On the Use of Coordinated Control of Power System ...

TRITA-ETS-2001-06ISSN 1650-674X

On the Use of Coordinated Control of Power

System Components for Power Quality

Improvement

Valery Knyazkin

Stockholm 2001

Technical Licentiate

Royal Institute of Technology

Department of Electrical Engineering

Akademisk avhandling som med tillstand av Kungl Tekniska Hogskolan framlag-ges till offentlig granskning for avlaggande av teknisk licentiat examen tisdagenden 11 december 2001 kl 13.00 i sal H1, Teknikringen 33 Kungl Tekniska Hog-skolan, Stockholm.

c© Valery Knyazkin, 2001

Universitetsservice US-AB, Stockholm 2001

Abstract

Deregulation and re-structurization of electrical power systems demand an im-provement of power delivery and quality of service. One of the fundamentalquantifiers of the latter is power quality. The issue of power quality has been inthe focus of research of electrical engineers for the last decade. The research hasresulted in various techniques aimed at enhancing the quality of electric power.

This thesis work presents a contribution to the existing methods of powerquality improvement. Primarily, the possibility to enhance power quality byusing coordination of power system equipment that possesses appropriate controlcapabilities is investigated.

The main power quality phenomena studied in the project are voltage sagsand frequency deviations. A unified approach has been developed that allowseffective alleviation of these power quality phenomena. It is shown that informa-tion known in advance can be utilized in order to facilitate the design of advancedcontrollers and to perform certain preventive measures prior to the occurrenceof a disturbance. This unified approach is termed “disturbance scheduling”.

Two case studies exemplify the concept of disturbance scheduling. The firstcase study deals with the voltage sag mitigation at a steel mill. The sensitive loadof the mill is subjected to voltage sags caused by a start up of a large synchronousmotor. With the aid of computer simulations, it is demonstrated that the voltagesag problem can be effectively alleviated by employing the disturbance schedulingmethod and an advanced voltage controller. The second case study is concernedwith the quality of supply frequency. The local power grid of a paper mill isstudied. Main objectives here are an improvement in the reliability of the system,enhanced frequency control, and formulation of a method for better utilization oftechnical and economical potentials of the system. The last objective demandsa new control algorithm to allow the operation of backup generators providingbackup power and having market signals incorporated in their operation. This isagain achieved by utilization of the disturbance scheduling method and advancedauxiliary control. This combination assures proper, economically beneficial, andreliable operation of the local power system that is especially important in thenew environment of deregulated electricity markets.

Implementation of the aforementioned disturbance scheduling method re-quires a model of the studied power system. There are two ways of obtainingpower system models: to analytically derive a model or to apply a system iden-tification technique. For a number of reasons that are discussed in the thesis,the latter is particularly well suited for power system applications. Therefore,a survey of several system identification and linear controller design proceduresare presented and discussed. Furthermore, a new nonlinear load estimation algo-rithm is developed. The main equations of the algorithm are in a compact, closedform and can be relatively easy implemented in a numerical analysis package.

TRITA-ETS-2001-06 • ISSN 1650-674X

iii

Acknowledgments

This Licentiate thesis finalizes the work which was started at the Department ofElectric Power Engineering, Royal Institute of Technology (KTH), in February1999.

First of all, I wish to express my sincere gratitude to my supervisor ProfessorLennart Soder for the discussions, helpful suggestions, and encouragement thattook place during the whole period of work on the project.

Special thanks go to Dr. Mehrdad Ghandhari for his valuable comments andconstructive suggestions, which to a great extent improved this thesis.

The invaluable help of the secretaries of the department, Mrs. Lillemor Hyl-lengren and Margaretha Surjadi is also highly appreciated. My colleagues at thedivision of Electric Power Systems are acknowledged for creating warm ambi-ent, which played an important role in making the work on this document mucheasier.

Many thanks go to Mr. Per Ivermark, the manager for Electrical Maintenanceof “Billerud” in Grums, Sweden for very productive collaboration and diversehelp in obtaining and interpreting the data for one of the case studies presentedin the thesis.

The financial support of the project from the Competence Centre in ElectricPower Engineering at the Royal Institute of Technology is acknowledged.

There are a lot of people who I would really like to acknowledge; the completelist would take a chapter of the thesis. I thank only a few of you, my dear friends,but you all are, of course, meant!

I am grateful to Dr. Lawrence Jones of Alstom EMM, for numerous valuablediscussions we have had on almost every aspect of this thesis. These discussionswere a source of inspiration and encouragement for my work.

I am also greatly indebted to my friend and colleague Mr. Waqas M. Ar-shad, who was always helpful in all conceivable aspects which outspread fromengineering to various practical questions.

Finally, I wish to write my gratitude to my dear wife, Olga for her care andsupport, love and friendship. You have harmonized and enriched my life in theway I could never dream of; yet I will always be grateful to you for standing bymy side throughout all the sorrow and happiness.

v

Contents

1 Introduction 31.1 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . 6

I Theoretical Foundations 9

2 Main Mathematical Definitions and Concepts 112.1 Definition of the system . . . . . . . . . . . . . . . . . . . . . . . 112.2 Norms and their calculation . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 p-norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 The Hardy space and H∞ -norm . . . . . . . . . . . . . . 142.2.3 Calculation of some norms . . . . . . . . . . . . . . . . . . 15

3 Power Quality Phenomena 193.1 Motivation for investigating the power quality phenomena . . . . 203.2 Voltage sag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Quantifying under- and overvoltage tolerance . . . . . . . 233.3 Conventional methods of voltage sag mitigation . . . . . . . . . . 24

3.3.1 Dynamic voltage restorer . . . . . . . . . . . . . . . . . . 263.3.2 Step-dynamic voltage regulator . . . . . . . . . . . . . . . 273.3.3 Uninterruptible power supply . . . . . . . . . . . . . . . . 283.3.4 Superconducting magnetic energy storage device . . . . . 283.3.5 Motor-generator sets . . . . . . . . . . . . . . . . . . . . . 293.3.6 Other equipment used for voltage sag mitigation . . . . . 29

3.4 Concluding remarks on voltage sag mitigation equipment . . . . 30

4 Power System Modeling 334.1 Main components of power systems . . . . . . . . . . . . . . . . . 33

4.1.1 Linear and nonlinear systems . . . . . . . . . . . . . . . . 344.1.2 Modeling of synchronous machines . . . . . . . . . . . . . 364.1.3 Modeling the excitation system . . . . . . . . . . . . . . . 37

vii

viii Contents

4.1.4 Modeling the turbine and governor . . . . . . . . . . . . . 384.2 Other power system components . . . . . . . . . . . . . . . . . . 39

4.2.1 Shunt devices . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.2 Series devices . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Algebraic constraints in power systems . . . . . . . . . . . . . . . 424.4 Notes on linearization of power system DAE’s . . . . . . . . . . . 43

4.4.1 Analytical linearization . . . . . . . . . . . . . . . . . . . 434.4.2 Measurement-based linearization . . . . . . . . . . . . . . 48

5 System Identification 495.1 Prony method based system identification . . . . . . . . . . . . . 505.2 ARMAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3 State space identification methods . . . . . . . . . . . . . . . . . 525.4 Subspace identification methods . . . . . . . . . . . . . . . . . . . 535.5 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.5.1 Constant impedance load identification . . . . . . . . . . 555.5.2 Complex dynamical structure identification . . . . . . . . 60

5.6 Concluding remarks on system identification . . . . . . . . . . . . 63

6 Automatic Control Systems Synthesis 656.1 Linear control systems . . . . . . . . . . . . . . . . . . . . . . . . 656.2 Linear optimal control . . . . . . . . . . . . . . . . . . . . . . . . 676.3 Robust control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.3.1 Frequency domain solutions to H∞ control problem . . . 706.3.2 Time domain solutions to H∞ control problem . . . . . . 73

6.4 Control Lyapunov function . . . . . . . . . . . . . . . . . . . . . 766.5 Noncasual control . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

II Applications 81

7 Oxelosund Case Study 837.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.1.1 An Example of Simplified Short-Circuit Calculation . . . 867.1.2 Selecting the size of the choke . . . . . . . . . . . . . . . . 88

7.2 Controller Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . 907.2.1 Design strategy . . . . . . . . . . . . . . . . . . . . . . . . 907.2.2 System modeling . . . . . . . . . . . . . . . . . . . . . . . 917.2.3 Disturbance scheduling . . . . . . . . . . . . . . . . . . . 987.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8 Gruvon Case Study 1098.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.1.1 Economical Motivation . . . . . . . . . . . . . . . . . . . 1098.1.2 System studied . . . . . . . . . . . . . . . . . . . . . . . . 109

Contents ix

8.1.3 Fast Switches and Current Limiting Devices . . . . . . . . 1128.1.4 New mode of operation . . . . . . . . . . . . . . . . . . . 112

8.2 Coordinated Control . . . . . . . . . . . . . . . . . . . . . . . . . 1128.3 System modeling and controller design . . . . . . . . . . . . . . . 113

8.3.1 System modeling . . . . . . . . . . . . . . . . . . . . . . . 1138.3.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . 115

8.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9 Conclusions and Future Work 1219.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A Identification of Eigenvalue Clusters in Large Power Systems 125A.0.1 Multiple eigenvalue issue . . . . . . . . . . . . . . . . . . . 126A.0.2 Transfer function identification . . . . . . . . . . . . . . . 129A.0.3 Newton’s method for signal identification . . . . . . . . . 133A.0.4 Gradient and Hessian matrix . . . . . . . . . . . . . . . . 136

A.1 Application examples . . . . . . . . . . . . . . . . . . . . . . . . . 139A.1.1 Distinct poles (Large separation) . . . . . . . . . . . . . . 139A.1.2 Closely spaced poles . . . . . . . . . . . . . . . . . . . . . 142

A.2 Discussion on the results . . . . . . . . . . . . . . . . . . . . . . . 144A.2.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . 145

B Main data for the Oxelosund and Gruvon case studies 147

C Comparison of the software used in the thesis work 151

D PID Controller Tuning 153D.1 Main data for the Gruvon case study . . . . . . . . . . . . . . . . 154

E Power System Identification 155E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

E.1.1 Technical details . . . . . . . . . . . . . . . . . . . . . . . 157E.1.2 What models can be obtained? . . . . . . . . . . . . . . . 158

E.2 Preprocessing of the data and qualitative analysis . . . . . . . . . 159E.2.1 Rotor angular speed of G1 and G3 . . . . . . . . . . . . . 159E.2.2 System voltage . . . . . . . . . . . . . . . . . . . . . . . . 160E.2.3 System currents . . . . . . . . . . . . . . . . . . . . . . . 161E.2.4 Field voltages of G1 and G3 . . . . . . . . . . . . . . . . . 165

E.3 Load identification . . . . . . . . . . . . . . . . . . . . . . . . . . 167E.3.1 Identification of a linear model . . . . . . . . . . . . . . . 169E.3.2 Identification of a nonlinear model . . . . . . . . . . . . . 169E.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 172

List of Figures

3.1 A typical voltage sag . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 ITIC—voltage tolerance curves . . . . . . . . . . . . . . . . . . . 243.3 Qualitative relationship between distance and cost of equipment 253.4 Schematic diagram of a dynamic voltage restorer . . . . . . . . . 263.5 One-line schematic diagram of step-dynamic voltage regulator . . 273.6 A typical uninterruptible power supply . . . . . . . . . . . . . . . 283.7 Schematic diagram of a superconducting magnetic energy storage

device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 Domain of attraction of the system (4.2) . . . . . . . . . . . . . . 354.2 IEEE Type DC1 exciter system with saturation neglected . . . . 384.3 Thyristor-controlled reactor . . . . . . . . . . . . . . . . . . . . . 404.4 Thyristor-switched capacitor . . . . . . . . . . . . . . . . . . . . 404.5 Single-Machine Infinite-Bus system . . . . . . . . . . . . . . . . . 454.6 Block diagram of the system studied . . . . . . . . . . . . . . . . 464.7 The excitation system block diagram . . . . . . . . . . . . . . . . 46

5.1 A simple test system . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Constant impedance load identification . . . . . . . . . . . . . . . 575.3 System’s output–current ILoad used for identification . . . . . . . 585.4 Zoomed version of Fig. 5.3 . . . . . . . . . . . . . . . . . . . . . . 585.5 System’s input–voltage VLoad used for identification . . . . . . . 595.6 Set-up for nonlinear load identification . . . . . . . . . . . . . . . 615.7 Block “Load” in Fig. 5.6 . . . . . . . . . . . . . . . . . . . . . . . 625.8 Load and system voltages in the presence of an SVC . . . . . . . 625.9 Comparison of estimated and measured currents . . . . . . . . . 63

6.1 Standard diagram of the plant and controller . . . . . . . . . . . 706.2 SISO plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.3 Nyquist plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.4 Parameterization of all suboptimal H∞ controllers . . . . . . . . 756.5 Linear system controlled by a noncasual controller . . . . . . . . 79

7.1 Simplified model of the Oxelosund steel plant . . . . . . . . . . . 84

xi

xii List of Figures

7.2 Equivalent scheme of the Oxelosund plant . . . . . . . . . . . . . 867.3 Short circuit data vs. the choke’s inductance . . . . . . . . . . . 897.4 EMTDC model of electrical network of the Oxelosund steel plant 927.5 Step response of the actual and the identified plants . . . . . . . 947.6 Step responses of the four identified plants . . . . . . . . . . . . . 957.7 Comparison of step responses: Linear model vs. EMTDC . . . . 967.8 Voltage Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.9 Information flow for the control scheme used . . . . . . . . . . . 1017.10 Step responses of the compensated plants . . . . . . . . . . . . . 1037.11 Comparison of the voltage profiles at a motor start . . . . . . . . 1047.12 Step response of the linear model . . . . . . . . . . . . . . . . . . 1067.13 Bode diagrams of the full and reduced-order plants . . . . . . . . 1067.14 EMTDC simulation: Voltage control. H∞ versus PID controller 108

