKF est.

0 2 4 6 8 101.5

2.2

2.9

3.6

4.3

5

Time (s)

VR2(k)

Real state

PF est.

UKF est.



0 2 4 6 8 100.4

0.52

0.64

0.75

E′ d3(k)

Real state

PF est.

UKF est.

0 2 4 6 8 100.55

0.64

0.73

0.82

0.9

E′ q3(k)

Real state

PF est.

UKF est.

0 2 4 6 8 100.995

1.007

1.019

1.03

ω3(k)

Real state

PF est.

UKF est.

0 2 4 6 8 101.2

1.6

2

2.4

Efd3(k)

Real state

PF est.

UKF est.

0 2 4 6 8 100.22

0.27

0.31

0.36

Rf3(k)

Real state

PF est.

UKF est.

0 2 4 6 8 101

2

3

4

5

Time (s)

VR3(k)

Real state

PF est.

UKF est.




0 2 4 6 8 100

0.4

0.8

1.2

1.6×10−2

w1:N

1(k:k)

0 2 4 6 8 100

1.75

3.5

5.25

7×10−3

w1:N

2(k:k)

0 2 4 6 8 100

0.3

0.6

0.9

1.2×10−2

Time (s)

w1:N

3(k:k)

Figure 1.13: Variation of particle filter weights with time in 9-bus system.Figure 5.13: Variation of particle filter weights with time in 9-bus system.


1

G10

39

9

G2

31

1311

32

G3

10

12

3

4 14

18

Slack

G8

G137

2530

217

26

27

28

G9

38

15

16

G6

3522

21

24

34

G5

36

G7

23

33

G4

20

19

Figure 5.14: IEEE 39-bus, 10-generator (New England) test system.

5.4.3 Multi-machine IEEE 39-bus 10-generator test system

Here we consider the IEEE 39-bus, 10-generator (New England) test system

shown in Figure 5.14. The ten generation units can be categorized into three main

types, type I with manual excitation system without PSS, type II with IEEE DC1A

AVR without PSS and type III with IEEE ST1A AVR with PSS. In Figure 5.14,

generation units 7-10 are of type I, generation units 1-5 are of type II and generation

unit 6 is of type III. All generation units, i.e., l ∈ 1, . . . , 10 have the following

discrete dynamic [28],

E ′dl(k + 1)=

∆t

T ′qol

(−E ′

dl(k) +(Xqi −X ′

ql

)iql(k)

)+ E ′

dl(k),

E ′ql(k + 1)=

∆t

T ′dol

(−E ′

ql(k)−(Xdl −X ′

dl

)idl(k) + Efdl(k)

)+ E ′

ql(k),

δl(k + 1)=∆t(ωl(k)− ωsl

)ωbl + δl(k),


ωl(k + 1)=∆t

2Hl

ωsl

(Tml − E ′

dl(k)idl(k)− E ′ql(k)iql(k)

−(X ′

ql −X ′dl

)idl(k)iql(k)−Dl

(ωl(k)− ωsl

))+ ωl(k). (5.44)

The dynamic of the IEEE DC1A AVR without PSS applicable to generation units

1-5, i.e., l ∈ 1, . . . , 5 is given below [28],

Efdl(k + 1)=∆t

TEl

(−(KEl + Axle

BxlEfdl(k))Efdl(k) + VRl(k)

)+ Efdl(k),

Rfl(k + 1)=∆t

TFl

(−Rfl(k) +

KFl

TFl

Efdl(k)

)+Rfl(k),

VRl(k + 1)=∆t

TAl

(−KAlKFl

TFl

Efdl(k)− VRl(k) +KAlRfl(k) +KAl

(vrefl − vl(k)

))+VRl(k). (5.45)

The dynamic of the IEEE ST1A AVR with PSS applicable to generation unit 6, i.e.,

l = 6 is given below [28],

Efdl(k + 1)=∆t

TRl

(KAl

(vrefl +

KpslT1lT3l

T2lT4l

(ωl(k)− ωsl

)+y1l(k) + y2l(k) + y3l(k)− vl(k)

)− Efdl(k)

)+ Efdl(k),

y1l(k + 1)=∆t

Twl

(T ′l

(ωl(k)− ωsl

)− y1l(k)

)+ y1l(k),

y2l(k + 1)=∆t

T2l

(T ′′l

(ωl(k)− ωsl

)− y2l(k)

)+ y2l(k),

y3l(k + 1)=∆t

T4l

(T ′′′l

(ωl(k)− ωsl

)− y3l(k)

)+ y3l(k), (5.46)

where

T ′l =

−KpslT2wl+KpslTwlT1l+KpslTwlT3l−KpslT1lT3l

(Twl−T2l)(Twl−T4l),

T ′′l =

−KpslTwlT1lT2l+KpslTwlT1lT3l+KpslTwlT22l−KpslTwlT2lT3l

T2l(Twl−T2l)(T2l−T4l),

T ′′′l =

KpslTwlT1lT3l−KpslTwlT1lT4l−KpslTwlT3lT4l+KpslTwlT24l

T4l(Twl−T4l)(T4l−T2l).

Note that for type I manual generation units, the field excitation voltage Efdl(k) in

(5.44) is chosen manually. The parameters of the ten generators, PSS and exciters

are shown in Table 5.2. Assume stator winding resistance Rsl of all the generators is

0. By applying Kirchhoff’s voltage law to the circuit in Figure 5.2, terminal currents

of the lth generator according to (5.19) and (5.20) can be written as,

idl(k)=1

X ′dl

(E ′

ql(k)− vl(k)cos(δl(k)− θl(k)

)),

iql(k)=1

X ′ql

(−E ′

dl(k) + vl(k)sin(δl(k)− θl(k)

)). (5.47)


Table 5.2: Generator, PSS and exciter parameters in 39-bus system.


No. Hl Dl Xdl X ′dl Xql X ′

ql T ′dol T ′

qol Axl Bxl

1 42 4 0.1 0.031 0.069 0.028 10.2 1.5 0.07 0.91

2 30.3 9.75 0.295 0.0697 0.282 0.170 6.56 1.5 0.07 0.91

3 35.8 10 0.2495 0.0531 0.237 0.0876 5.7 1.5 0.07 0.91

4 28.6 10 0.262 0.0436 0.258 0.166 5.69 1.5 0.07 0.91

5 26 3 0.67 0.132 0.62 0.166 5.4 0.44 0.07 0.91

6 34.8 14 0.254 0.05 0.241 0.0814 7.3 0.4 0.07 0.91

7 26.4 8 0.295 0.0491 0.292 0.186 5.66 1.5 0.07 0.91

8 24.3 9 0.29 0.057 0.280 0.0911 6.7 0.41 0.07 0.91

9 34.5 14 0.2106 0.057 0.205 0.0587 4.79 1.96 0.07 0.91

10 248 33 0.296 0.006 0.0286 0.005 5.9 1.5 0.07 0.91

PSS Parameters

No. KA6 Kps6 T16 T26 T36 T46 Tw6 TR6 Vref6

6 200 5 0.1 0.2 0.1 0.25 40 0.01 1.0605

Exciter Parameters

No. KAl TAl KEl TEl KFl TFl vrefl

1 40 0.2 1 0.785 0.063 0.35 1.087

2 40 0.2 1 0.785 0.063 0.35 1.097

3 40 0.2 1 0.785 0.063 0.35 1.069

4 40 0.2 1 0.785 0.063 0.35 1.074

5 40 0.2 1 0.785 0.063 0.35 1.369

By substituting (5.24) and (5.47) into relevant generator dynamic equations (5.44)-(5.46),

the dynamics of all generation units can be rewritten in the form of (5.25) as follows,

xl(k + 1)=fl

(xl(k), vl(k), θl(k), Tml, vrefl, ηl,3(k), ηl,4(k)

). (5.48)

Using (5.48) and a process noise covariance value σ2µl, we approximate the probability

density p (x(k + 1)|x(k)) as follows,

p

(xl(k + 1)|xl(k)

)=N

(xl(k + 1), fl

(xl(k), vl(k), θl(k), Tml, vrefl, 0, 0, 0, 0

)+ µl(k), σ

2µl

).