8.1 Electrical network of the paper mill of “Billerud” . . . . . . . . . 1108.2 System studied . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118.3 Structure of the controller based on CLF . . . . . . . . . . . . . . 1168.4 System controlled by the CLF regulator . . . . . . . . . . . . . . 1168.5 System voltages with and without extra controllers . . . . . . . . 1178.6 Frequency deviation caused by islanding of the system . . . . . . 1188.7 Phase portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.1 State Feedback Controller . . . . . . . . . . . . . . . . . . . . . . 128A.2 Collision of the upper half-plane eigenvalues . . . . . . . . . . . . 129A.3 Change of eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . 130A.4 Train of impulses . . . . . . . . . . . . . . . . . . . . . . . . . . . 133A.5 Order selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138A.6 Convergence illustration . . . . . . . . . . . . . . . . . . . . . . . 141A.7 PDF of a sum of superimposed sinus waves . . . . . . . . . . . . 141A.8 Distinct poles. Noisy data . . . . . . . . . . . . . . . . . . . . . . 142

B.1 EMTDC schematic diagram of the case study . . . . . . . . . . . 149

D.1 Optimization of a convex function . . . . . . . . . . . . . . . . . 154

E.1 Rotor mechanical speed of generators G1 and G3 . . . . . . . . . 159E.2 System voltage at test 1 . . . . . . . . . . . . . . . . . . . . . . . 161E.3 System currents at test 1 . . . . . . . . . . . . . . . . . . . . . . 162E.4 System active powers at test 1 . . . . . . . . . . . . . . . . . . . 163E.5 Current ICable at test 1 . . . . . . . . . . . . . . . . . . . . . . . 164E.6 Current IG1, A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164E.7 Current IG3, A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165E.8 Test 1: Field voltages of G1 and G3 in Volts . . . . . . . . . . . . 166E.9 Spectra of Efd,1 and Efd,3 . . . . . . . . . . . . . . . . . . . . . . 166E.10 Active power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

List of Tables

2.1 Results from Example 2.8 . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Categories of Power Systems EM Phenomena . . . . . . . . . . . 213.2 Classification of unsymmetrical voltage sags . . . . . . . . . . . . 23

5.1 Comparison of “true” and identified impedances . . . . . . . . . 60

6.1 Eigenvalues of the closed-loop system . . . . . . . . . . . . . . . 686.2 Comparison of eigenvalues . . . . . . . . . . . . . . . . . . . . . . 76

7.1 Equivalent circuit parameters . . . . . . . . . . . . . . . . . . . . 867.2 Comparison of the short circuit currents . . . . . . . . . . . . . . 887.3 Coefficients of the numerator in descending order . . . . . . . . . 937.4 Coefficients of the denominator in descending order . . . . . . . . 937.5 Four operational conditions . . . . . . . . . . . . . . . . . . . . . 957.6 Coefficients K1, . . . ,K6, K7δ, K7E′q . . . . . . . . . . . . . . . . . 957.7 Summands of the numerator (H∞ controller, descending order) . 1057.8 Summands of the denominator (H∞ controller, descending order) 1057.9 Summands of the numerator (Reduced-order H∞ controller, de-

scending order) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.10 Summands of the denominator (Reduced-orderH∞ controller, de-

scending order) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8.1 Characteristics of commonly used circuit breakers [86] . . . . . . 112

A.1 Movement of eigenvectors . . . . . . . . . . . . . . . . . . . . . . 130A.2 Convergence illustration . . . . . . . . . . . . . . . . . . . . . . . 140A.3 Distinct poles, noisy data . . . . . . . . . . . . . . . . . . . . . . 143A.4 Convergence illustration . . . . . . . . . . . . . . . . . . . . . . . 143A.5 Closely spaced poles, noisy signal . . . . . . . . . . . . . . . . . . 144A.6 Rate of convergence . . . . . . . . . . . . . . . . . . . . . . . . . 144

B.1 Case study. Main data for the transformer . . . . . . . . . . . . . 147B.2 Case study. Main data for the synchronous generator and motor 148

xiii

xiv List of Tables

C.1 Comparison of the software packages used . . . . . . . . . . . . . 152

D.1 Main data for the case study . . . . . . . . . . . . . . . . . . . . 154

E.1 Data for system identification . . . . . . . . . . . . . . . . . . . . 156

In memory of my dear father Victor

Chapter 1

Introduction

The modern society is heavily dependent upon electricity. Be it household ap-pliances or large industries, reliable power supply is essential. In many cases,it is not enough to provide reliably electricity, the customers now begin seeingelectricity as a commodity which must possess certain pre-specified quality.

The issue of power quality arose because of several reasons, but perhapsthe most important reason was the increased financial loss due to electricityof “bad” quality. Nowadays, financial losses caused by a fairly short poweroutage or a voltage sag may amount to millions of dollars. One such example isgiven in this thesis. It is therefore important to maintain power quality in thechanging environment of distribution networks. It should be noticed that theoverwhelming majority of power quality problems occur on distribution networks,which to a certain extent determines the location of mitigation equipment–itshould be placed close to the customer.

It is widely recognized by electrical engineers that one of the most commonpower quality phenomena is voltage sag. A voltage sag is a temporary reductionof the feeding voltage. The definition of voltage sag might give a false messagethat voltage sags do not have a significant impact on customers’ equipment. Ofcourse, there are more severe events that might occur on power systems, e.g., apower outage; however, voltage sags are of frequent occurrence on distributionnetworks which makes them quite important.

There are a number of various solutions to power quality phenomena. Mostof the solutions solely rely upon power electronic devices. Performing reasonablywell, these pieces of equipment still remain quite expensive–the higher the powerrating, the more expensive the device is. Both high installation and maintenancecosts comprise these expenses.

This consideration suggests that a search for alternative solutions whichwould be i) able to maintain power quality and ii) inexpensive compared withthe existing solutions.

3

4 Chapter 1. Introduction

In this thesis, it is conjectured that one solution to the voltage sag problemcould be to more fully utilize the potential of the existing power system com-ponents for enhancing power quality. The main instruments for achieving thisgoal are coordination of power system components and advanced auxiliary con-trol. It appears that not only does coordination of some components alleviatevoltage sags, but also other power quality problems can be solved. For instance,in one of the case studies in the thesis, it is demonstrated that in some cases thefrequency of an islanded distribution network may be improved by using coordi-nated control. The next section provides further explanations of the concept ofcoordinated control.

1.1 Thesis outline

This thesis is organized in two mutually complementing parts entitled “Theo-retical Foundations” and “Applications”. As the title suggests, the first partestablishes a theoretical background of the work reported in the thesis.

As was mentioned above, the two cornerstones of our concept for powerquality improvement are coordination of existing power system components andadvanced control. The “modern” control theory is a very complicated tool forbuilding “advanced controllers”. Regardless of the degree of simplification in-volved in the exposition of the methods of modern control theory, it is a profoundtask to demonstrate its machinery to engineers which might not have the appro-priate background. It is, however, believed that properly designed exposition ispossible, provided all basic concepts are explained and all unnecessary detailsare omitted.

The thesis is therefore opened by presentation of some very basic notions frommathematics, control and system theory. These issues are synoptically coveredin Chapter 2.

One of the main objectives pursued in this project is to develop techniquesand algorithms that can be applied in the industry in order to enhance the qualityof electric power. To have a basis for comparative analysis, basic power qualityphenomena and some conventional solutions to the phenomena are surveyed inChapter 3.

To be able to perform system analysis and design any controller, a model ofthe system is needed that reasonably reflects all major features of the systemunder consideration. Therefore, modeling of key components of a power systemis the main subject of Chapter 4.

Electric power systems have a number of properties that make them veryspecial. For example, power systems often have very large dimensions. Here, notonly the huge geographical size of power systems is meant, but also the numberof equations describing such systems. Another feature of power systems is theiruncertain nature–both current state and some parameters of the power systemsmay be unknown. This fact makes power system modeling quite an elaboratedexercise. To simplify the modeling, system identification may be utilized. The

1.2. Main contributions 5

use of system identification in power system applications is briefly presented inChapter 5.

Provided a model of the power system is available, the analyst can performnecessary analysis and map out possible configurations for a controller which iscapable of fulfilling the control objectives. At this stage, it is essential to revisethe existing options for automatic controller design. Chapter 6 shortly reviewssome techniques relevant to the problem at hand. This chapter also concludesthe theoretical part of the thesis.

In the second part two case studies are presented. The purpose of the casestudies is to demonstrate possible applications of the theories described in Part Ito real-life industrial processes. In spite of some technical differences, both casestudies are in fact very similar—conceptually one approach works in both cases—coordinated control is used to alleviate power quality problems.

It is worth mentioning that when saying “coordination” not necessarily onlycoordination of some pieces of equipment or their respective controls is meant.On the contrary, often it does not suffice to coordinate the controls of exist-ing components, as in principle these are far too slow to be able to mitigate avoltage sag. In almost all cases it is much more reasonable to coordinate thecontrols of the system’s equipment with some information that might be knownbeforehand. In this project, a new concept called “disturbance scheduling” isintroduced which is defined as “incorporation of information known beforehandin the control algorithms of the existing components”. In the case studies, itwas shown by means of computer-aided simulations that disturbance schedulingis a viable technique that facilitates the control of such quantities as the volt-age level and frequency of a network, which achieves the ultimate goal of theproject–enhancement of quality of electric power.

Appendices contain some intermediate results which are important in theirown rights. In particular, in Appendix A identification of eigenvalue clusters inpower systems is presented. Possible impacts of eigenvalue clusters on the oper-ation of power grids are discussed. As an example, consider a controller designprocedure. A controller that is designed based on a false model that does notadequately represent the real process, e.g., an eigenvalue cluster was approxi-mated by a composite eigenvalue, can eventually lead to poor performance ofthe controller when installed on the actual power system. Moreover, computersimulation would unlikely indicate any problem unless the model used in thesimulation program is very detailed.

Another interesting result is reported in Appendix E. In this Appendix, afield measurements are used to estimate the parameters of a real-life load locatedin Grums, Sweden. A linear model of the load is first derived and a procedurefor nonlinear load identification is given in every detail.

1.2 Main contributions

The main contributions of the project are listed below:

6 Chapter 1. Introduction

• A new concept termed “disturbance scheduling” is introduced. Two appli-cations of disturbance scheduling in real-life industries have demonstratedits main features. By means of computer simulations and analysis it wasshown that this concept can be effective in mitigating some common powerquality phenomena.

• It is shown that utilization of disturbance scheduling makes the operationand control of a local distribution power system more effective and reliable,which is especially valuable in the deregulated environment of electricitymarkets.

• Application of some “linear” system identification techniques to power sys-tem load identification is demonstrated. In addition, a new set of equationsfor nonlinear estimation of load parameters is developed. The equationsare formulated in closed-form and can easily be implemented using a nu-merical analysis package such as, for instance, Matlab.

• The issue of eigenvalue cluster identification in large power systems istreated. A new algorithm for eigenvalue cluster identification is presented.The new algorithm is essentially a combination of several existing methods;however, none of these “old” methods is capable of reproducing the resultsof the new algorithm, i.e., identification of several closely spaced poles.

• A comparison between several power system simulation packages is pro-vided. Each of the packages used in the work has its advantages as wellas some disadvantages; therefore, such a comparison might be informa-tive for those who intend to perform power system dynamical studies withemphasis on development and implementation of various control schemes.

1.3 List of publications

• V. Knyazkin, L. Soder, “The Use of Coordinated Control for Voltage SagMitigation Caused by Motor Start”, The Proceedings of the 9th Interna-tional Conference on Harmonics and Quality of Power, vol. 3, pp. 804–809,2000.

• V. Knyazkin, “The Oxelosund Case Study”, A–EES–0010, Internal report,Electric Power Systems, Royal Institute of Technology, Sweden, August,2000.

• V. Knyazkin, “The Use of the Newton Optimization for Close Eigenval-ues Identification”, A–EES–0012, Internal report, Electric Power Systems,Royal Institute of Technology, Sweden, September, 2000.

• L. Jones, G. Andersson, and V. Knyazkin, “On Modal Resonance and In-terarea Oscillations in Power Systems”, Presented at “Bulk Power SystemDynamics and Control V” in August, 2001 in Onomichi, Japan.

1.3. List of publications 7

• V. Knyazkin, L. Soder, “Mitigation of Voltage Sags Caused by Motor Startsby Using Coordinated Control and a Fast Switch”, Presented at PowerTech2001, held September 9–13 2001 in Porto, Portugal.

Part I

Theoretical Foundations

9

Chapter 2

Main MathematicalDefinitions and Concepts

2.1 Definition of the system

The notion of “system” is one of the key concepts of this thesis. Being widelyused, this concept is often introduced based on appealing to the intuition of theengineer. In order to establish a solid basis for the subsequent discussion, theissue of “system” and related concepts will now be explained.

We begin with an informal statement due to [14]:

• A system is a set of objects together with relationships between the objectsand between their attributes.

• Objects are defined as “the parts or components of the system.”

• Attributes are “properties of objects.”

• Relationships are physical constraints which “tie the system together.” Theword “constraints” is understood here in a broad sense, e.g. an empiricallaw.

Rigorously speaking, the notation of “system” presented above is just a model ofa physical process. A clear distinction must be made between a physical processand a model of the process—no matter how detailed the model is, it will not beexactly the same as the process. To clarify this statement, it is enough to remindthat any physical process comprises an infinite set of parameters. However, forengineering purposes, most of the processes can be satisfactorily approximatedby finite-dimensional models. Now, a definition of a model can be stated as in[14]:

11

12 Chapter 2. Main Mathematical Definitions and Concepts

Definition 2.1. Model is an ordered pair M =< D,F >, where D is a domainof M (nonempty set) and F is a function assigning to the predicate variables ofL relations of corresponding rank on D. 2

Moving on from general to specific, the concept of system is now presented asit is understood in modern control theory. First, some mathematical foundationsare established [79].