(5.49)

From (5.18), it follows for l ∈ 1, 2, . . . , 10,[il(k)

γl(k)

]=

√i2dl(k) + i2ql(k)

tan−1(

−idl (k)

iql (k)

)+ δl(k)

. (5.50)


Using (5.24) and (5.47) in (5.50), we can write the observation equation in the form


yl(k) =

[yl,1(k)

yl,2(k)

]=

[il(k) + ηl,1(k)

γl(k) + ηl,2(k)

]=

[il(k)

γl(k)

]

= hl

(xl(k), vl(k), θl(k), ηl,1(k), ηl,2(k), ηl,3(k), ηl,4(k)

).

(5.51)

The measurement noise ηl,i(k), i ∈ 1, . . . , 4 is normally distributed with 0 mean

and covariance σ2ηl,i

, i.e., ηl,i(k) ∼ N (·, 0, σ2ηl,i

). Using (5.10) and (5.51) we approxi-

mate the probability density p (yl(k)|xl(k)) using variances σ2iland σ2

γlas follows,

p

(yl(k)|xl(k)

)=p

(il(k)|xl(k)

)× p

(γl(k)|xl(k)

)=N

(hl

(xl(k), vl(k), θl(k), 0, 0, 0, 0

), il(k), σ

2il

)×N

(hl

(xl(k), vl(k), θl(k), 0, 0, 0, 0

), γl(k), σ

2γl

). (5.52)

The initial conditions for the states of all generators are found by performing a load

flow considering the steady state behavior of all the generators and also considering

the active and reactive power data for all the buses in the IEEE 39-bus, 10-generator

test system data given in Matpower toolbox [29].

Implementation: Initial particles are drawn from the following initial distribution,

xil(k = 0)∼N

(·, [xl(0)]nl×1, [diag(10

−6)]nl×nl

), i ∈ 1, . . . , N, (5.53)

where nl is the number of states of generator l, l ∈ 1, . . . , 10. To implement de-

centralized estimators, ten distributed particle filters are used to estimate the states

of the ten generators. All ten estimators are independent and are designed sepa-

rately. Based on the number of states that require estimation, we choose N = 400

for type I, N = 500 for type II and N = 600 for type III generators with a re-

sampling threshold Nthr =50%. A resampling threshold of Nthr =50% is generally

chosen in particle filters, see [30]. The choice of the number of particles for the

different type of generators were based on the minimum number of particles re-

quired to produce an accurate estimate of the states. Variances in (5.52) which

approximate the likelihood density function for all 10 generators are chosen as

σ2il= σ2

γl= 2 × 10−3, l ∈ 1, . . . , 10 and also covariances in (5.49) which approx-

imate the transitional density function for type I, type II and type III generators


Table 5.3: Noise variances for all PMU voltage and current measurements in 39-bus

system.

Time period 1 Time period 2 Time period 3

(0 to 3 seconds) (3 to 8 seconds) (8 to 15 seconds)

Variance of ηl,1(k) 10−10 10−8 10−6

Variance of ηl,2(k) 10−8 10−6 10−5

Variance of ηl,3(k) 10−10 10−8 10−6

Variance of ηl,4(k) 10−8 10−6 10−5

are chosen as σ2µ3

= [diag(10−20, 10−20, 10−11, 10−11, 10−11, 10−11, 10−20)]7×7, σ2µ6

=

[diag(10−8, 10−8, 10−11, 10−11, 10−14, 10−15, 10−15, 10−15)]8×8, σ2µ10

= [diag(10−20, 10−20

, 10−11, 10−11)]4×4, respectively. In fact, those covariances can be regarded as the de-

sign parameters of the particle filter algorithm and were held constant for the whole

duration of the state estimation process. For comparison we also implemented the

UKF algorithm presented in [15] (same algorithm as in Chapter 4) with parame-

ters chosen as, Ql(k) = [diag(10−10, 10−8)]2×2 and Rl(k) = [diag(10−10, 10−8)]2×2, l ∈1, . . . , 10 for the entire duration. Here we present the state estimation results for

each type of generator: Type I (generator number 10), Type II (generator number

6) and Type III (generator number 3). Also note that the three considered gener-

ators have the same generator parameters as those used in [15], and also the noise

covariances Ql(k) and Rl(k) are also the same as those reported in [15]. During the

process of state estimation we varied the noise covariance is in three time frames

while still keeping the parameters of the UKF algorithm [15] and also the param-

eters of particle filter algorithm unchanged for the whole duration of time. White

noise variance of voltage and current measurements from PMUs for the three time

periods are shown in Table 5.3. We also triggered a fault at 1.5 s on the line con-

necting bus 14 to bus 15 which was cleared by removing that line. A sampling time

of ∆t = 20ms was chosen for PMU measurements in Type I generator (generator

10) and Type III generator (generator 6), the estimation of states by particle filter

and UKF are shown in Figure 5.15 and Figure 5.16. A sampling time of ∆t = 30ms

was chosen for Type II generator (generator 3), the estimation of states by particle

filter and UKF for this Type II generator are shown in Figure 5.17.

As can be seen in Figures 5.15-5.17, the UKF estimations deteriorate from the

start of the second time frame and progressively getting worse in the third time

frame with changing the noise covariances in the PMU measurements while the PF

estimates remain smooth and continuous in all time frames implying an ability to


0 3 6 9 12 15

0.214

0.216

0.217

0.218

E′ d10(k)

0 3 6 9 12 15

10146

10148

10150

10152

×10−4

E′ q10(k)

Real state

PF est.

UKF est.

0 3 6 9 12 15

0

1

2

3

δ 10(k)

0 3 6 9 12 15

10000

10005

10010

×10−4

Time (s)

ω10(k)

Real state

PF est.

UKF est.

Figure 5.15: State estimation of generator 10 with manual excitation in 39-bus

system.


0 3 6 9 12 15

0.481

0.483

0.485E

′ d6(k)

0 3 6 9 12 15

1.04

1.045

1.05

E′ q6(k)

0 3 6 9 12 15

1

2

3

4

δ 6(k)

Real state

PF est.

UKF est.

0 3 6 9 12 15

10000

10005

10010

×10−4

Time (s)

ω6(k)

Figure 5.16: State estimation of generator 6 with IEEE ST1A AVR and PSS in

39-bus system.


0 3 6 9 12 15

2.1

2.2

2.3

2.4E

fd6(k)

0 3 6 9 12 15

−0.8

−0.4

0

×10−3

y 16(k)

Real state

PF est.

UKF est.

0 3 6 9 12 15

−3.6

−2.4

−1.2

0

×10−3

y 26(k)

Real state

PF est.

UKF est.

0 3 6 9 12 15

0

3.5

7

×10−3

Time (s)

y 36(k)

Real state

PF est.

UKF est.

Figure 5.16: State estimation of generator 6 with IEEE ST1A AVR and PSS in

39-bus system (continued).


0 3 6 9 12 15

0.444

0.447

0.45E

′ d3(k)

Real state

PF est.

UKF est.

0 3 6 9 12 15

1.008

1.014

1.02

E′ q3(k)

Real state

PF est.

UKF est.

0 3 6 9 12 15

1

2

3

4

δ 3(k)

Real state

PF est.

UKF est.

0 3 6 9 12 15

10000

10004

10008

×10−4

Time (s)

ω3(k)

Real state

PF est.

UKF est.

Figure 5.17: State estimation of generator 3 with IEEE DC1A AVR in 39-bus sys-

tem.


0 3 6 9 12 15

2.234

2.252

2.27

2.288

Efd3(k)

Real state

PF est.

UKF est.

0 3 6 9 12 15

0.402

0.405

0.408

0.411

Rf3(k)

Real state

PF est.

UKF est.

0 3 6 9 12 15

3.43

3.51

3.59

Time(s)

VR3(k)

Real state

PF est.

UKF est.

Figure 5.17: State estimation of generator 3 with IEEE DC1A AVR in 39-bus system

(continued).


Table 5.4: Comparison of PF and UKF computational time in 39-bus system.

UKF Particle Filter

No. of particles - 400 500 600

Relative computation time units 1 23.2 33.3 42.7

Matlab computational time (ms) 0.3 6.96 10.0 12.81

cope with unknown changes in noise covariances. Furthermore, the UKF algorithm

may fail because error covariance matrix P (k) is no longer positive definite and no

solution can be found for the Cholesky factorization. In our simulation study of

the UKF we observed that while P (k) did not become negative definite, at times

it was positive semi-definite, for instance P (0) is positive semi-definite according

to the UKF algorithm in [15]. As evident from Figures 5.15-5.17, the performance

of the PF algorithm is relatively much better than UKF in presence of unknown

changes in noise covariance. The performance of the particle filter also did not

deteriorate by increasing the sampling time up to 30ms in estimating states of the

Type II generator, see Figure 5.17. An ability to estimate the states accurately

with a higher sampling period of 30ms in comparison to 20ms also imply a 33.3%

reduction in sampling time which is significant in a real-time estimation scheme. The

particle filter require more computational time than the UKF, Table 5.4 presents a

relative comparison of the computational time for PF and UKF based on a Matlab

implementation. While MATLAB is not a favourable choice (hence not considered)

for real-time implementation of PF or UKF algorithms, in Table 5.4 we show the

associated computational time achievable in MATLAB on a Intel Core i7 CPU

personal computer to present a relative comparison of the computations associated

with PF and UKF algorithms. As is evident from Table 5.4, the computational time

of PF algorithm is 23.2 to 42.7 times greater in comparison to that of UKF, and it is

largely due to the 400 to 600 particles that PF needs to process in comparison to 12

to 20 sigma points that UKF needs to process. However, real-time implementation of

the PF algorithm can be carried out using a low-level language on faster processors

that can make the computational times smaller than 10ms, see for instance [31] and

references therein where entire studies are dedicated to the real-time implementation

of PF algorithms. Those papers also report cases of implementing 5000 particles

which require less than 10ms to process, and processing 400 to 600 particles inside a

sampling period of 20ms or 30ms is certainly feasible given the faster processors that

are accessible with the current technology. Given that the required computations

5.5. Dealing with Bad Data 152

can be carried out within the sampling period, computational time associated with

the PF algorithm does not present a disadvantage, and it is computationally feasible

to implement in a real-time processing system. The simulation results reported in

this chapter for the three types of generators are indicative of the observed results

for all the other generators as well. While the performance of UKF algorithm in

DSE of generators can be satisfactory in certain scenarios (see [15]), there are also

instances where its performance can be far from satisfactory as identified in this

section, hence the proposed PF algorithm can be viewed as an useful alternative in

DSE of generators.

5.5 Dealing with Bad Data

Gross error which is due to bad data (outlier) is another issue in PMUs that needs

to be addressed apart from measurement noise. A bad data detection algorithm is

necessary to rectify such bad data. Let us define discrete Bernoulli random variable

ol,d(k), d = 1, 2 to represent whether the dth measured variable yl,d(k) from PMU in

the lth generation unit is a bad data or not as follows,

ol,d(k) =

0 yl,d(k) is not a bad data

1 yl,d(k) is a bad data(5.54)

Taking p (ol,d(k) = 1) = εl as prior probability of bad data being present in generator

l and observation d, we can write the probability of ol,d(k) = 1 given yl,d(k) as

follows [32],

pl,d (ol,d(k) = 1|yl,d(k))=N∑i=1

wil (k|k)

εlN(eil,d(k), 0, β

2l,dς

2l,d

)(1− εl)N

(eil,d(k), 0, ς

2l,d

)+ εlN

(eil,d(k), 0, β

2l,dς

2l,d

) ,(5.55)

where

eil,d(k) = yl,d(k)− hl,d

(xil(k), vl(k), θl(k), 0, 0

), (5.56)

and ς2l,d is generally the variance of measurement noise and βl,d is a factor that

accounts for the substantially larger standard deviation associated with the outliers

and in specific application these parameters can be chosen for a good detection. For

instance choosing a larger β value will account for detecting outliers further from a

true measurement. Let the estimated measurement yl(k) of the lth generation unit

can be written as follows,

yl(k) =N∑i=1

wil (k|k)hl

(xil(k), vl(k), θl(k), 0, 0

). (5.57)


We can categorize the dth measurement as bad data if pl,d (ol,d(k) = 1|yl,d(k)) > 0.5

where the value of 0.5 is chosen arbitrarily (and can be changed if necessary), if

a higher value is chosen then bad data may not be detected and on the other

hand if a lower value is chosen then there is the possibility of false alarms. Since

yl(k) = il(k)∠γl(k), any bad measurements (outliers) of il(k) and γl(k) can be de-

tected based on the proposed method. Bad measurements of vl(k) and θl(k) can also

be detected based on the proposed method because any bad measurements (outliers)

of vl(k) and θl(k) will affect estimation of the states, and also the estimated obser-

vation according to (5.57) will differ significantly from the measured values yl(k).

When bad data is detected, estimated value of the measurements yl(k) according

to (5.57) are picked up as the correct value and corresponding measurement yl(k)

is tagged as an outlier. In order to rectify bad measurements Algorithm 5.1 needs

to be modified. Accordingly a complementary detection strategy and rectification

is augmented to the beginning of step 2 of Algorithm 5.1 which is presented in Al-

gorithm 5.2. To show the effectiveness of the bad data rectification algorithm, the

Algorithm 5.2 Particle filter bad data rectification

Step 2A: Mean estimation and bad data detection/rectification

• Estimate state mean according to x(k) =∑N

i=1 wi(k|k)x

i(k).

• Calculate marginal posterior probability pl,i (ol,i(k) = 1|yl,i(k)) for each mea-

surement j according to (5.55).

• If pl,1 > 0.5 and pl,2 < 0.5 then calculate yl,1(k) from (5.57), put yl,1(k) =

yl,1(k) and recalculate and normalize the weights.

• If pl,2 > 0.5 and pl,1 < 0.5 then calculate yl,2(k) from (5.57) and put yl,2(k) =

yl,2(k) and recalculate and normalize the weights.

• If pl,1 > 0.5 and pl,2 > 0.5 then calculate yl,1(k) and yl,2(k) from (5.57),

put yl,1(k) = yl,1(k), yl,2(k) = yl,2(k), calculate the estimates of (vl(k), θl(k))

according to (5.47), take those estimates as the measurements (vl(k), θl(k))

and recalculate and normalize the weights.

• Go to Step 2 of Algorithm 5.1.