Set: A set is a collection of objects of an arbitrary nature. If a set contains atleast one member (even 0), it is called non-empty; otherwise the set is calledempty and is denoted “∅”. Certain sets can be described by listing theirmembers, e.g. X = xini=1. To avoid ambiguity, sets are often describedby their properties, e.g. X = x : List of properties.

Time: In most of the models of this thesis, time is an independent variable,restricted to the set of non-negative members. That is, T = t | 0 6 t.

Input: The input UI is understood as a set of maps: UI = u | u : I → U.State space: The non-empty set of internal dynamic variables X of a model

M is denoted as the state space.

Now an elementary definition of a system applicable to studies in control theorycan be made as states

Definition 2.2. A quadruplet consisting of a time set T , state space X , inputspace U , and a transition map φ : Dφ → X is called a systemM.If the system has outputs, two extra concepts must be introduced: a non-emptyoutput space Y and a map h : T × X → Y. This defines a system with outputs.

2

The symbol × stands for the Cartesian product1. The definition of the transitionmap along with a more detailed presentation on modern mathematical controltheory can be found in [79].

In this thesis, we often refer to the so-called time-invariant systems. Thus inthe lines of [79], the concept of time-invariant system is presented by

Definition 2.3. The system M is referred to as time-invariant when for eachu ∈ U [σ,τ), each x ∈ X , and each µ ∈ T , the following holds: if u is admissiblefor x then also the translation

uµ ∈ U [σ+µ,τ+µ), uµ(t) := u(t− µ)

is admissible for x, and

φ(τ, σ, x, u) = φ(τ + µ, σ + µ, x, uµ),

where φ is defined as above. 2

1The Cartesian product: A1×A2×· · ·×An form the set of all ordered tuples (a1, a2, . . . , an).

2.2. Norms and their calculation 13

Time-invariant systems are explored in more detail in [89] and [79].

Note 2.4. It should be emphasized that so far, despite the existing fundamentaldifferences, no distinction was made between linear and nonlinear systems. Theconcepts of linear and nonlinear systems will be appreciated below in the text.

2.2 Norms and their calculation

A substantial part of this thesis is dealt with the design of controllers for powersystem applications. The concept of “norm” plays an important role in boththe design and analysis of such controllers. This section is therefore devoted tonorms and to computational issues related to the norms.

In spite of the apparent ease, a rigorous definition of a norm would neces-sarily involve the introduction of some underlying concepts which support thenorm. That is, we would have to subsequently define ‘the function’, ‘the set ofall real (complex) functions’, ‘the space’, ‘the compact space’, and some othermathematical objects. To avoid the burden of defining the endless concepts, itis supposed that the reader already possesses certain degree of familiarity withthe subject of real analysis. Thus the norm can readily be defined as states

Definition 2.5. Let X be a linear space over the field C of complex numbers.Then, a norm on X is a function x→ ‖x‖ from X to the field R of real numbershaving the following four properties

1. ‖x‖ > 0,

2. ‖x‖ = 0⇔ x = 0,

3. ‖cx‖ = |c| ‖x‖ ,∀c ∈ C,

4. ‖x+ y‖ 6 ‖x‖+ ‖y‖.2

Having defined the norm, one establishes the grounds for treating the metrics,open (closed) sets, limit points, convergent sequences, and other objects.

2.2.1 p-norms

Now the notion of the so-called p-norm of a vector valued function f ∈ Rn isintroduced. The same argument will also be applicable to vectors x ∈ Cn.

Definition 2.6. Let f belong to Rn. Then the p-norm is defined as

‖f(x)‖p =

∞∫

0

|f(x)|p dx

1/p

1 6 p 6∞. (2.1)

2


Table 2.1: Results from Example 2.8

‖ · ‖p f = (1 + x)−1 yT = [1, 1]

p = 1 ∞ 2

p = 2 1√2

p =∞ 1 1

Although defined for any p, only three norms are normally used in controlapplications, namely, p = 1, 2, and ∞.

Note 2.7. When calculating the norm of a vector x, the integral in equation ( 2.1)should be replaced by summation from i = 1 to i = n. For example, the∞-normof a vector xT = [a1, a2, . . . , an] is the modulus of the greatest component aj .

In system analysis, it is often irrelevant what norm is used2, however, thereare some subtle points about norms, which must be accounted for, while dealingwith various norms. This statement is exemplified below.

Example 2.8. Given the function f = (1 + x)−1 and the vector yT = [1, 1], weshall calculate their p-norms for p = 1, 2, and ∞. Applying the definition (2.1),immediately yields the result shown Table 2.1. As the table reveals, the resultsmay differ significantly when different norms are meant. 2

Note 2.9. The norms presented are the most often encountered in control andsystem theory applications, however, there are some other norms in use. Forinstance, the Frobenius, Hankel, or “µ” norms [76]. The latter is often wrongfullyreferred to as a norm, but it is not, at least not in the true sense of the definitionof a norm.

So far, only vector norms have been introduced. Now matrix norms arebriefly reviewed. Matrix norms are exceptionally important in the theory oflinear robust control. The overwhelming majority of the results from the modernlinear automatic control make extensive use of the matrix norms, e.g., see thecentral theorem of linear control—the small gain theorem [92].

Consider a matrix A ∈ Cn×m. The 1, 2, and ∞ norms of the matrix aredefined as follows

‖A‖p , supx6=0

‖Ax‖p‖x‖p

, p = 1, 2,∞. (2.2)

2.2.2 The Hardy space and H∞ -norm

Another concepts of paramount importance—the Hardy space—is now intro-duced.

2It is known that all norms are topologically equivalent. That is, k1‖x‖α 6 ‖x‖β 6 k2‖x‖α,for some positive k1, k2 and for all x ∈ Rn.


Definition 2.10. The Hardy space is the set of all complex-valued functionsthat are analytic and bounded in the open right half plane.

∃α : |f(s)| 6 α <∞, <(s) > 0, (2.3)

where s = σ + jω, j =√−1, and σ, ω ∈ R. Then, the H∞ -norm can be defined

on the Hardy space as

‖f‖∞ , ess sup |f(s)| : <(s) > 0 , (2.4)

where “ess sup” for simplicity can be thought of as “the least upper bound”. 2

By virtue of the maximum modulus theorem3, the last equality can be rewrit-ten in a slightly different form

‖f‖∞ , ess sup |f(jω)| : ω ∈ R . (2.5)

2.2.3 Calculation of some norms

While the calculation of vector and matrix norms is fairly straightforward, thecomputing of the H∞ -norm is somewhat more involved. Indeed, the compu-tation of the norm requires an iterative search for the least upper bound of|f(jω)|. In the frequency domain, this search is performed on a grid of frequen-cies 0 < ω1 < ω2 < · · · < ωN , for a sufficiently large N . It should however benoticed that almost always the H∞ -norm is calculated in the time domain usingthe procedure described below in this section.

Although not mentioned explicitly, the function f(s) in this thesis representsthe transfer function of a dynamical system, unless stated otherwise. Let usbriefly introduce the notion of the transfer function.

Definition 2.11. Consider the linear model of a Single-Input-Single-Output(SISO) process

∑aidix

dti=∑

bkdku

dtk(2.6)

Taking the Laplace transform [39] of the both sides of equation (2.6) and assum-ing zero initial conditions, we obtain the transfer function f , which is defined asthe ratio X(s)/U(s):

X(s)∑

aisi = U(s)

∑bks

k (2.7)

⇒ f(s) ,X(s)

U(s)=

∑bks

k

∑aisi

. (2.8)

2

3 Maximum modulus theorem: if f(s) is not constant in a domain D, it has neither amaximum nor a minimum in D. The maximum and the minimum are necessarily on theboundary of D [45].


As was written above, most of the norm computations in control applicationsare done in the time domain due to more efficient numerical algorithms, i.e., thetransfer function is translated to the time domain and then all computations arecarried out. The conversion of a transfer function to the so-called state spaceform is very well documented in the literature on this subject [76], [92] and there-fore will not be presented here. The formalism of the H∞ -norm computation isgiven by the following

Lemma 2.12. Let γ ∈ R+ and

G =

[A BC D

]∈ RL∞. (2.9)

Then ‖G‖∞ < γ iff σ(D) < γ and H has no eigenvalues on the imaginary axis.

H ,

[A+BR−1D∗C BR−1B∗

−C∗ (I +DR−1D∗)C −(A+BR−1D∗C

)∗]

(2.10)

and R = γ2I −D∗D. 2

In equations (2.9) and (2.10), the following notation is used: σ(A) is thelargest singular value of A; B∗ is complex conjugate transpose of B, and RL∞is the Lebesgue space of real-rational functions bounded on (−∞,∞).

A proof of the lemma can be found in [92]. This lemma provides a useful toolfor calculation of theH∞ -norm. That is, to calculate theH∞ -norm of a transferfunction/matrix, one chooses an initial γ and then calculates the eigenvalues ofmatrix H. If there are no <(λ(H)) = 0, the value of γ is reduced. The procedurecontinues until there is <(λ(H)) = 0. The last value of γ thus approximates theupper bound of the H∞ -norm of the system G.

Note 2.13. Since in Lemma 2.12 no assumption was made on the dimensions ofthe matrices A,B,C, and D, the result is valid for systems having an arbitrarynumber of inputs and outputs.

Example 2.14. We now illustrate the procedure of H∞ -norm calculation. Con-sider the system f(s) = 1/(s + 1). The calculation of ‖f‖∞ begins with trans-lation of f to the state space form. This system has the following state spacerealization: A = −1, B = C = 1, D = 0. The matrix H is

H =

[−1 γ−2

−1 1

]. (2.11)

The eigenvalues of H are given by λ(H) = ±jγ−1√−γ2 + 1. As can be easily

seen, the eigenvalues of H are zeros when γ is equal to one4. Thus, ‖f(s)‖∞ = 1.

2

4The negative root γ = −1 must be rejected since by definition a norm is a nonnegativereal number.


This example concludes the first chapter which has introduced the basic defi-nitions and concepts that form an elementary mathematical basis of the materialpresented in the subsequent chapters that deal mainly with electric power systemissues. Among others, one of the most important issues is the “power quality”issue.

Chapter 3

Power Quality Phenomena

Power Quality (PQ) is quite a broad concept, summarizing different electricalpower characteristics. Ideally, the delivered power should have perfect currentand voltage waveshapes and be one hundred percent reliable. The fact that thecurrent is normally determined by customers, implies that the power quality seenfrom the utility side is essentially the same as the voltage quality1. Thus, theutilities should supply power having the voltage waveshape as close to ideal aspossible.Although power quality phenomena have become important for many customers,there is no unique definition of power quality. Many authors proposed variousdefinitions emphasizing different aspects of the problem caused by poor PQ.Below we reproduce two definitions [22], [8] which are particularly well suitedfor our study.

Definition 3.1. Power quality is a summarizing concept, including differentcriteria to judge the technical quality of an electric power delivery. 2

Definition 3.2. Power quality problem is any power problem manifested involtage, current, or frequency deviation that results in failure or misoperation ofcustomer equipment. 2

The ambiguousness of the definition of PQ is mainly due to the fact thatdifferent pieces of equipment may have different sensitivities to the same PQphenomenon. For example, a voltage reduction to 90% of the nominal maynormally be tolerated by an induction motor but be dangerous to its drive.Therefore the quality of electric power should be regarded as “good” if the cus-tomers’ equipment does not misoperate. This consideration is emphasized inDefinition 3.2.

1The term PQ assumes that the matter in question is the quality of electric power that isthe product voltage × current. The utility has control over the voltage; conversely, it has nocontrol over the current, for the load determines it. Thus, the utility can only maintain thevoltage quality. This consideration levels the concepts of power and voltage quality.

19

20 Chapter 3. Power Quality Phenomena

3.1 Motivation for investigating the powerquality phenomena

Over the last two decades the power quality issue has become a matter of growingconcern. There are several causes that have initiated and later stimulated thisconcern. Some of the causes are listed below [58].

• The rapid development of microelectronics has changed the characteristicsof the load; the sensitivity of the load increases, requiring the deliveredpower to meet more stringent specifications. At the same time, the newlyinstalled sensitive load often reduces the power quality, e.g., by injectingsignificant harmonic currents in the distribution grid.

• The negative impact of PQ phenomena on customers’ equipment. Forexample, harmonics may reduce the overall efficiency and cause aging andheating of transformers, motors, etc. One way to increase the efficiencywould be to deal with the harmonics.

• Power systems are highly interconnected which makes their componentsdependent on each other. A failure of one of the components may triggerfailures of some others. This avalanche-like event is called a cascadingfailure. Costs of a cascading failure can be quite high since large areas andlarge number of customers can be affected. One way to minimize the riskof cascading failures is to maintain the quality of electric power.

• Another motivation is the “awareness” of the customers of the PQ relatedissues.

In summary it can be stated that economical considerations are the mecha-nism that governs the growing interest of both the utilities and customers.

Major PQ phenomena are classified due to [22] and presented in Table 3.1.This table gives almost a complete picture of the phenomena that occur in elec-tric power systems and sometimes cause a serious damage to equipment andeconomical loss. For example, a loss of power supply for a short time at the pa-per mill “Billerud” may cause shutdown of the major processes of the mill. Therestoration of the processes takes approximately 3 days and the total financialloss due to the shutdown is estimated to be $ 7.5 M. Another example of eco-nomic losses due to unsatisfactory PQ was reported in [5]. In this example, thedamage at a process interruption amounted to $ 200 K. The economical lossesdue to “bad” power quality can be far more severe for some other industries,e.g., semiconductor manufacturers.

3.2 Voltage sag

In this thesis work, the main emphasis is put on voltage quality or, more specif-ically, on one of the most important voltage-related problems—voltage sag.

3.2. Voltage sag 21

Table 3.1: Categories and Characteristics of Power Systems EM PhenomenaTypical Typical Typical

Categories spectral voltagecontent duration magnitude

1. Transients1.1 Impulsive1.1.1 Nanosecond 5 ns rise < 50 ns1.1.2 Microsecond 1 µs rise 50 ns – 1 ms1.1.3 Millisecond 0.1 ms rise > 1 ms

1.2 Oscillatory1.2.1 Low frequency < 5 kHz 0.3− 50 ms 0− 4 p.u.1.2.2 Medium frequency 5− 500 kHz 20 µs 0− 8 p.u.1.2.3 High frequency 0.5− 5 MHz 5 µs 0− 4 p.u.

2. Short duration variation2.1 Instantaneous2.1.1 Interruption 0.5− 30 cycles < 0.1 p.u.2.1.2 Sag (dip) 0.5− 30 cycles 0.1− 0.9 p.u.2.1.3 Swell 0.5− 30 cycles 1.1− 1.8 p.u.

2.2 Momentary2.2.1 Interruption 30 cycles – 3 s < 0.1 p.u.2.2.2 Sag (dip) 30 cycles – 3 s 0.1− 0.9 p.u.2.2.3 Swell 30 cycles – 3 s 1.1− 1.4 p.u.

2.3 Temporary2.3.1 Interruption 3 s – 1 min < 0.1 p.u.2.3.2 Sag (dip) 3 s – 1 min 0.1− 0.9 p.u.2.3.3 Swell 3 s – 1 min 1.1− 1.2 p.u.

3. Long duration variation3.1 Interruption, sustained > 1 min 0.0 p.u.3.2 Undervoltage > 1 min 0.8− 0.9 p.u.3.3 Overvoltage > 1 min 1.1− 1.2 p.u.

4. Voltage unbalance Steady state 0.5− 2%

5. Waveform distortion5.1 dc offset Steady state 0− 0.1%5.2 Harmonics 0− 100th harm. Steady state 0− 20%5.3 Interharmonics 0− 6 kHz Steady state 0− 2%5.4 Notching Steady state5.5 Noise Broadband Steady state 0− 1%

6. Voltage fluctuation < 25 Hz Intermittent 0.1− 7%

7. Power frequency variations < 10 s


1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08V

olta

ge r

ms,

p.u

.

0 2 4 6 8 10 12Time, seconds

Motor starting. Field measurement

-2

Figure 3.1: A typical [measured] voltage sag caused by a motor start. As therating of the motor is rather large, the voltage recovery takes quite long time

Definition 3.3. A voltage sag is a temporary decrease in the rms voltage mag-nitude. 2

Note 3.4. It should be stressed that the voltage sag is defined in terms of rms,that is, voltage waveshape distortions which decay in a time period shorter than1 cycle (sub-cycle voltage distortions) are regarded as transients and do not fallinto the category of voltage sags. Typical window sizes for calculation of the rmsvalues are 1/2 and 1 cycle.

It is commonly assumed [22], [58] that there are two causes of voltage sags,namely, line-to-ground faults and motor starting. Line-to-ground faults result involtage sags that are shorter (from approximately 10 ms) and have magnitudesdown to 10%. Motor starting produces sags having smaller magnitudes but theylast longer, up to 600 ms. Fig. 3.1 depicts a typical voltage sag caused by startingof a large synchronous motor.

In spite of the fact that the PQ problem caused by a motor start is usuallyreferred to as minor, sometimes motor starting causes some protection equipmentto trip. This action of the protection leads to a costly process disruption. Sincemotor starts are quite frequent, there is a need for technically and economicallyeffective solutions to mitigate the voltage sags.

There are more severe events that can occur in power systems, e.g., sustainedinterruptions, but these are substantially less frequent. The great likelihood ofoccurrence of voltage sags in power systems makes this type of PQ problem

3.2. Voltage sag 23

Table 3.2: Classification of unsymmetrical voltage sags. The definition ofvariables V and F can be found in [91]

Single line toground fault

Line toline fault

Double lineto ground fault

3-phase fault

Va V F F V

Vb − 12V − j√32 F − 12F − j

√32 V − 12F − j

√32 V − 12V − j

√32 V

Vc − 12V + j√32 F − 12F + j

√32 V − 12F + j

√32 V − 12V + j

√32 V

especially important [58]. In this work, the term “voltage sag” is commonlyused for denomination of a balanced sag2, unless otherwise stated.

Three-phase sags are satisfactorily characterized by their duration and mag-nitude; however, the phase angle jump due to unbalanced faults is completelyignored by these characteristics. The need for a better assessment of the impactof unsymmetrical sags on the power systems, required a new, more elaboratedcharacterization of voltage sags. Since unsymmetrical sags are beyond the scopeof this thesis, we only cite some references in which sags caused by unsymmetri-cal faults are classified and considered in every detail [16], [91]. Table 3.2 showsa classification of the main unsymmetrical voltage sags as proposed in [91]. Inconclusion it is to be noted that mitigation of unsymmetrical sags generally re-quires the use of rather expensive power electronic devices capable of controllingall three phases independently.

3.2.1 Quantifying under- and overvoltage tolerance

To quantify the tolerance of customers’ equipment to voltage variations, theComputer Business Equipment Manufacturers Association (presently known asthe Information Technology Industry Council) has introduced two curves relat-ing the tolerable magnitude and duration of voltage fluctuations. This set ofcurves was named the “CBEMA curve” and “has become a de facto standardfor measuring the performance of all types of equipment and power systems”[22]. Although meant for establishing a standard for single-phase, low voltageequipment tolerance, this curve is widely used for assessing the tolerable volt-age variations at different voltage levels including 3-phase voltage supplies. In1996 the CBEMA curve was revised and the continuous curves were replacedby piecewise linear segments, see Fig. 3.2. In this figure, the magnitude of avoltage fluctuation is related to its duration. Values of magnitude-duration en-closed by the thick lines (area 1) characterize voltage phenomena which shouldnot cause any damage to mainframe computers and similar equipment. Con-versely, voltage excursions beyond those limits (areas 2 and 3) may trigger apower quality-related problem. Since we are chiefly interested in voltage sags

2The terms “balanced”, “symmetrical”, and “three-phase” sag are used here synonymously.


10−4

10−3

10−2

10−1

100

101

102

0

50

100

150

200

250

300

350

400

450

500

Vol

tage

mag

nitu

de, p

erce

nt o

f no

min

al

Time, seconds

ITI (CBEMA) Tolerance Curves. (Revised 1996)

140 120 110

90 80 70

3. Overvoltage area

1. Normal operation area 2. Undervoltage area

Applicable to 120, 120/208,and 120/240 nominal voltages

Figure 3.2: ITIC—voltage tolerance curves. Values enclosed by the thick linesare normally tolerable by most of electronic equipment

rather than in overvoltages, the lower part of the ITIC curve is mainly of inter-est. Most often the ITIC graph is used when the equipment in question does nothave any voltage tolerance specifications. In this case, the ITIC plot providesan initial estimate of allowable variations of the voltage level. For example, thevoltage of power supply should not exceed 120% of nominal for a time durationgreater than 3 msec. If this excessive voltage is applied for a longer time, a givenpiece of equipment may misoperate or be permanently damaged.

3.3 Conventional methods of voltage sagmitigation

In this section, we overview the existing methods for voltage sag mitigation.Also a brief description of new approaches to voltage sag correction developedin this thesis are outlined. A more detailed presentation of those will be givenin Chapter 6.

Two different strategies are normally used in order to reduce the vulnerabilityof sensitive equipment to voltage sags. Those are briefly discussed below.

Perhaps the simplest way to alleviate the impact of a voltage sag is to increasethe ride-through capability of the most sensitive load. A good example wouldbe an industrial process controlled by a programmable logic controller (PLC).If the process itself can ride through a voltage sag but the PLC cannot, the

3.3. Conventional methods of voltage sag mitigation 25

installation of a new controller that is less sensitive to voltage variations wouldsolve the problem of voltage sags. We will not pursue this method of voltagesag mitigation in this thesis any further, since in spite of being valuable for elec-tronics manufacturers it does nothing to power quality improvement. Instead,the main emphasis is put on solutions that actively participate in voltage qualityenhancement.

The second strategy is an approach which in [16] is called “Change of PowerSystem.” As the name suggests, a change of the topology of the network is thesubject of this strategy. That is, this method is solely reliant upon building newfeeders and/or installing backup generators capable of supplying the critical loadwhen a voltage sag occurs.

The last strategy of PQ improvement is more complicated as it involves activeintervention in the voltage control. As is often the case, active PQ correctorsare more sophisticated and expensive; however, the number of customers whobenefit from cleaner power will increase significantly. That is, if for example astatic VAr compensator is connected to a weak network, all the customers fedby the feeder will enjoy a more stiff voltage supply experiencing thereby no sagsor more shallow ones.

Now the conventional methods used for voltage sag mitigation will be syn-optically presented.

Equipment for mitigation of power quality phenomena in general, and volt-age sag in particular, can be placed at several locations “between” the utilityand the customer. Since the cost of the equipment increases as the mitigating

Cos

t of

solu

tion

Distance from customer

Figure 3.3: Qualitative relationship between the distance from end-customerand cost of equipment for power quality conditioning

equipment is placed further from the customer, see Fig.3.3, often it is more effi-cient to install that equipment closer to the customer [22]. For example, if thedistribution network is weak and some customers are subjected to voltage sags,it is more reasonable to improve the power quality locally instead of changingthe topology of the grid. This type of solution is especially suitable if mainly thecustomer’s equipment contributes to the power quality deterioration. Hence, so-lutions at the customer side can reduce the necessity for substantial investments,if no significant load increase is planned. The consequent local voltage quality


Converter

Energystorage

To loadTo net

Figure 3.4: Schematic diagram of a dynamic voltage restorer

improvement therefore has significant potential and is the principle subject ofstudy of this thesis.

Power quality improvement can be performed in a variety of ways. Most of theexisting solutions to power quality problems rely heavily upon power electronics.Some of them are briefly outlined below [22], [23].

3.3.1 Dynamic voltage restorer

The Dynamic Voltage Restorer (DVR) is one of the most effective solutions to theproblem of voltage sag. A DVR is a power electronic device designed to injectpower during a sag thus protecting the sensitive load from undervoltage, seeFig. 3.4. Not only does a DVR mitigate voltage sags, but it can also correct forthe phase jumps associated with the voltage sag [9]. As can be seen from Fig. 3.4,a DVR is a series device which consists of a boost transformer, converter, energystorage, and associated control and protection circuits. Most often, capacitorbanks are used as the energy storage. This limits the spectrum of possibleapplications of DVR, as it cannot protect the load from a power outage; however,power outages are infrequent in comparison to voltage sags. A DVR has thecapability to control the magnitude, phase, and frequency of the converter outputvoltage in real time, which makes the DVR a very flexible voltage control tool.

Normally, the response time of a DVR is less than 1/4 cycle and the maximalachievable voltage boost is around 50%, i.e., a voltage reduction down to a halfof the nominal can successively be mitigated. The ride-through time dependson the rating of the energy storage and is usually in the range 10− 30 cycles ofpower frequency. The range of loads that can be protected by an average DVRvaries between 4 and 100 MVA. The DVR is quite an expensive solution, whichamounts to approximately $ 250, 000− 600, 000 per 1 MVA of protected load.

Typically dynamic voltage restorers are used to protect loads at paper mills,semiconductor plants, and some other industries [9]. The technology of DVRhas already materialized in real life installations. The list below presents afew worldwide installations of dynamic voltage restorers which were put intooperation over the last 4 years.

• Florida Power Corp., 2 MVA, 1996.


Seco

ndar

yvo

ltage

Prim

ary

volta

ge

Figure 3.5: One-line schematic diagram of step-dynamic voltage regulator

• Salt River Microprocessor fabrication, 6 MVA, 1998.

• Southeast Asian Utility, semiconductor plant, 4 MVA, 1998.

• Powercor, Australia Ltd., Bonlac Foods, 2 MVA, 1997.

3.3.2 Step-dynamic voltage regulator

One of the most simple and inexpensive methods of controlling voltage in adistribution network is to use transformers equipped with tap changers. Thisconcept is well-known and widely used in power systems. However, the majorityof the tap changers are of a mechanical type and therefore a little help when fastvoltage regulation is required for reliable operation of the load. To overcomethe lentitude of the mechanical tap changers, a tap changer with an electronictap was developed. This set was named “Step-Dynamic Voltage Regulator” (S-DVR). The S-DVR consists of a three-phase autotransformer, a thyristor tapchanger, and associated control and protection equipment, see Fig. 3.5. Thestep-dynamic voltage regulator is capable of both boosting and bucking the out-put voltage in the range from +50 to −10% of the nominal voltage in steps of3 or 5%. The response time of a S-DVR is about 1/2 cycle, which is acceptablefor most applications. The S-DVR does not require any energy storage, as theenergy needed for sag mitigation is taken from the feeder and injected in theload circuits. As a result the S-DVR can mitigate sags with magnitudes of 50%for indefinite time; however, this is done at the cost of drawing higher currentsfrom the feeder. Typical ratings of an S-DVR are in the range of several tens ofMVA.

Step-dynamic voltage regulators are best suited for the protection of largeloads such as sensitive load at paper mills, semiconductor plants, pharmaceuticalinstallations, etc [9].


AC/DC DC/AC

S

Energy storage

Figure 3.6: A typical uninterruptible power supply

3.3.3 Uninterruptible power supply

The uninterruptible power supply (UPS) is one of the most commonly useddevices for electric power conditioning. A typical UPS consists of a rectifier,inverter, battery bank, and control and protection equipment, see Fig. 3.6. Thereare different configuration of UPS available for commercial use. Among thoseare i) standby, ii) on-line, and iii) hybrid UPS systems [22]. Having somedifferences3 these configurations, however, in general share the same principleof operation, which can be describes as follows. In a UPS, the ac power of thesupply is converted to dc which is used for charging the battery bank. Thenthis dc power is inverted to ac and supplied to the sensitive load. Not only cana UPS mitigate voltage sags, it is also capable of supplying the load during apower outage.