39-bus system presented in case study 5.4.3 is considered with a given outlier value

of -0.2 in the measurement v3(k) at time t = 4s and another outlier value of +0.2


0 3 6 9 12 150

0.5

1p3,1(k)

0 3 6 9 12 150

0.5

1

Time (s)

p3,2(k)

Figure 5.18: Probability of bad data in measurement 1 and measurement 2 of gen-

erator G3 in 39-bus system.

in the measurement y3,1(k) = i3(k) at time t = 9s. Measurement noise covariances

were also altered in the three time frames as described in Section 5.4. The design

parameters have been chosen such that ς23,1 = ς23,2 = 0.03, β23,1 = β2

3,1 = 10 and

εl = 0.05. The probabilities p3,1(o3,1(k) = 1|i3(k)

), p3,2

(o3,2(k) = 1|γ3(k)

)and the

estimation of rotor angular velocity of the 3rd generator are shown in Figures 5.18

and 5.19 respectively. As evident from Figure 5.18 the proposed estimation scheme

can detect the measurement outliers at t = 4s and t = 9s very well. As evident from

Figure 5.19 the measurement outliers at t = 4s and t = 9s have been rectified with

the particle filter estimation remaining smooth and continuous throughout including

at those time instances. Other methods of dealing with bad data include generation

of residual signal when bad data is present in the output measurements, see [33].

However, this technique of residual signal based bad data detection is studied based

on a linear model of the system under consideration and further work is required to

make such techniques applicable to nonlinear systems as considered in this chapter.

5.6. Conclusion 155

0 3 6 9 12 15

10000

10004

10008

×10−4

Time (s)

ω3

Real state

PF est.

Figure 5.19: Estimation of w3 in generator G3 in the presence of bad data and

variable noise covariance in 39-bus system.

5.6 Conclusion

This chapter has developed a powerful decentralized method for dynamic state

estimation of a power system based on the particle filter approach. The proposed

estimation scheme complements the previous schemes which are based on Kalman

filter approach. It is shown that the proposed scheme in this chapter can provide

accurate estimation of the states even when the noise covariance deviates from known

initial values. Moreover, the proposed scheme is applicable for systems subjected to

Gaussian as well as non-Gaussian noise.

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160CHAPTER 6

DSE Approach to Fault Detection with

Application to Wind Turbines

This chapter is based on a published article with the following details:

Kianoush Emami, Tyrone Fernando, Brett Nener, Hieu Trinh

and Yang Zhang, “A Functional Observer Based Fault Detection

Technique for Dynamical Systems,” Journal of the Franklin Insti-

tute, vol. 352, no. 5, pp. 2113–2128, 2015.


This chapter presents a functional observer based fault detection

method. The fault detection is achieved using a functional observer

based fault indicator that asymptotically converges to a fault indi-

cator that can be derived based on the nominal system. The asymp-

totic value of the proposed fault indicator is independent of the func-

tional observer parameters and also the convergence rate of the fault

indicator can be altered by choosing appropriate functional observer

parameters. The advantage of using this new method is that the ob-

served system is not necessarily needed to be observable; therefore,

the proposed fault detection technique is also applicable for systems

where state observers cannot be designed; moreover, the functional

observer fault detection scheme is always of reduced order in com-

parison to a state observer based scheme.

6.1 Introduction

Fault detection is an important area of study because many processes, if not

all, are subject to faults at some point in its lifetime. Some of these system faults

that can occur may be catastrophic, for instance, according to the US office of the

Secretary of Defence, about 80 per cent of flight incidents concerning unmanned

aerial vehicles are due to faults occurring in the actuators, sensors or faults due to

changes in inner parameters of the system dynamics [1]. Fault detection is not only

important in aerospace applications but also in many other applications such as


automobiles, trains, chemical and process systems, power generation applications to

state a few. In such applications, especially when safety is of paramount importance,

detecting faults in a timely manner and then taking corrective action to mitigate

the fault is the key to avoiding unwanted consequences. The subject of taking

corrective action once a fault is detected is a separate area of study in itself, and

in the literature it is often refereed to as Fault Tolerant Control (FTC). The focus

of this chapter is not on FTC, rather on the detection of faults, in particular faults

occurring in actuators and faults due to changes in inner parameters of the system

dynamics.

While the controllers can compensate for many types of disturbances, there are

some changes in the process which the controllers cannot handle adequately, these

changes are called faults. More precisely, a fault is any kind of malfunction in

the actual dynamic system, the plant, that leads to an unacceptable anomaly in

the overall system performance [2]. A fault may happen in every system due to

an internal event, change in environmental conditions, error in the design of the

system, sensor failure or actuator malfunctioning. Fault detection can be achieved

by a residual generator, and typically this residual generator constitute an observer.

For other methods of fault detection such as data-based techniques see [3, 4] or

fault detection based on key performance indicators see [5]. The methods based on

process of residual generation uses a model of the system, known input signals, and

the measured outputs to predict the behaviour of the system which is the method

reported in this chapter. A residual signal which we will refer to as fault residual

can be generated by comparing the measured system output with the predicted

system output. Obviously, the fault residual should be close to zero in fault-free

condition, and deviate from zero in the presence of a fault. Given that residual

signal generation is at the core of fault detection, a considerable effort has been

devoted to the generation of the fault residual signal using various forms of observers:

state observers [6–10], high gain observers [11, 12], sliding mode observers [13–15],

extended Luenberger observers [16], H∞ observers [17, 18] and adaptive observers

[19, 20]. For an excellent survey of fault detection and identification methods, see

[1, 2].

In this chapter, we consider the generation of the fault residual signal using

a functional observer. A functional observer is a more general form than a state

observer, and by using a functional observer for residual signal generation it is

possible to bring many of the advantages associated with functional observers to

fault detection. Functional observer is essentially a reduced order observer [21, 22],

6.2. Fault Residual In Actual System 162

using a functional observer we only estimate a desired linear function of the state

vector in comparison to estimating the entire state vector as in a state observer,

consequently, the order of a designed functional observer can be much less than the

order of a state observer. Furthermore, in the process of fault detection via residual

signal generation, only the estimate of the output is required and it is more logical to

estimate the desired estimate of the output directly using a functional observer than

estimating all the individual states, and then linearly combining those individual

estimates of the states to construct an estimate of the output. Apart from having

higher order, a state observer has other disadvantages as well, the system needs

to observable in order to estimate all the system states whereas state observability

is not a requirement in the design of functional observers, the system only has to

be functional observable to be able design a functional observer, see [21, 22]. The

functional observability requirement is less stringent than the state observability

requirement. This chapter is the extension of the previous work in [23] when the

system is not functional observable.

The results presented in this chapter are motivated originally by the work in [24].

In [24] a state observer based fault detection algorithm is reported which can detect

faults in the actuator and faults due to changes in inner parameters of the system

dynamics. Unlike the results in [24], in this chapter we propose to use a functional

observer to detect system faults, and it is not limited to observable systems. With

the proposed method we will show in this chapter it is always possible to design

a fault detection observer irrespective of the system being observable or not with

at least one output. The proposed fault residual also has an asymptotic value

independent of the observer parameters and the rate of convergence of the fault

residual can be altered by choosing appropriate observer parameters. The residual

generating functional observer has a reduced structure than a state observer and

has reduced dynamics.

The chapter is organised as follows: Section 6.2 discusses two fault residuals,

a nominal system based fault residual and also a functional observer based fault

residual. In Section 6.3, a modified fault residual is introduced and its properties

are discussed. In Section 6.4 we consider two unobservable systems to numerically

illustrate the theory presented in this chapter.