UPS systems are most suitable for the protection of relatively small loadsin the range of kW, as the costs of a UPS increases rapidly with the rating.This consideration confines the use of UPS to protecting low-power electronicequipment such as mainframe computers, automatic control systems, and thelike.

3.3.4 Superconducting magnetic energy storage device

A Superconducting Magnetic Energy Storage Device (SMES) is a piece of powerelectronic equipment designed for conditioning the electric power. As the namesuggests, a superconducting magnet is used in an SMES to store energy. Themain functionalities of an SMES and UPS are similar and thereby they will notbe treated here in detail.

An SMES has a number of advantages compared to a UPS [22]:

• The dimensions of an SMES are considerably smaller due to the use of asuperconducting magnet.

3Normally the load protected by an on-line UPS is always isolated from the supply and isfed exclusively by the UPS. On the contrary, a standby UPS is only activated (automaticallywithin a few milliseconds) if the incoming power falls short to fulfill some pre-specified criteria;otherwise the load is fed by the supply.


AC/DC Converter

DC supply

Magnet

Figure 3.7: Schematic diagram of a superconducting magnetic energy storagedevice

• The SMES is generally more reliable.

• The SMES requires less maintenance.

There are different designs of SMES devices for a range of sizes. One of thedesigns is shown in Fig. 3.7.

To the best knowledge of the author, there exists to date no publicly availableinformation concerning a commercial installation of an SMES, however, severalpilot installations were successfully tested [22].

3.3.5 Motor-generator sets

A typical motor-generator set (MGS) consists of a motor, a synchronous genera-tor that is capable of producing constant output power irrespective of reasonablevariations of the speed of the machine, a flywheel, and control and protectioncircuits.

The flywheel normally has a relatively large inertia that “allows the generatorrotor to keep rotating at speeds above 3150 rpm4 once power shuts off” [22]. Thusan MGS isolates the sensitive load from power disturbances. Depending on thedesign, an MGS can supply full load for about 15 seconds [22].

3.3.6 Other equipment used for voltage sag mitigation

The list of presented devices that are used for power conditioning, can be ex-tended even further. The ones that are relevant to industrial applications arebriefly described below.

Magnetic Synthesizers (MS). A magnetic synthesizer is an electromagneticdevice in which the incoming ac power is conditioned and supplied back to theload. In the process of power conditioning, saturated transformers and capacitorsare utilized for energy storage in a magnetic synthesizer. This energy is thenused to synthesize a good quality power. In addition to voltage regulation, an

4In this example, the system frequency is 60 Hz.


MS has another favorable feature–it filters harmonics from the incoming power.Typically MS devices serve in protecting large loads, starting from a few kVA.

Ferroresonant transformer (FT). A ferroresonant transformer is a transformerwhich is operated in the saturated region of its B-H characteristic. In this region,the output voltage of the transformer is insensitive to the current variations.Thus load fluctuations do not cause significant output voltage excursions fromthe nominal values. FTs are normally designed to have a transformation ratioof 1, since the primary goal of an FT is to protect sensitive load from voltagesags. Ferroresonant transformers are most effective in protecting small-size loadsfrom slow input voltage variations. An FT can successfully mitigate voltage sagshaving magnitudes as low as 30% [22]. However, an FT also has some drawbackssuch as inability to mitigate rapid voltage variations, relatively high losses andthe dimension problem–it is a common practice to use transformers having therated power 5 times that of the load being protected.

3.4 Concluding remarks on voltage sagmitigation equipment

In this chapter several devices commonly used for voltage sag mitigation havebeen presented. Many of these solutions seem promising; however, most of the so-lutions inherit one common shortcoming–the installation and maintenance costscan be exceedingly high. Thus, other possibilities should be sought. This keyidea was the leitmotiv of this thesis–to investigate methods of voltage sag mit-igation in which already existing power system equipment could be effectivelyused. The benefits of such a method are evident: improved power quality, moreeffective use of the power system components, and financial savings through theavoidance of redundant and more expensive solutions. These savings can be in-vested in safety arrangements and/or improving other equipment which wouldfurther increase the overall efficiency of the manufacturing process.

The methods5 of voltage sag mitigation developed in this thesis are mostlyaimed at neutralization of the impact caused by symmetrical 3-phase sags. Al-though the consideration of these types of voltage sags may seem to representonly a part of the problem, in many cases from real life the solutions to the volt-age sag problem developed in this thesis may be a sound alternative syncretizingthe technical elegance and economical profitableness. For instance, most of thevoltage sags caused by a motor start or switching of a large load are symmetrical.On the other hand, such large loads/motors are often found at large industrialfactories or mills which may have (and often do have) backup generators of a sig-nificant capacity. Thus, introduction of new control philosophies/strategies for

5Our method for voltage quality improvement essentially relies upon coordination of thecontrols of power system equipment and use of certain information known beforehand. Thatis, one can speak of two concepts: (i) “controller–controller” coordination and (ii) “controller–information” coordination. More detailed treatment of this idea is presented in Chapter 6.

3.4. Concluding remarks on voltage sag mitigation equipment 31

operating these generators opens new perspectives for power quality improve-ment (via mitigation of sags), economical benefit (through more efficient use ofthe backup units), and increased overall reliability of the power supply.

In this study an attempt was made to investigate the possibility of volt-age sags mitigation by introducing coordinated control of existing power systemcomponents that have voltage control capabilities. In the design of the control-ler some information known in advance was utilized to enhance the process ofvoltage regulation. This theme will be considered in every detail in Chapter 6.

Prior to considering the design of a controller, the controlled object shouldbe thoroughly studied. Following this axiom, we proceed with a description ofcontrollable power system components which are the sole subject of the nextchapter.

Chapter 4

Power System Modeling

“Obtaining maximum benefits from installed assets on an interconnected powersystem is becoming increasingly dependent on the coordinated use of automaticcontrol systems. The ability to optimize the configuration of such control devicesand their settings is dependent on having an accurate power system model, aswell as controllers themselves” [3].This compendious but neat quotation form a CIGRE report is cited here tosignify the importance of having an accurate model of the system studied. In-deed, the development of a good quality model of the process is an essential partof engineering work. This chapter is therefore devoted to describing the basicprinciples of electric power system modeling.

4.1 Main components of power systems

The modern power systems are characterized by growing complexity and size.For example, the energy consumption in India doubles every 10 years, which alsoapplies to some other countries [62]. As the size of the power systems increases,the dynamical processes are becoming more complicated for analysis and under-standing the underlying physical phenomena. In addition to the complexity andsize, power systems do exhibit nonlinear and time-varying behavior.

In an electrical system the power cannot be stored1, at each time instant thereshould be balance between the total produced and consumed power. Mathemat-ically this balance is expressed by differential and algebraic equations. The pres-ence of algebraic equations significantly complicates both analytical and compu-tational aspects of work when tackling with power systems.

To obtain a meaningful model of the power system, each component of thepower system should be described by appropriate equations be it algebraic equa-tions, differential equations, or both. For example, there are different models of

1There are some exceptions e.g., a pump storage; however in those energy rather than power

is stored.

33

34 Chapter 4. Power System Modeling

an electrical generator; depending on the application a model of suitable exact-ness and complexity should be chosen to represent the generator in the study.On the one hand, very simple models of a generator are rarely used in power sys-tem studies when accuracy of the results is a great concern. On the other hand,if a system consists of 300 generators, each modeled by a set of three differen-tial equations, the system analyst would have to process at least 900 differentialequations describing the system as well as quite a few algebraic equations, thenumber of which depends on the topology of the network. The presence of otherequipment, e.g., high voltage direct current (HVDC) systems, contributes to theaforementioned number of equations. Clearly, it is barely possible to carry outany analytical study on such systems.

To overcome the problem of high dimension, the order of the system has tobe reduced. This can be done in several ways:

• Based on the physical insights, several generators are aggregated in a groupof coherent generators [63].

• Having set up the system equations, one applies a model reduction tech-nique and eliminates the states that have little effect on the system dy-namics [63].

• Using field measurements, one applies a system identification technique toobtain an equivalent model of the system [52].

Depending on the case study, any of the methods or a combination of them canbe used to obtain best dynamical models.

4.1.1 Linear and nonlinear systems

As was already mentioned, the nature of power systems is essentially nonlinear.Nonlinear systems are known to be very hard to manage. To work around thisproblem, when studying the behavior of a power system in a neighborhood of anequilibrium point, it is a common assumption that the power system is a linear,time-invariant system [72]. That is, the initial nonlinear system is approximatedby a linear one. In many cases of practical importance, this assumption worksquite well yielding numerous advantages. However, when transient stability2 ofthe system is investigated, the use of a linear model may not be justified. Thereare several reasons for questioning the validity of the linear model; the mainreason is the dependence of the qualitative behavior of a nonlinear model on thelevel of disturbance. This statement is further illustrated by

2A power system is said to be transiently stabile if after a disturbance it remains withinsome domain of attraction [63].

4.1. Main components of power systems 35

-6 -4 -2 0 2 4 6-2

-1

0

1

2

x2

x 2x

..

2 ) + 0.01t(Domain of attraction of x

2(t()+ sin

2x t) = 0.

.

Figure 4.1: Domain of attraction of the system (4.2)

Example 4.1. Consider the following two systems described by second order ho-mogenous differential equations (DE)

x1(t) + 0.01x1(t) + x1(t) = 0, (4.1)

x2(t) + 0.01x2(t) + sinx2(t) = 0. (4.2)

Equation (4.1) is a linear DE, while (4.2) is a nonlinear differential equation. Wenow determine the qualitative behavior of the solutions of (4.1) and (4.2).

Since the first equation is a linear equation with the eigenvalues having neg-ative real part (<(λ1,2) = −1/200), the domain of attraction is the whole plane.This means, for any choice of initial state of the system, the system state vari-ables will always converge to the origin. This is confirmed by the explicit solutionof (4.1)

x(t) = exp(−t/200)C sin (ω1t+ φ) ,

where ω1 is the imaginary part of the eigenvalue, the constants C and φ aredetermined by initial conditions. Clearly, lim

t→∞x(t) = 0,∀ C, φ.

Now equation (4.2) is examined. Despite its apparent simplicity, there existsno closed-form solution to this equation. The difficulty in finding closed-formsolutions to nonlinear differential equations3 has stimulated the search for othermethods which allow the analyst to obtain a qualitative characteristics of asolution without actually having to solve the equation.

For the moment, we will not pursue this approach, but content ourselveswith finding a domain of attraction of this system. This is done by integrationof the equation backwards in time [85]. The domain of attraction4 is the regionwhich includes the origin, see Fig. 4.1. If the initial state of the system waschosen inside the region, the system states will eventually converge to the origin,otherwise the origin will never be reached. 2

3Only in some exceptional cases there exist closed-form solutions to nonlinear systems [84].4The Bendixson theorem indicates that this domain is open.


It is evident that for small deviations from the origin, equations (4.1) and(4.2) are equivalent (sinx ≈ x); as the deviations grow in magnitude, the differ-ence in the behavior becomes more expressed. This example concludes the noteson qualitative difference between linear and nonlinear models.

Appreciating the importance of proper modeling of power system compo-nents, we first present the nonlinear and then linearized equations describing thebasic components of power systems.

4.1.2 Modeling of synchronous machines

Synchronous machines are one of the most important power system components.They are also among the oldest pieces of electrical equipment in use. We com-mence by considering the equations describing a synchronous machine.

Depending on the nature of a study, several models of a synchronous gen-erator, having different levels of complexity, can be utilized [46], [63]. In thesimplest case, a synchronous generator is represented by a second-order differ-ential equation, while studying fast transients in the generator’s windings wouldrequire the use of a more detailed model, e.g., 7th order model.

In this project, fast dynamics of synchronous generators and the networkare neglected and the generators are modeled by the two-axis model [74], i.e., itis assumed that the dynamical characteristics of a generator can be accuratelyrepresented by four differential equations, see (4.3)–(4.6).

dδidt

= ωi − ωs (4.3)

Midωidt

= TM, i −(E′q,i −X ′

d,iId,i)Iq,i

−(E′d,i +X ′

q,iIq,i)Id,i −Di (ωi − ωs) (4.4)

T ′do,i

dE′q,i

dt= −E′

q,i −(Xd,i −X ′

d,i

)Id,i + Efd,i (4.5)

T ′do,i

dE′d,i

dt= −E′

d,i −(Xq,i −X ′

q,i

)Iq,i (4.6)

In the equations above, the following symbols are used to denote:

• δi: The rotor shaft angle of the ith generator. Normally this angle isexpressed in radians or degrees.

• ωi, ωs: The rotor angular velocity of the ith generator. This velocity is com-monly expressed in radians per second or per unit. ωs is the synchronousspeed of the system which usually takes two values ωs = 2π50, (2π60)rad/sec.

• Mi: The shaft inertia constant of the ith generator which has the units ofseconds squared.

• TM,i: The mechanical torque applied to the shaft of the ith generator.

4.1. Main components of power systems 37

• E′q,i, E

′d,i: These symbols denominate the transient EMF’s of the machine

in the q and d axes, respectively.

• Iq,i, Id,i: Are the equivalent currents of the synchronous machine in the qand d axes, respectively.

• Di: The damping coefficient of the ith generator.

• T ′do,i, T

′qo,i: Are transient time constants of the open circuit and a damper

winding in the q-axis. These time constants are commonly expressed inseconds.

• Xq,i, Xd,i, X′q,i, X

′d,i: These four symbols stand for the synchronous reac-

tance and transient synchronous reactance of the ith machine.