6.2 Fault Residual In Actual System

In the following subsections we introduce actual and nominal systems, respec-

tively:


6.2.1 Actual system

The following uncertain system with p outputs and m inputs with unpredictable

fault signals f(t) ∈ Rm entering from the system input is defined as the actual

system and it is expressed as follows:

x(t) =(A+∆A

)x(t) +

(B +∆B

)u(t) +Df(t),

y(t) = Cx(t),(6.1)

where A ∈ Rn×n, B ∈ Rn×m and C ∈ Rp×n are known nominal system parameters,

∆A ∈ Rn×n and ∆B ∈ Rn×m are unknown deviations from nominal values A and

B respectively, and D ∈ Rn×m is a constant unknown matrix. x(t) ∈ Rn, y(t) ∈ Rp

and u(t) ∈ Rm are the state, output and the input vectors respectively. It is also

assumed that limt→∞ f(t) = f(∞) is the final value of the fault, and it is an unknown

constant.

6.2.2 Nominal System

Since the nominal values of the system parameters A,B and C are known, we

can consider the following system dynamics without any fault

x∗(t) = Ax∗(t) +Bu(t),

y∗(t) = Cx∗(t),(6.2)

where x∗(t) ∈ Rn and y∗(t) ∈ Rp are the state and output of the faultless system

(6.2). The output y∗(t) for any given input u(t) can be computed if the initial system

states x∗(0) is known. Obviously, for any given u(t) we have x(t) = x∗(t), y(t) =

y∗(t) ∀ t > 0 if ∆A = 0, ∆B = 0 and f(t) = 0. Furthermore, given that the

functional observer is only used for the purpose of fault detection, we can assume

that A is stable and also invertible because if not, a control system can be designed

to ensure that A is stable and also invertible before proceeding with the task of fault

detection.

6.2.3 Fault Residuals

Now consider the following functional observer of the form

˙z(t) = Nz(t) + Jy(t) +Hu(t), (6.3)

where z(t) ∈ Rr is an asymptotic estimate of

z(t) = Lx(t), (6.4)


and L ∈ Rr×n is any given known matrix, N ∈ Rr×r, J ∈ Rr×p and H ∈ Rr×m are

functional observer parameters. We may define functional observer estimation error

as

e(t) = z(t)− z(t) = Lx(t)− z(t), (6.5)

and considering (6.1) and (6.3), the estimation error dynamics can be written as

follows

e(t) = Ne+(LA+ L∆A−NL− JC

)x(t) +

(LB + L∆B −H

)u(t) + LDf(t).

(6.6)

If

∆A = 0,∆B = 0, f(t) = 0, (6.7)

LA−NL− JC = 0, (6.8)

LB −H = 0, (6.9)

then (6.6) becomes,

e(t) = Ne(t). (6.10)

Now, if we let

L = C, (6.11)

then

z(t) = y(t), (6.12)

and we can define y(t) as

y(t) = z(t), (6.13)

which is an asymptotic estimate of y∗(t) if (6.7)-(6.9) are satisfied and N is Hurwitz.

We can now define two fault residuals ro(t) and rn(t) as follows

ro(t) = y(t)− y(t), (6.14)

and

rn(t) = y(t)− y∗(t). (6.15)

Note that the fault residual ro(t) is based on an estimate of the system output from

the functional observer, and the second fault residual rn(t) is based on the nominal

faultless model. Figure 6.1 shows the computation of the two fault residuals ro(t)

and rn(t).


Figure 6.1: Fault detection process

Remark 6.1. If x(0) is known then rn(t) = 0, ∀t ≥ 0 if (6.7) is satisfied. However,

x(0) is not known, therefore even if (6.7) is satisfied rn(t) 6= 0, ∀ 0 ≤ t < ∞.

Furthermore, even if (6.7) is satisfied, the rate of convergence of rn(t) cannot be

altered. On the other hand if (6.7)-(6.9) are satisfied then from (6.10) it is clear that

the convergence of ro(t) can be altered by choosing the eigenvalues of the functional

observer parameter N .

Remark 6.2. The functional observer (6.3) considered in this chapter is a special

form of the functional observer reported in [25], in particular by letting E = 0 in

the functional observer structure proposed in [25], we obtain the functional observer

form (6.3).

The necessary and sufficient functional observer existence conditions reported

in [25] can then be easily rewritten as follows

rank

LA

C

L

= rank

[C

L

], (6.16)

and

rank

sL− LA

CA

C

= rank

CA

C

L

∀s ∈ C, Re(s) ≥ 0. (6.17)


Condition (6.17) is equal to observability of (F,G) where

F = LAL+ − LAC+CL+ (6.18)

G =(I − CC+

)CL+ (6.19)

and ’+’ denotes the Moore-Pnenrose generalized inverse, and A and C are

A = A(I − L+L

)C = C

(I − L+L

). (6.20)

The functional observer parameter N and also the matrix K can be obtained by

any pole placement method from

N = F −KG, (6.21)

and furthermore, the functional observer parameter J and H can be obtained from

J = LAC+ +K(I − CC+

)H = LB

(6.22)

Remark 6.3. For further detail on the functional observer existence conditions

(6.16) and (6.17), and the derivation of the functional observer parameters N and

J , we refer the reader to [25]. It is also shown in [25] that (6.16) and (6.17) are in

fact the necessary and sufficient conditions for the satisfaction of (6.8) and (6.9).

6.2.4 Asymptotic Value of Fault Residuals

In this subsection we consider the asymptotic value of the fault residuals ro(t)

and rn(t), i.e. ro(∞) = limt→∞ ro(t) and rn(∞) = limt→∞ rk(t). Let x be the differ-

ence between x and x∗,

x(t) = x(t)− x∗(t) → ˙x(t) = x(t)− x∗(t),

and from (6.1) and (6.2) we have,

˙x(t) =(A+∆A

)x(t) +

(B +∆B

)u(t) +Df(t)− Ax∗(t)−Bu(t)

= Ax(t) + ∆Ax(t) + ∆Bu(t) +Df(t).(6.23)

Lemma 6.4. The asymptotic value of rn(t) is

rn(∞) = limt→∞

rn(t) = −CA−1Ft, (6.24)

where Ft is obtained from

Ft = limt→∞

(∆Ax(t) + ∆Bu(t) +Df(t)

). (6.25)


Proof. Considering (6.23) in frequency domain we have

sX(s) = AX(s) + ∆AX(s) + ∆BU(s) +DF (s),

where L(x) = X(s) and L(·) is the Laplace transform of (·), which leads to

X(s) =(sI − A

)−1(∆AX(s) + ∆BU(s) +DF (s)

). (6.26)

Using the final value theorem we get

limt→∞

x(t) = lims→0

sX(s)

= lims→0

(sI − A

)−1

lims→0

s(∆AX(s) + ∆BU(s) +DF (s)

)= −A−1 lim

t→∞∆Ax(t) + ∆Bu(t) +Df(t)

= −A−1Ft.

(6.27)

Since rn(t) = y(t)− y(t)∗ = Cx(t), we obtain

rn(∞) = limt→∞

rn(t) = −C limt→∞

x(t) = −CA−1Ft. (6.28)

Remark 6.5. The result in Lemma 6.4 is reported in [24] without its proof, although

the proof is trivial we have shown it here for completeness.

Now let

z(t) = Lx(t)− z(t), (6.29)

then from (6.1)-(6.4) we get

˙z(t) = Lx(t)− ˙z(t)

= Lx(t)− ˙z(t)

= L((

A+∆A)x(t) +

(B +∆B

)u(t) +Df(t)

)−Nz(t)− Jy(t)−Hu(t) +NLx(t)−NLx(t)

= Nz(t) +(LA−NL− JC

)x(t) +

(LB −H

)u(t)

+ L(∆Ax(t) + ∆Bu(t) +Df(t)

).

(6.30)

Using (6.18)-(6.22) in (6.30) we obtain

˙z(t) = Nz(t) + L(∆Ax(t) + ∆Bu(t) +Df(t)

). (6.31)


Lemma 6.6. The asymptotic value of ro(t) is

ro(∞) = limt→∞

ro(t) = −N−1LFt, (6.32)

where Ft is defined as in (6.25).

Proof. Taking the Laplace Transform of (6.31) we get

Z(s) =(sI −N

)−1

C(∆AX(s) + ∆BU(s) + Ff(s)

).

If L = C as in (6.11) then z(t) = y(t) as in (6.13), and furthermore z(t) in (6.29)

reduces to

z(t) = ro(t). (6.33)

Since the observer matrix N can be chosen to be Hurwitz and also invertible, using

the final value theorem [26], we get

limt→∞

ro(t) = limt→∞

z(t) = lims→0

sZ(s)

= −N−1L limt→∞

(∆Ax(t) + ∆Bu(t) +Df(t)

)= −N−1LFt.

(6.34)

Remark 6.7. Asymptotic value of the nominal system based fault residual, i.e.,

rn(∞) and the asymptotic value of the functional observer based fault residual, i.e.,

ro(∞) are not the same. In fact, ro(∞) is also dependent on the functional observer

parameter N . Clearly, triggering faults based on ro(t) is influenced by the choice of

observer parameter N which is undesirable.

6.2.5 Modified Fault Residual

In this subsection we propose a new solution by modifying the functional observer

based fault indicator ro(t) so that

i) The asymptotic value of the modified fault indicator is independent of the

functional observer parameters, and

ii) The asymptotic value of the modified fault indicator is identical to the asymp-

totic value of the fault indicator based on the nominal system, i.e., rn(∞),

and

iii) The rate of convergence of the modified fault indicator can be altered by

choosing the eigenvalues of the functional observer parameter N .


The proposed solution is stated in the following theorem:

Theorem 6.8. If functional observer based fault residual ro(t) is pre-multiplied by

the factor

M = CA−1(N−1L

)++ Z

(I −

(N−1L

)(N−1L

)+), (6.35)

where Z is any arbitrary matrix, then the asymptotic value of Mro(t) is

limt→∞

Mro(t) = rn(∞) = −CA−1Ft. (6.36)

Proof. The factor M is given by the solution to the following equation

M(N−1L) = CA−1. (6.37)

According to general solution of linear matrix equations [27], the solution of (6.37)

exists in the form of (6.35) if and only if

CA−1(I −

(N−1L

)(N−1L

)+)= 0 (6.38)

or equivalently

rank

[CA−1

N−1L

]= rank

[N−1L

]. (6.39)

Performing elementary row operations on both sides of (6.39) we get

rank

[CA

L

]= rank

[L], (6.40)

which is the observer existence condition (6.16) with L = C that is satisfied.

Remark 6.9. The asymptotic value of the modified fault residual Mro(t) as in (6.36)

is independent of the observer parameters, and also it is equal to the asymptotic value

of rn(t) as in (6.28). Furthermore, the convergence rate of Mro(t) can be altered by

choosing the eigenvalues of N .

Remark 6.10. Choosing L = C gives a functional observer of order p provided

that (6.16) and (6.17) are satisfied. However, (6.16) and (6.17) may not be always

satisfied and if not satisfied then we need to seek a functional observer of order

greater than p. In the next section we discuss how to design a functional observer

of order greater than p for the purpose of fault detection.

6.3. Higher Order Functional Observers For Fault Detection 170

6.3 Higher Order Functional Observers For Fault Detection

Here we consider the existence of an observer order greater than p, let us first

consider a (p + q)-order functional observer where 0 ≤ q ≤ n − p. In section 2, L

was specifically chosen as C, and the order of the observer was equal to the number

of rows of L which is p. In this section we allow L to be a built matrix that includes

C in its rows. We choose L to have two components, C and R (where R has the

same number of columns as C) so that

L =

[C

R

]. (6.41)

The role of R (if required) is to ensure that the built matrix L is such that (6.16)

and (6.17) are satisfied. The presence of C in L ensures that z(t) = Lx(t) will

contain the required estimate y(t) in its rows, hence an estimate of z(t) constitutes

an estimate of y(t). Since the order of the functional observer is equal to the number

of rows of L, with L as in (6.41), the order of the functional observer is greater than

p.

Lemma 6.11. [28] - The triple (A,C, L) is Functional Observable if and only if

rank

C

CA...

CAn−1

L

LA...

LAn−1

= rank

C

CA...

CAn−1

. (6.42)

Remark 6.12. If L = C 6= ∅ then according to lemma 6.11, the triple (A,C, L)

is always functional observable, i.e., we can always design a functional observer to

estimate y(t). Furthermore, from lemma 6.11 it follows that in order to design a

functional observer, it is necessary to choose R such that

(A,C,

[C

R

])is also

Functional Observable.

Remark 6.13. The functional observability condition stated in lemma 6.11 was

derived for a more general form of observer than the one considered in (6.3), the

functional observability condition (6.42) is also still applicable for the form consid-

ered in (6.3).


Let us define Y and χ as follows

Y =

C

CA...

CAn−1

, (6.43)

χ = rank(Y )− p. (6.44)

Lemma 6.14. The triple (A,C,X) is Functional Observable if and only if rows of

X belong to R(Y ) where R(·) is the rowspace of (·).

Proof: If R(X) ⊆ R(Y ) then R(XAj) ⊆ R(Y ) ∀j ∈ J, so lemma 6.11 is satisfied

for L = X. On the other hand if R(X) 6⊆ R(Y ) then rank

[X

Y

]6= rank(Y ), hence

left-hand-side of lemma 6.11 is greater than its right-hand-side for L = X.

Let

κ ∈ Rχ×rows(Y ) (6.45)

and[[Y]]

be a matrix of row basis vectors for the row space of Y .

Lemma 6.15. If and only if

L =

[C

κ[[Y]] ] , (6.46)

is full row rank then L can satisfy conditions (6.16) and (6.17) with left and right-

hand-sides of (6.16) and (6.17) equalling to rank(Y ).

Proof: If L is full row rank and also L =

[C

κ[[Y]] ] then the right-hand-side of

(6.16) equal to rank(Y ). The left-hand-side of (6.16) is also equal to rank(Y ) be-

cause for all X such that R(X) ⊆ R(Y ) we have R(XAj) ⊆ R(Y ) ∀j ∈ J which es-

tablishes the sufficiency of satisfying (6.16). On the other hand if rows(κ) < χ then

rank

[C

κ[[Y]] ] < rank(Y ) which establishes the necessity of satisfying (6.16).

Furthermore, if L =

[C

κ[[Y]] ] cannot satisfy condition(6.17) then it presents a

contradiction to lemma 6.14 because the R

([C

κ[[Y]] ]) = R (Y ) and conse-

quently L cannot be built any further ensuring (A,C, L) is functional observable.


With L as in (6.46) also ensures the existence of M according to (6.35). If L is

chosen according to (6.46) then

z(t) =

[y(t)

κ[[Y]]x(t)

], (6.47)

and we can then define y(t) as

y(t) =[Ip 0

]z(t). (6.48)

The fault threshold selection is arbitrary and it is based on the specific case consid-

ered and system experience. We can now propose a fault detection algorithm that

triggers a fault when the magnitude of the observer based fault residual exceeds a

chosen threshold level of δ > 0 as follows:

Algorithm 6.1 A functional observer based fault detection design.