Sometimes equation (4.6) is eliminated yielding the third-order model of thesynchronous generator. In the equations above, the index i runs from 1 to n,where n is the number of synchronous generators in the system. In our casestudies, the number of machines does not exceed 2; yet in many studies thisnumber may exceed several hundred.

4.1.3 Modeling the excitation system

Control of the excitation system of a synchronous machine has a very strong in-fluence on its performance, voltage regulation, and stability [20]. Not only is theoperation of a single machine affected by its excitation, but also the behavior ofthe whole system is dependent on the excitation system of separate generators.For example, inter-area oscillations are directly connected to the excitation ofseparate generators [42]. These are only a few arguments justifying the neces-sity for accurate and precise modeling of the excitation system of a synchronousmachine. This subsection therefore presents the modeling principles of the exci-tation system. A detailed treatment of all aspects of the modeling is far beyondthe scope of the thesis; we only synoptically present a literature survey on thesubject.

There are different types of excitation systems commercially available inpower industry. However, one of the most commonly encountered models is theso-called “IEEE Type DC1” excitation system. The main equations describingthis model are listed below.

TE,idEfd,idt

= − (KE,i + SE,i (Efd,i))Efd,i + VR,i (4.7)

TA,idVR,idt

= −VR,i +KA,iRf,i −KA,iKF,i

TF,iEfd,i

+KA,i (Vref,i − Vi) (4.8)

TF,idRf,idt

= −Rf,i +KF,i

TF,iEfd,i (4.9)

In these equations, the parameters and variables used are:


Stabilizingfeedback

1

E EK sT+FDE∆RV

Voltage regulatorrefV

ΣtV+

1A

A

K

sT+

1F

F

sK

sT+

−−

Exciter

Figure 4.2: IEEE Type DC1 exciter system with saturation neglected

• TE,i,KE,i, Efd,i, SE,i,: Time constant, gain, field voltage, and saturationfunction of the excitor.

• VR,i, TA,i,KA,i: Exciter input voltage, time constant and gain of the volt-age regulator (amplifier), respectively.

• Vref,i, Vi: The reference and actual voltage of the ith node.

• Rf,i,KF,i, TF,i: Transient gain reduction circuit parameters—state, gain,and time constant.

A block diagram of the exciter given by equations (4.7)–(4.9) is shown in Fig. 4.2.As is evident from (4.7)–(4.9), each excitor of the type DC1 adds three statevariables to the state matrix.

4.1.4 Modeling the turbine and governor

The number of poles of a synchronous generator and the speed of the primemover determine the frequency of the ac current produced by the generator. Inorder to control the primer mover, turbine with associated controls are used inpower systems. There exist two types of turbines—hydro and steam turbines.Only steam turbines will be presented here, since those were encountered in thecase studies of this thesis.

There are several models of the steam turbines in operation in power systems.We confine ourselves to exhibiting the simplest first-order models of the turbineand speed governor. The equations which model the dynamics of these devicesare shown below [74].

TCH,idTM,i

dt= −TM,i + PSV,i (4.10)

TSV,idPSV,idt

= −PSV,i + PC,i −1

Ri

(ωiωS

). (4.11)

The model (4.10)–(4.11), corresponds to a steam turbine with no reheater. Thevariables and parameters of equations (4.10)–(4.11) are given in [74]. While

4.2. Other power system components 39

being important pieces of power system equipment, the dynamics of the turbineand governor are normally much slower5 than that of the exciter. This fact isoften used as an argument for neglecting the dynamics of these devices.

4.2 Other power system components

In this section some auxiliary equipment, which does not directly participatein active power generation but strongly affects the voltage regulation, is brieflysurveyed.

4.2.1 Shunt devices

Shunt reactors (ShR)

Shunt reactors are used to reduce overvoltages that are caused by lines/cablescapacitance or load level variations [54]. The physical task of shunt reactorsis to absorb the reactive power that has been generated by the lines, these areconnected to. These pieces of equipment are either permanently connected tothe net or are switchable. There exists a wide variety of shunt reactors designedfor a range of voltages. ShR can be directly connected to the net or via a step-down transformer. ShR equipped with a tap changer provides the possibility tovary the reactance of the reactor. Thyristor tap changers can be successfullyused for rapid overvoltage control. In the context of the present document shuntreactors should be called “semi-controlled power system component” since it doesnot seem to be profitable to integrate their controls into the coordinated control.From this perspective much more promising is the concept of coordinated controlof SVS or adjustable speed drives of electrical machines.

Shunt capacitors (ShC)

Transfer of large amounts of reactive power is generally uneconomic and causesboth active and reactive power losses, reduces power factor, and entails unwantedvoltage drop along the power lines [54]. Partially, the reactive power can beproduced by shunt capacitors at the line terminals or in the middle position ofthe line.

Shunt capacitors can not be continuously controlled because of possible over-currents. However, if several capacitor banks are installed, then the number ofthe banks connected can be varied, provided an adjustment of the produced re-active power. One vital shortcoming with shunt capacitors is the dependence ofthe produced reactive power upon the bus voltage, these are connected to. Inspite of this drawback, shunt capacitors are widely used for the power factor cor-rection both of a large plant and/or individual device. Like shunt reactors, shuntcapacitors are connected directly to lines or by means of step-down transformers.

5In [74] the following figures are given: TSV = 2 sec. and TCH = 4 sec. On the other hand,TA is approximately 10 to 100 times smaller.


V

T 1

T 2

L

Figure 4.3: Thyristor-controlledreactor

V

T 1

T 2

C

Figure 4.4: Thyristor-switchedcapacitor

Synchronous condensers (SC)

Synchronous condensers are power systems components which are used to pro-duce or absorb the reactive power thus controlling the voltage of the buses theyare connected to [46]. Basically, a synchronous condenser is an unloaded syn-chronous machine. The excitation system completely determines the regime ofan SC: whether it absorbs or generates the reactive power. Sometimes SC aresupplemented by shunt capacitors which reduce the installation costs. SC’s maybe operated at the line voltage or transformers can be utilized to connect the SCto a different voltage level. Being an expensive power system component, theSC has the following advantages over Static Var Systems (SVS):

• The reactive power production is not dependent on the bus voltage.

• The use of SC allows increasing the system Short Circuit Ratio (SCR.)During transients they can produce considerable amount of the reactivepower.

Regardless of their advantages the SC’s high price has made it unable to be acompetitor to the modern SVS’s.

Static Var Systems

A wide class of power systems devices is included into the concept “SVS”. Fig. 4.3and Fig. 4.4 show the main components of an SVS, namely Thyristor ControlledReactor (TCR) and Thyristor Switched Capacitor (TSC.) Based on them, vari-ous SVS’s can be made up [23], [46], [54]. Main objective with SVS is to controlthe voltage at the connection point by means of fast and precise reactive powercontrol. These apparatus are controlled and can be subjected to some controls

4.2. Other power system components 41

modification in order to improve the performance. Among the main advantagesof SVS are:

• Continuous and precise voltage regulation.

• The response of the SVS can be as fast as 3 ms.

• SVS may produce or absorb the reactive power.

However, one of the most severe shortcomings with the SVS is the injectionof harmonics to the power system. These harmonics must be suppressed byrelatively costly filters.

4.2.2 Series devices

Tap-changing and regulating transformers

The primary task of any transformer is to transfer power from/to different volt-age levels [54]. Since the load level is always changing, transformers are oftenequipped with a tap changer to reduce voltage deviations due to the load fluc-tuations. In addition to the voltage magnitude regulation, some transformerscan control the voltage phase which may be necessary if the power flow in ameshed grid should be re-directed without a voltage magnitude change. Thetransformation ratio in such a case is a complex number.

The tap-changing transformer is normally constructed in two designs thatallow on- or off-load operation modes. Tap is usually driven by motors. It isdesirable to minimize the number of tap position changes which conduce to thetransformer ageing. Being very effective devices for mitigation of slow voltagevariations, this power system component cannot be used for fast power qualityphenomena compensation, unless an electronic tap changer is installed. Theelectronic tap changer is capable of quick and precise voltage regulation; however,modern electronic tap changers are rather expensive pieces of equipment.

Series capacitors

Series capacitors are intensively used for offsetting the inductive reactance of thetransmission line [46]. Installation of series capacitors increases the efficiencyof power transmission–it reduces power losses, decreases voltage drop along theline, and increases the maximal power transfer. The overall power system sta-bility improves also since both voltage stability and electromechanical stabilityenhance.

The degree of line compensation typically varies between 25-70%. Furtherincrease of the degree is no longer beneficial [54], for the cost of the protec-tion equipment becomes higher, the power flow is very sensitive to the phaseangle differential, and there is a risk of subsynchronous resonance. Normally,capacitor banks are installed at the ends of the transmission line to simplify themaintenance process.


A fixed series capacitor is not controlled. In a certain sense this equipmentis self-regulated, since the amount of generated reactive power is proportionalto the current flowing through the capacitor. Fast response and relatively lowprice make series capacitors an appropriate solution to some flicker problems. Itwas reported in [57] that as much as 60 to 80 percent of the voltage flicker canbe successfully suppressed; however, such a measure effectively works with lightflicker level only.

4.3 Algebraic constraints in power systems

As was briefly explained in Section 4.1 on page 33, the power systems are de-scribed by a set of differential and algebraic equations. The origins of the dif-ferential equations have already been discussed, while those of the algebraicequations are the main subject of this section.

The main equations relating the algebraic variables of the power system aregiven below.

0 = Viejθi + (Rs,i + jX ′

d,i)(Id,i + jIq,i)ej(δi−π

2 ) −− [E′

d,i + (X ′q,i −X ′

d,i)Iq,i + jE′q,i]e

j(δi−π2 ) (4.12)

0 = −Pi − jQi + Viejθi (Id,i − jIq,i) e−j(δi−

π2 ) +

+ PL,i(Vi) + jQL,i(Vi) (4.13)

0 = −Pi − jQi + PL,i(Vi) + jQL,i(Vi) (4.14)

0 = −Pi − jQi +

n∑

k=1

ViVkYi,kej(θi−θk−αi,k) (4.15)

In equations (4.12)–(4.15), the following notation is adopted:

• Vi, θi The magnitude and and phase angle of the ith node.

• Pi, Qi The active and reactive power injection in the ith node.

• PL,i, QL,i The active and reactive power of the load connected to the ith

node. These quantities generally are nonlinear functions of the node volt-age.

• Yi,kejαi,k The complex admittance of the branch connecting the ith andkth nodes.

• Rs,i The resistance of the stator of ith generator.

The rest of the variables and parameters have been introduced earlier in thischapter. The number of the algebraic equations is dependent on the topologynetwork, though the structure of the equations is generic and always correspondsto that shown in this section.

4.4. Notes on linearization of power system DAE’s 43

It is preferable to eliminate as many algebraic variables as possible and dealwith differential equations only, which is much simpler. In the special case ofconstant impedance loads, which is always the case in this thesis, it is possibleto reduce the total number of algebraic variables to 2n equations. That is,the only remaining variables are the complex nodal voltages. To eliminate thestator currents one has to solve equation (4.12) for Iq,i and Id,i. After somemanipulation, the following expressions are obtained:

Iq, i = −X ′d, iVi sin(θi − δi) + ViRs, i cos(θi − δi) +X ′

d, iE′d, i − E′

q, iRs, i

X ′d, iX

′q, i +R2

s, i

(4.16)

Id, i =ViRs, i sin(θi − δi) + E′

d, iRs, i −X ′q, iVi cos(θi − δi) +X ′

q, iE′q, i

X ′d, iX

′q, i +R2

s, i

(4.17)

Having done this, one can substitute equations (4.16) and (4.17) into (4.12)–(4.15). Moreover, one could reduce the number of algebraic states by eliminatingthe active P and reactive Q power from equations (4.12) – (4.15).

4.4 Notes on linearization of power systemDAE’s

Linear systems are much more manageable in comparison to most of nonlinearsystems. When studying the behavior of the system in a neighborhood of anequilibrium point, one often attempts to linearize the initial nonlinear systemand perform analysis on the linearized counterpart.

The use of the linear model enables the engineer to apply all the analyticaltools that were developed over the last 6 decades. Among these tools are theClassical Control, Linear Optimal Control, H∞ -control, µ-synthesis, etc.

It should be noticed, however, that the use of linear models is justified unlesscertain linearization conditions are violated or the system under considerationpossesses intrinsic nonlinearities and cannot be linearized at a given equilibriumpoint.

In this section we shortly describe two methods that can be applied to lin-earize the corresponding nonlinear system. The first method is utilized whensystem equations are known and all the parameters that enter these equationsare assumed to be perfectly known. If some pieces of information are unavailable,the second method is a reasonable tool for performing linearization.

4.4.1 Analytical linearization

Given a set of nonlinear and possibly linear equations describing the systemunder consideration, the main task of analytical linearization is to obtain a setof equivalent linear equations which satisfactorily approximate the behavior of


the nonlinear system when the system’s trajectory lies in some region close tothe equilibrium point. Consider the following affine system.

x = f(x) +m∑

i=1

gi(x)ui

y = hi(x), 1 6 i 6 p,

where m is the number of inputs, p is the number of outputs, ui are the inputsignals, yi are the outputs of the system, x is the vector of state variables. Theobjective is to derive a linearized version of the equations above:

∆x = A∆x+B∆u

∆y = C∆x. (4.18)

The analytical linearization method is based on truncation of all nonlinear termsof the Taylor series expansion of the function:

f(x0 +∆x) = f(x0) + Jf (x0)∆x+∆xTHf (x0)∆x+O(∆x3) (4.19)

hi(x0 +∆x) = hi(x0) + Jh(x0)∆x+∆xTHh(x0)∆x+O(∆x3), (4.20)

where Jf (x0), Jh(x0) and Hf (x0), Hh(x0) denote the Jacobian and Hessian ma-trices, evaluated at the equilibrium point x0, respectively. That is,

[Jf (x0)]ij =∂fi∂xj

∣∣∣∣x0

, A, [Jh(x0)]ij =∂h·,i∂xj

∣∣∣∣x0

, C, (4.21)

[Hf (x0)]ij =∂2f

∂xi∂xj

∣∣∣∣x0

, [Hh(x0)]ij =∂2h

∂xi∂xj

∣∣∣∣x0

. (4.22)

Here x0,∆x, f ∈ Rn. If the system were not affine, the similar argument wouldhave been used to obtain the matrix B. The symbol “∆” should be used in alinearized version of dynamic equations, but in some equations in this thesis work∆ is omitted for conciseness. For more detailed treatment of the mathematicaltheory behind linearization, the reader should consult the classical manuscripts[39] or [79].