1. Choose an appropriate value for the threshold detection level δ > 0 and also

choose L = C. Check the existence of the functional observer using (6.16) and

(6.17), if satisfied continue, otherwise jump to step 4.

2. Find observer matrices N , H, J according to (6.21) and (6.22) and M accord-

ing to (6.35).

3. Find z according to (6.3) and (6.4), and find y(t) according to (6.13) and ro(t)

according to (6.14). Jump to step 7.

4. Choose L =

[C

κ[[Y]] ], according to lemma 6.15 the functional observer

existence conditions (6.16) and (6.17) are always satisfied.

5. Find observer matrices N , H, J according to (6.21) and (6.22) and (6.22) and

M according to (6.35).

6. Find z(t) according to (6.3) and (6.4), and find y(t) according to (6.48) and

ro(t) according to (6.14).

7. Trigger a fault if the modified functional observer based fault residual

‖Mro(t)‖ > δ.

6.4. Numerical Examples 173

6.4 Numerical Examples

6.4.1 Example 1

A) Consider the following nominal values for A,B,C and D

[A B D

C

]=

−4 −1 0.34 −0.34 0 2

0 −3 0.34 1.67 0 −2

1 1 −3.34 0.34 0 4

1 1 0.67 −3.67 1 0

−2 −2 −2 −2

−3 0 −3 0

.

Also consider the following deviations ∆A and ∆B from the nominal values of A

and B respectively

[∆A ∆B

]=

0.2 0.3 0.2 −0.5 −0.2

0.1 0.2 −0.1 −0.6 −0.2

0.8 −0.45 0.2 −0.3 −0.2

−0.4 0 0.8 −0.3 0.2

.

Choosing L = C, does satisfy the functional observer existence conditions (6.16)

and (6.17). The observer parameters N, J and H can be computed according to

(6.21) and (6.22) as follows

[N J H

]=

[−50.0 0.0 48.0 0.0 −2.0

0.0 −40.0 0.0 37.0 0.0

],

and M according to (6.35) is

M =

[25.0 0.0

0.0 13.34

].

Note that the pair (A,C) is not observable, yet following the proposed algorithm in

this chapter it is possible to design a functional observer to perform the task of fault

detection. The fault detection threshold was set at 5 and the fault was simulated at

time t = 4. As can be seen from Figure 6.2, the modified fault residual Mro(t) can

detect the fault very quickly whereas the unmodified residual ro(t) cannot detect

the fault. Furthermore, the asymptotic value of Mro(t) is the same as that of rn(t)

and the rate of convergence of Mro(t) can be altered by choosing the eigenvalues of

N to have a faster response. The order of the observer is p < n.


0 2 4 6 8 10

0

2

4

6

8

Nominal system based

fault residual for output 1

Nominal system based

fault residual for output 2

Modified functional observer

based fault detection time

∆t =

0.06

0.03

Fault commencement time

Fault is not detected if unmodified functional

observer based scheme is used

Fault Threshold

Nominal system based fault

detection time∆t =

0.82

0.38

Time(s)

Fau

ltR

esid

uals

rn(t)

ro(t)

Mro(t)

1

Figure 6.2: Three fault residuals of the first example.

B) To compare the proposed scheme with previous method proposed in [24], the

pair (A,C) must be observable. Thus, matrix C is altered as C =[1 −1 2 −3

].

Matrices A, B, ∆A, ∆B and D are the same as in part A. The observer parameters

are calculated as follows,

[N J H

κ

]=

−6 3 0 0 6 −3

8 0 3.34 0 −8 3.22

−9.6 0 0 10 9.6 −3.5

2.64 −4.62 −7.1 −14 −3.84 1.38

0 0.334 0.1 0.01

,M =[2.5 0 0 0

].

Furthermore, a time varying fault signal f(t) = (1−e−0.1(t−tfault))u(t− tfault) where

u(t) is the step function and tfault is the time of fault was used in the simulation

study. Figure 6.3 shows a comparison between the state observer based scheme and

the functional observer based scheme. As evident in Figure 6.3 the full order state

observer based method and functional observer based method have comparable per-

formance and can detect faults equally well. However, as in part A, the functional

observer method can be used to detect faults even when the state observer based

scheme cannot be used.


The observer parameters are calculated as follows,

N J H

κ

=

−6 3 0 0 6 −3

8 0 3.34 0 −8 3.22

−9.6 0 0 10 9.6 −3.5

2.64 −4.62 −7.1 −14 −3.84 1.38

0 0.334 0.1 0.01

,M =

[2.5 0 0 0

].

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

1.5

2

2.5

Fault commencement

time

Fault Threshold

Time(s)

Fau

ltR

esid

uals

Nominal system based residual

Functional obs. based residual

Modified funct. obs. based residual

State obs. based residual

Modified State obs. based residual

Figure 3: State observer and functional observer based scheme comparison.

Furthermore, a time varying fault signal f(t) = (1− e−0.1(t−tfault))u(t− tfault)

where u(t) is the step function and tfault is the time of fault was used

in the simulation study. Figure 3 shows a comparison between the state

observer based scheme and the functional observer based scheme. As evi-

dent in Figure 3 the full order state observer based method and functional

observer based method have comparable performance and can detect faults

equally well. However, as in part A, the functional observer method can be

used to detect faults even when the state observer based scheme cannot be used.

18

Figure 6.3: State observer and functional observer based scheme comparison.

6.4.2 Example 2

Here we consider a linearized model of a 15m radius three-blade variable speed

wind turbine working in 12m/s wind-speed and generating 220V as follows [29]:

[A B D

C

]=

−5 0 0 0 0 5 2

0 0 1 0 0 0 4

−10.5229 −1066.67 −3.38028 23.5107 0 0 0

0 993.804 3.125 −23.5107 0 0 2

0 0 0 10 −10 0 0.2

0 0 0 1 1

.


Also consider the following deviations ∆A and ∆B from the nominal values of A

and B respectively[∆A ∆B

]=

0 0.5218 −0.26583 −0.25 −0.57725 −0.1

0.2 0.5 0.231 −0.100 −0.1578 0.1

−0.2 −0.499 −0.1483 0.1 0 0.15

0 0.1 −0.4455 0 −0.211 0.1

0.1 0.2495 0.664 0.5 −0.3692 0.3

.

Note that the pair (A,C) is not observable, yet following the proposed algorithm

in this chapter it is possible to design a functional observer to perform the task of

fault detection. Choosing L = C, does not satisfy the functional observer existence

conditions (6.16) and (6.17). Hence, according to the design algorithm we build

matrix L to include κ[[Y]]

where Y according to (6.43) is

Y =

C

CA...

CA4

.

and we can find[[Y]]

as follows

[[Y]]

=

C

CA...

CA3

.

Choosing

κ =[0 10−2 10−3 10−4

],

makes

L =

[C

κ[[Y]] ] ,

full row rank. The observer parameters N, J and H can be computed according to

(6.21) and (6.22) as follows:

N =

−14.109 100 0 0

8.6608 0 10 0

−9.5382 0 0 10

−65.86 −70.7649 −120.7127 −31.891

, J =

14.109

−8.6608

9.5382

65.0035

,


0 20 40 60 80 100 120 140 160

−200

−150

−100

−50

0

Fault commencement time

Poles=[−10 − 11 − 20 − 30]

Poles=[−10 − 11 − 12 − 13]

Poles=[−7 − 8 − 9 − 10]

Fault may not be detected if unmodified

functional observer based scheme is used

Fault Threshold

Fault may be detected while there is no fault if

unmodified functional observer based scheme is used

Time(s)

Fau

ltR

esid

uals

rn(t) Mro(t) ro(t)

Figure 4: Three fault residuals of the second example.