Example 4.2. Let us derive equations that describe a linearized model of thesingle-machine infinite-bus system (SMIB), see Fig. 4.5. Fig. 4.6 shows blockdiagram of the SMIB. For this work is essentially concerned with voltage regu-lation, it was decided to exclude the dynamics of the turbine and governor, butaccurately model the excitation system of the generator. To start with, onlyequations (4.3)–(4.5), and (4.7) are used.


First, the basic dynamics of the system is put into matrix form

∆ω

∆δ

∆E′q

∆EFD

=

A

∆ω∆δ∆E′

q

∆EFD

, (4.23)

A =

0 −K1/M −K2/M 0ωb 0 0 00 −K4/T ′

do −1/(T ′doK3) 1/T ′

do

0 −KAK5/TA −KAK6/TA −1/TA

. (4.24)

However, as was stated above, certain care must be exercised while modeling theexcitation system. This means that in addition to the voltage regulator variablesKA, and TA, new state variables (ζ and VR) should be added to incorporate atransient gain reduction block and the exciter itself, see Fig. 4.6 and Fig. 4.7.Thus, the following 6th order system is obtained:

Ac =

0 −K1

M1

−K2

M10 0 0

ωb 0 0 0 0 00 −K4

T ′do1

−1T ′do1

K30 1

T ′do1

0

0 −KATcK5

TATb−KATcK6

TATb−1TA

0 KA

TA

(1− Tc

Tb

)

0 0 0 1TE

−KE

TE0

0 −K5

Tb−K6

Tb0 0 −1

Tb

. (4.25)

The functional relationships for coefficients Ki, i = 1, . . . , 6 can be found in [46],[63], or [74].

Finally, the new state vector becomes x =[ω, δ, E′

q, VR, EFD, ζ]T

. Graphicalinterpretation of equation (4.25) is shown in Fig. 4.6 and Fig. 4.7. Note thatthere is no Vref. in Fig. 4.6 since the equations are linearized at a steady-statepoint.

Stronggrid

GS

G1

ChokeL1

L2

LoadVt

OT20-10

Figure 4.5: Single-Machine Infinite-Bus system


([FLWDWLRQ

V\VWHP

8 'Y

X

.

. .

.

.

'7 '7 '7

'Z 'G

qEc'

FDE'

3

11 do

K

sT c

1 1

1

D sM

2 f

s

S

Figure 4.6: Block diagram of the system studied. The block “Excitationsystem” is shown in Fig.4.7

1

E EK sT+FDE∆

WV

Trans. gainreductionrefV

Σ

tV

+ 1

1c

b

sT

sT

++ 1

A

A

K

sT+−

Exciter

RV

Voltageregulator

Figure 4.7: The excitation system block diagram


In order to apply the standard robust control routine, the equations presentedin this section should be rewritten in the standard state space form. Partly thishas already been done c.f., equation (4.25); the complete form is given by:

x = Acx+Buy = Cx+Du

(4.26)

In this study, matrices D, C, andB have the form

D = 0, C = [0,K5,K6, 0, 0, 0] , B =

[0, 0, 0,

KATcTATb

, 0,1

Tb

]T(4.27)

It is noteworthy mentioning that the state space model of equations (4.26)–(4.27) relates the exciter voltage and the generator terminal voltage. A minormodification of the matrix C in equation (4.27) is required to obtain the voltageat bus OT20-10. The equations below are used to calculate the new coefficientsK7δ and K7E′q . Equation (4.28) is a nonlinear equation which must be linearizedat a steady-state operating point in order to be used in a linear model. Thelinearization is performed in several steps as shown below:

VOT20−10 = Vt − jxchokeIG1= Vt,d + jVt,q − jxchoke (Id + jIq)

= Vt,d + xchokeIq + j (Vt,q − xchokeId)Vt,d = xqIq, Vt,q = E′

q − x′dId

= xqIq + xchokeIq + j(E′q − x′dId − xchokeId

)

= (xq + xchoke) Iq + j(E′q − (x′d + xchoke) Id

)(4.28)

Id = YdE

′q + Fdδ, Iq = YqE

′q + Fqδ

= (xq + xchoke)(YdE

′q + Fdδ

)︸︷︷︸

VLoad,d

+ j(E′q − (x′d + xchoke)

(YqE

′q + Fqδ

))︸︷︷︸

VLoad,q

= VLoad,d + jVLoad,q. (4.29)

The voltage at bus OT20-10 is described by an algebraic equation and should beexpressed in terms of the state variables δ and E ′

q. First, equation (4.29) mustbe linearized:

VOT20−10 =√V 2OT20−10,d + V 2

OT20−10,q

∆VOT20−10 =VOT20−10,d,0

VOT20−10,0∆VOT20−10,d +

VOT20−10,q,0

VOT20−10,0∆VOT20−10,q

K7δ =VOT20−10,d,0

VOT20−10,0(xq + xchoke)Fq +

VOT20−10,q,0

VOT20−10,0(1− Fd (x

′d + xchoke))

K7E′q=

VOT20−10,d,0

VOT20−10,0(xq + xchoke)Yq +

VOT20−10,q,0

VOT20−10,0(1− Yd (x

′d + xchoke)),

(4.30)


where all the variables with the subscript “0” are obtained from the load flowcalculation, coefficients Yd, Yq, Fd, and Fq can be found in [63]. Small deviationsof the voltage OT20-10 from a steady-state are given by the following equality:

∆VOT20−10 = K7δ∆δ +K7E′q∆E′q. (4.31)

Thus, the new matrix Cc becomes:

Cc =[0,K7δ,K7E′q , 0, 0, 0

]. (4.32)

2

Note 4.3. It is interesting to note that linearization obviously yields linear mod-els, however, often it also gives rise to time-variant linear systems [39].

4.4.2 Measurement-based linearization

The analysis of real-life power systems is even more complicated than the studyof idealized yet sophisticated models. The main reasons are:

• In the privatized electricity market environment the data about powersystems is considered to be valuable property. Because of this, the ownerof the data is not willing to share this property. On the other hand, themodern power systems are interconnected, which means that the analystmust have the complete information in order to perform a credible analysisand design.

• Ageing of power system components makes the systems in some sense un-certain. For instance, the lifetime of a transformer is around 25 years.Clearly, its characteristics change with time and may significantly deviatefrom those of the new transformer.

• Missing data is another reason that devalues the analytical linearizationapproach, since certain parameters of the system may be unavailable.

• When linearized, electric power systems result in high-order models. Thatis, if a power system consists of 10 generators, each described by 4 differ-ential equations, than the system will be, at least, of 40th order. Such asystem is barely manageable, therefore a model reduction technique has tobe applied in order to obtain models meaningful for analysis and controllerdesign.

One way to overcome the aforementioned difficulties with the analytical lineariza-tion is to apply system identification. With system identification, a dynamicalequivalent of the system is obtained based on the analysis of input-output rela-tionships of the system. The modern system identification theory is a substan-tive branch of science which is nearly impossible to cover in a short section. Thenext chapter is therefore chiefly devoted to an elementary introduction to systemidentification techniques.

Chapter 5

System Identification

In this chapter provides a brief overview of several time domain identificationtechniques used in the framework of this thesis. Only key-points from a literaturesurvey on the subjects—without proofs or in-depth statistical properties analysisof the identification methods—are presented in this chapter.

First, a suitable notation is introduced, which will be used throughout thechapter, unless otherwise stated.

• N is the number of data samples available.

• q and q−1 are the forward and backward shift operators.

• δi,k is the Kronecker delta.

• T is a sampling period smaller than the Nyquist period.

• j =√−1 is the imaginary unit.

• s = σ + jω is the Laplace complex variable. Both z and s domains canbe used in this work, thus no difference is made between them, but thefollowing mapping is assumed: s = T−1 loge z.

• u [y] is an input [output] signal, having the following representation in thes and z domains: U(z) [Y (z)] or U(s) [Y (s)].

• n is the order of a model, that is the minimal number of differential ordifference equations necessary for describing the dynamics of the load.

• E(x) stands for the expected value of x.

• i, k, l,m ∈ N are indices taking nonnegative integer values.

• ai, bk are respectively the ith coefficient of the denominator polynomialand kth coefficients of the numerator polynomial of a transfer function.

• Although not always true, in most of the cases in this chapter, variablesdenoted by capital (lower case) letters stand for matrices (vectors).

49

50 Chapter 5. System Identification

5.1 Prony method based system identification

Linear time-invariant model of a process obtained with a Prony method-basedidentification is of the form:

H(s) =

n∑

i=1

Ris− λi

+D, (5.1)

where, in general, Ri andλi ∈ C; D ∈ R. Model identification is usually done inseveral steps. First the eigenvalues λi of the system are identified (signal identi-fication) followed by estimation of the residues Ri. Evaluation of the throughputterm D completes the model identification. The first part of identification willbe presented here as an illustration of the method. The rest of the identificationfollows a similar pattern and will be omitted.

Signal identification

The task is:

Given a measurement y(t), find the parameters Ai, λi of an estimate y(t) suchthat the estimate is optimal in a Least Squares sense.

Let us assume that y(t) may be represented in a form of weighted exponentialfunctions

y(t) =

n∑

i=1

Ai eλit, t = 0, 1, 2, . . . , N − 1. (5.2)

where, in general, Ai andλi ∈ C and y(t) is the estimate of the output signaly(t). Time t is sampled at a sampling period Ts smaller than the Nyquist period.For convenience, the following mapping ψ is used:

ψ : λi → zi,zi = eλiTs

(5.3)

Applying ψ, equation (5.2) may be rewritten as follows:

y(kTs) =

n∑

i=1

Aizki , k = 0, . . . , N − 1. (5.4)

Assuming that N samples of signal y are available, equation (5.4) can be writtenin the following form:

1 1 · · · 1z1 z2 · · · zn...

.... . .

...

zN−11 zN−1

2 · · · zN−1n

A1A2...AN

=

y(0 · Ts)y(1 · Ts)

...y((N − 1) · Ts)

. (5.5)

Being written in matrix form, equation (5.5) becomes ZA = Y. To be able to findthe unknown Ai’s, one has to determine zi’s which are roots of the characteristic

5.2. ARMAX 51

polynomial given by

n−1∑

j=0

a′jzji = 0, i = 1, . . . , n. (5.6)

Division of both sides of equation (5.6) by a′1 and denoting aj = −a′j+1

a′1give:

n−1∑

j=0

ajzji = 0, a0 := +1, i = 1, . . . , n. (5.7)

The values of aj may now be evaluated by using the Least Squares method. Todo so, a new vector aj is to be defined

aj , [0, 0, . . . , 0︸︷︷︸j zeros

, an, an−1, . . . , 1, 0, 0, . . . , 0︸︷︷︸N−n−j−1 zeros

]T . (5.8)

Multiplication of equation (5.5) yields the desired result: aTj Y = aTj ZA = 0,

j = 0, 1, . . . , N−n−1. ⇒ aTj Y = 0. The last equation must be solved for aj thatwill lead to the characteristic polynomial of equation (5.6). zi’s will be availablethrough factoring the characteristic polynomial. Finally, zi’s can be substitutedinto equation (5.5) to find the unknown A’s. The system of equations givenby (5.5) is an overdetermined system and should also be solved in the LeastSquares sense: A = (ZTZ)−1ZTY 1. So far only the signal identification waspresented. To obtain the full model of a plant, a more extensive algebra must beinvolved which will not be derived here. The reader may consult [4], [71], [77],or Appendix A for more detail on the theory behind this identification method.

5.2 AutoRegressive Moving Average witheXternal input (ARMAX) method

In its simplest form, the procedure of process identification may be formulatedas follows:

Given two vectors u, y find three sets of parameters ai, bk, cl, i = 1, n, k =1,m, l = 1, p of a transfer function such that the model output y best fits themeasured data y, being subjected to the same excitation signal u.

1The solution A = (ZTZ)−1ZTY should not however be blindly implemented as it is nu-merically ill-conditioned. Instead, the Singular Value Decomposition (SVD) should be utilized.


The desired parameters can be found as shown below. Assume that the processcan be described by the model (5.9)

y(t) + a1y(t− 1) + · · ·+ any(t− n)︸︷︷︸A(q)y(t)

= b1u(t− 1) + · · ·+ bmu(t−m)︸︷︷︸B(q)u(t)

+ e(t) + c1e(t− 1) + · · ·+ cpe(t− p)︸︷︷︸C(q)e(t)

(5.9)

or equivalently,A(q)y(t) = B(q)u(t) + C(q)e(t), (5.10)

with the parameter vector

θ = [a1, . . . , an, b1, . . . , bm, c1, . . . , cp]T.