The fault detection threshold was set at -100 and the fault was simulated at

time t = 10. For input and fault signals an unit step signal was used. As can be

seen from Figure 4, the modified fault residual Mro(t) can detect the fault in a

timely manner and also modified fault residual is independent of the observer

poles. The Figure 4 also shows the unmodified fault residual signal which is

dependent on the poles of the observer and changing the poles location of the

observer can cause faults to be undetected. If the threshold is set at -150 then

the unmodified residual signal will produce a false alarm as evident in Figure 4.

Furthermore, the asymptotic value of Mro(t) is the same as that of rn(t). The

order of the observer is rank([[Y]])

< n.

5. CONCLUSION

This paper has presented a functional observer based fault detection method.

The order of the proposed functional observer is either p or rank(Y ) which

is less than than the order n of a full state observer. Observability is not a

21

Figure 6.4: Three fault residuals of the second example.

H =

0

0

−0.1644

−4.7045

,

and M according to (6.35) is

M =[2.0033 0 0 0

].

The fault detection threshold was set at -100 and the fault was simulated at time

t = 10. For input and fault signals an unit step signal was used. As can be seen from

Figure 6.4, the modified fault residualMro(t) can detect the fault in a timely manner

and also modified fault residual is independent of the observer poles. The Figure 6.4

also shows the unmodified fault residual signal which is dependent on the poles of

the observer and changing the poles location of the observer can cause faults to be

undetected. If the threshold is set at -150 then the unmodified residual signal will

produce a false alarm as evident in Figure 6.4. Furthermore, the asymptotic value of

Mro(t) is the same as that of rn(t). The order of the observer is rank([[

Y]])

< n.

6.5. Conclusion 178

6.5 Conclusion

This chapter has presented a functional observer based fault detection method.

The order of the proposed functional observer is either p or rank(Y ) which is less

than than the order n of a full state observer. Observability is not a requirement

to designing the proposed functional observer and the proposed functional observer

always exists provided the system has at least one output. The asymptotic value

of the proposed fault indicator is independent of the observer parameters and the

convergence rate of the fault indicator can be altered by choosing appropriate func-

tional observer parameters. Detailed Matlab simulation results illustrates how to

design the proposed functional observer and also how to trigger faults based on those

functional observers.

References 179

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182CHAPTER 7

Conclusion and Future Directions

Dynamic state estimation (DSE) is one of the fundamental tasks in engineering

which involves in identification of internal states of a system from its measurements.

In modern power systems the estimated dynamic states are used to control and

validate the received data from widely dispersed locations. State estimation plays

a critical role in every energy management system (EMS). The control strategies

in EMS are made based on the validated states and models of the power system

provided by the distributed dynamic state estimators. The decentralized DSE tech-

niques proposed in this thesis can deliver estimation of states at the generation of

electrical power. Estimation of the states can be used for local monitoring and con-

trol actions or can be sent to EMS for security monitoring and central control. One

of the control actions in power system is load frequency control (LFC). LFC ensures

that all areas in a modern power system have a same frequency and power exchange

between areas remains unchanged.

7.1 Summary of the Results

This thesis utilize both deterministic and probabilistic approaches to the prob-

lem of dynamic state estimation of generators in power systems using the latest

technology available in phase measurement units (PMUs). Deterministic functional

observer based decentralized DSE approach presented in Chapter 3 has a simpler

structure in comparison to the state observer based schemes. The functional ob-

servability requirement is less stringent than the state observability requirement.

Moreover, the proposed functional observer design Algorithm 3.1 can provide the

least possible order of the functional observer. Probabilistic approaches including

unscented Kalman filter approach presented in Chapter 4 and particle filter ap-

proach presented in Chapter 5 can handle noise in the measurements. Bad data in

PMU measurements which can deteriorate DSE also can be detected and be recti-

fied by the proposed particle filter scheme. It has been shown that proposed particle

filter approach can provide accurate estimation of the states even when the noise

covariance deviates from known initial values. Implementing on different IEEE test

systems demonstrates the flawless performance of the proposed schemes.

Furthermore, it has been shown in this thesis that such estimation techniques

can be used in load frequency control (LFC) of highly interconnected power systems,

7.2. Future Directions 183

see Chapters 3 and 4. Incorporating the proposed decentralized DSE techniques to

LFC delivers the quasi-decentralized schemes because tie-line powers are measured

by PMUs in order to bring tie-line power deviations back to zero when disturbance

in power system occurs. The analysis and design of LFC controllers presented in

this thesis is different to traditional methods previously reported in the literature.

The method proposed in this thesis considers the entire network topology. The pro-

posed unscented transform based LFC technique has been implemented and tested

on IEEE 39-bus 10-generator 3-area system considering noise in PMU measure-

ments, see Chapter 4. The proposed functional observer based technique proposed

in Chapter 3 is also implemented on IEEE 39-bus 10-generator 3-area test system

using PMU measurements. Considering 10% deviation in generator parameters from

nominal values shows no effect on the presented scheme.

Finally, DSE is applicable to detect faults in dynamical systems. A functional

observer based fault detection technique for dynamical systems with application to

wind turbines is proposed in Chapter 6 of this thesis. In the proposed scheme,

observability of the system is not a requirement. The proposed functional observer

always exists provided the system has at least one output. The proposed fault

detection technique is independent of the observer parameters and the convergence

rate of the fault detection can be altered by choosing appropriate functional observer

parameters.

7.2 Future Directions

Design of conventional electrical infrastructure belongs to past century. The load

demand is increasing continuously as population grows which means that our electri-

cal grid is becoming outdated. This ongoing increase in the consumption is stressing

our current electrical network as governed by the increase in the System Average

Interruption Duration Index (SAIDI) and System Average Interruption Frequency

Index (SAIFI). Outcome of this issue is the growth of power complications such as

blackouts, voltage sags and overloads which means lower power quality and reliabil-

ity. Apart from the reliability and quality subject, the other concern is the carbon

dioxide emission by the current electrical network. By introducing renewable energy

resources like wind, solar (photovoltaic, solar thermal), combined heat and energy-

CHP (fuel cell, biomass and microturbines), hydroelectric, wave, geothermal and

tidal energy in power systems, energy networks have became smarter. Smart energy

network will be more resilient and will be able to avoid blackouts. The term DG

encompasses any small-scale electricity generation technology that provides electric

7.2. Future Directions 184

power at a site close to consumers. The size of DG units could range from a few

kilowatts to hundreds megawatts. As a large number of DG with diverse character-

istics have been installed in the distribution system, most engineering and operation

concerns are focused on the development of new control approaches and tools, which

include new monitoring schemes, new reliability and security analysis. The integra-

tion of DG would cause system reliability and stability problems; in addition, it

would impose new challenges on the planning of transmission and distribution sys-

tems. The DSE techniques proposed in this thesis has a broad range of application

in the above mentioned challenges of such modern and complicated power systems.

Furthermore, the proposed DSE techniques can have applications in protection

of microgrids. The term microgrid has became a popular topic within the power

community. A microgrid is defined as a subsystem of distributed energy resources

(DER) and their associated loads. As the value of currents flowing in microgrids

are not considerably high because renewable sources are responsible for providing

energy demands, microgrid protection layers including relays may not detect faults

when there is a fault or they may detect a fault when there is no fault. DSE schemes

proposed in this thesis can also be used in such protection applications as well as

many more in other areas of engineering.

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