Equation (5.10) can be reshaped by introducing y(t|θ)–an estimate of y(t):

C(q)y(t|θ) = B(q)u(t) + [C(q)−A(q)] y(t) (5.11)

and further rearrangement of (5.11) yields:

y(t|θ) = B(q)u(t) + [1−A(q)] y(t) + [C(q)− 1] [y(t)− y(t|θ)]︸︷︷︸ε(t|θ)

(5.12)

= φT (t|θ)θ. (5.13)

Minimization of the prediction error ε(t|θ) will yield the desired result–the pa-rameter θ, i.e., the parameters ai, bk, and cl. Once an objective function hasbeen chosen, the minimization can be done in many ways. If the analyst hasdecided that the objective function should be a quadratic function in θ, e.g.,1/2ε(t|θ)T ε(t|θ), then the optimization results in a closed form solution:

θ =

[1

N

N∑

t=1

φ(t)φT (t)

]−11

N

N∑

t=1

φ(t)y(t). (5.14)

5.3 State space identification methods

Now suppose that a model of the process is given by the state space model:x(t+ 1) = Ax(t) +Bu(t) + v(t)y(t) = Cx(t) +Du(t) + w(t),

(5.15)

where v(t) and w(t) are the process and measurement noise [78]. The followingstatistical characteristics are given:

E [v(t1)vT (t2)] = R1(θ)δt1,t2

E [v(t1)wT (t2)] = R12(θ)δt1,t2 (5.16)

E [w(t1)wT (t2)] = R2(θ)δt1,t2 ,

5.4. Subspace identification methods 53

where R are covariance matrices. The aim of the identification is to obtainthe matrices A,B, and C such that the response of model (5.15) best fits themeasured data.

The task is solved in several steps which involve solving a Riccati equationassociated with (5.15) and (5.16):

P = APAT +R1 −[APCT +R12

] [CPCT +R2

]−1 [CPAT +RT

12

]. (5.17)

Note that in (5.17) the argument of matrices R(θ) is suppressed for conciseness.

Having found P , one should compute the Kalman gain as

K =[APCT +R12

] [CPCT +R2

]−1. (5.18)

Next, one-step ahead predictions are calculated which can be further used forthe unknown parameter determination:

x(t+ 1|t) = Ax(t|t− 1) +Bu(t) +Ky(t)y(t) = Cx(t|t− 1) + y(t).

(5.19)

There is a direct link between models of the type (5.15) and polynomial transferfunctions:

x(t+ 1) = Ax(t) +Bu(t)

y(t) = Cx(t) +Du(t) + w(t)⇒

qIx(t) = Ax(t) +Bu(t)

y(t) = Cx(t) +Du(t) + w(t)(5.20)

y(t) = C [qI −A]−1Bu(t) +Du(t) + w(t). (5.21)

5.4 Subspace identification methods

Subspace identification methods are relatively new; however, they have alreadyproven to be a sound alternative to well-established identification algorithms.

The power of the subspace algorithms lies in the clear link to “old” methods,intrinsic Multi-Input-Multi-Output (MIMO) nature of subspace identification al-gorithms, the possibility to robustly estimate a possible order of the plant (donethrough inspection of singular values of the identified model), and lucid geo-metrical interpretation which actively connects human’s intuition [67]. Recentresearch [25] indicates that the use of subspace identification techniques can beutilized for the model-free Linear Quadratic Gaussian (LQG) controller design,which can be viewed as a very useful feature enabling the user to obtain an LQGcontroller without the need for identification of a process model.

Subspace identification methods exploit the so-called orthogonal projectionsof “the future outputs onto past and future inputs and the past outputs”. Math-ematically the task of system identification is solved in several steps [25].


Step 1Given two measurements ui and yj , i, j = 1, N , form the input matrices for the‘past’ and ‘future’ signals:

U1 =

u0 u1 · · · uk−1u1 u2 · · · uk...

......

...uBH−1 uBH · · · uBH+k−2

, (5.22)

U2 =

uBH uBH+1 · · · uBH+k−1uBH+1 uBH+2 · · · uBH+k

......

......

uBH+FH−1 uBH+FH · · · uBH+FH+k−2

, (5.23)

where k is the number of columns2, BH and FH stand for the backward andforward prediction horizons (number of data samples used for backward/forwardprediction), respectively. According to [67], the prediction horizons must be“large enough”. U1 and U2 denote matrices containing the ‘past’ and ‘future’signals. In a similar manner matrices Y1 and Y2 for the output yk are formed.

Step 2Define a new matrix W1 = [Y T

1 , UT1 ]

T and then calculate the matrices Lw andLu:

Y2/

[W1U2

]= Y2

[WT1 U

T2

] [ W1WT1 W1U

T2

U2WT1 U2U

T2

]† [W1U2

](5.24)

= LwW1 + LuU2. (5.25)

In equation (5.24), the operator ‘†’ is understood as the Moore-Penrose pseudo-inverse. For any square nonsingular matrix R, its pseudo-inverse R† = R−1.If R is a non-square matrix and the following holds: R†RR† = R†, RR†R =R, (R†R)T = R†R, (RR†)T = RR†, then R† is the pseudo-inverse of R.

Step 3The procedure terminates by computing the singular value decomposition (SVD)of Lw and estimating of the future output x1 [30]:

Lw = [Ua Ub]

[Sa 00 Sb

] [V Ta

V Tb

], (5.26)

where Ua, Ub are the output singular vectors, Va, Vb are the input singular vectors,and Sa, Sb are the singular values of the matrix Lw.

x1 = S1/2a V Ta W1. (5.27)

2In [25], it is noted that k has to approach infinity in order to attain unbiased estimates. Asin practice all data sequences are of finite length, unbiased estimates are obtained in a slightlymore sophisticated way.

5.5. Illustrative examples 55

5.5 Illustrative examples

In this section several numerical examples are presented to exemplify the use ofidentification algorithms for obtaining dynamical equivalents of a power systemload. All signals used for system identification have been generated with the helpof Matlab and Simulink [7], [11].

5.5.1 Constant impedance load identification

Obviously, identification methods can be either applied to field measurements orto data obtained by means of a computer simulation. In the former case, the dataare unavoidably corrupted by noise which has to be filtered before performingidentification. In this chapter only simulated data is analyzed, for the analysisof “clean” data allows us to better evaluate the performance of the identificationmethods. However, for the sake of consistency, white noise having certain shape(described later in this section) is added to the plant output y(t) in order tomodel a real-life case, i.e., the presence of measurement noise.

Throughout the section, irrespective of the case study, be it noisy or noise-freecase, linear models of a load are obtained as shown below:

1. A detailed model of the process is created. In the present study in all cases,Matlab and Simulink [11] were used for process modeling.

2. The system is set in motion by applying a pre-specified excitation signal.Here, one can either use a white noise or other small-magnitude excitation.In a power system, it is more practical and easy realizable to simply increasethe voltage than to install a white noise generator having enough powerto excite modes of interest. To model this situation in simulations, thesystem is excited by changing the supply voltage by a factor of 1.05 for ashort time period.

3. Time domain responses are recorded.

4. If needed, the response y(t) and u(t) can be preprocessed e.g., by filtering,removing trends, etc. Certain care must be exercised at this point–thefiltering can eliminate the high frequency contents of the signal. Thus, oneshould decide on bandwidth of the model to be identified.

5. An identification technique has to be applied.

6. Load models should be validated. If the match between the model andrecorded responses is unsatisfactory, specify a new excitation signal andrepeat the procedure from step 1.

In this work, the feeding voltage (Vs in Fig. 5.1) can be increased step-wise by,for example, 5%. This choice is dictated by the simplicity of voltage increasing–one can simply adjust the transformer tap-changer. However, since the system isessentially linear, a unity step is applied to the input of the system. As an output


System

T

ZLdVs

Is

Figure 5.1: A simple test system

one can use the current Is or power. Thus, signals used for the identification areVs (input) and Is (output).

The leakage reactance xT of transformer T and the Thevenin impedance ofthe system are assumed to be known and lumped into the impedance Zs. Supposethat the load dynamics can be satisfactorily described by a set of difference-algebraic equations (5.28):

x(t+ 1) = Ax(t) +Bu(t)y(t) = Cx(t) +Du(t) + w(t),

(5.28)

where x(t) ∈ Rn×1 is the state vector; u(t) ∈ Rm×1 is the input vector; y(t),w(t) ∈ Rp×n are the output and measurement noise vectors, respectively. A, B,C, and D are matrices of appropriate dimensions, associated with the aforemen-tioned variables x(t), u(t), and y(t). It should be emphasized that in generalthe order n of (5.28) is not known in advance, so it must be estimated. In thepresent case, m = p = 1, i.e., the load is a single-input-single-output (SISO)system. We also conjecture that the order of the load model is 2. At a laterstage, we shall decide whether this order is appropriate.The main objective of the identification is to find a set of parameters (A, B,C,D, and n) such that the loss function 1/2ε(t)T ε(t), where ε(t) is a predictederror (see page 52), is minimized [52].

Noise free case

Refer again to Fig. 5.1. In the first test the impedance Zs is equal to 1 + j10 Ω,Vs = 100 sin(100πt), and the load in this example is represented by an impedanceZL = 1 + j10 Ω. The test model is created in Simulink. We try to determinethe value of ZL by the identification algorithm N4SID. VLoad and ILoad wererecorded and processed in order to find matrices A,B,C, and D that minimizethe loss function 1/2ε(t)T ε(t). The minimization was performed with the helpof Matlab Identification Toolbox [53].

Fig. 5.2 shows the results of the identification. It is self-evident from Fig. 5.1that the match is very accurate, for this reason the value of the impedance ZLfound by the identification algorithm is not shown here. The good match should


0 0.05 0.1 0.15−8

−6

−4

−2

0

2

4

6

8

10

12

Time, sec.

Cur

rent

, AZ

s=1 + j10 Ω, Z

Ld= 1 + j10 Ω, V

s= 100sin(ω t)

Actual signal Identified signal

Figure 5.2: Constant impedance load identification

not come as a surprise since all calculations were conducted in Matlab with16 reliable digits after the decimal comma. Moreover, the load was essentiallylinear, and there was no noise to corrupt the data arrays u(t) and y(t). To verifythe ability of load identification in the presence of noise, the same experiment isrepeated, with white noise added to the input u(t) and the output signal y(t).

Noisy case

Figures 5.3–5.5 show the signals used for system identification. One can see thatboth the input u(t) = vLoad(t) and the output y(t) = iLoad(t) are corruptedby white noise (zero mean value and standard deviation σ =

√2). As in the

previous case, the Matlab Identification Toolbox was employed to obtain alinear model of the load. The ARMA method was the instrument used for loadidentification. Being represented in the state space form, the model is:

A =

[−71.8704 1−1291.3 0

], B =

[31.44841130.1

]

C =[1 0

], D = 0

(5.29)

Inspection of (5.29) shows that a pole-zero cancellation takes place in the left-halfplane. Thus, the minimal realization of (5.29) becomes:

A = −35.9352, B = 1130.5C = 0.0278, D = 0

(5.30)

Thus, the pre-selected order n = 2 was too high. But it was relatively easy tonotice and rectify the incorrectly guessed order. Let us compare the result with


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−6

−4

−2

0

2

4

6

8

Load current

Load

cur

rent

, [A

]

Time, [sec]

IdentifiedMeasured True

Figure 5.3: System’s output–current ILoad used for identification

0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55−8

−6

−4

−2

0

2

4

6

8

10Load current

Load

cur

rent

, [A

]

Time, [sec]

IdentifiedMeasured True

0.515 0.52 0.525

−6

−5

−4

−3

−2

−1

0

Figure 5.4: Zoomed version of Fig. 5.3


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−60

−40

−20

0

20

40

60Load voltage

Load

vol

tage

, [V

]

Time, [sec]

Figure 5.5: System’s input–voltage VLoad used for identification

the true characteristics of the load. From electric circuit theory it is known thatthe current of a simple inductive circuit is given by the differential equation:

Ldi

dt(t) +Ri(t) = v(t). (5.31)

The translation of equation (5.31) to the state space form is not needed, asit is already written in that form. Indeed, when equation (5.31) is written as

di

dt= −R

Li(t) +

1

Lv(t) = Ai(t) + Bv(t) (5.32)

and the output is explicitly shown

y(t) = i(t) = Ci(t), (5.33)

the state space form is readily obtained. Now let us compare the results fromthe identification with the correct impedance. Recalling that ZL = 1 + j0.1 Ω,the four matrices A, B, and C are

A = −RL = −1/(10π)−1 = −10π ≈ −31.1416

B = 1L ≈ 31.1416, C = 1.

(5.34)

Comparing the numerical values of equations (5.30) and (5.34), it is easy to

discover that the matrices B and C differ from B and C. This is neither a


Table 5.1: Comparison of “true” andidentified impedances

A CBTrue −31.416 −31.416

Identified −35.9352 −31.4279

numerical discrepancy of the identification method nor is it an error in the an-alytical derivation. This difference is due to non-uniqueness of the state spacerepresentation. The proof of this claim is straightforward and is therefore shownbelow. Consider an abstract system:

ξ(t) = Aξ(t) +Bu(t)z(t) = Cξ(t) +Du(t).

(5.35)

Suppose there exists a nonsingular matrix T . Then, application of a similaritytransformation (change of variables) ξ(t) = Tφ(t) results in an equivalent statespace system:

T φ(t) = ATφ(t) +Bu(t)z(t) = CTφ(t) +Du(t)

(5.36)

Finally,φ(t) = T−1ATφ(t) + T−1Bu(t) = Fφ(t) +Gu(t)z(t) = CTφ(t) +Du(t) = Hφ(t) +Du(t).

(5.37)

Therefore, systems (5.35) and (5.37) are absolutely equivalent when analyzingtheir input-output relations.

Note 5.1. If the matrix T is chosen to be the matrix of the eigenvectors of A, thenF becomes a diagonal matrix with the eigenvalues of A on the main diagonal3.This is the most common choice of T .

Now there is only check that has be done: calculate the product CB for bothsystems (5.30) and (5.34).

CB = 31.4279, CB = 31.416. (5.38)

Table 5.1 gives a comparison of the true and identified parameters of the impe-dance. As can be seen in the table, in spite of the presence of noise, the error inthe identification was very insignificant.

5.5.2 Complex dynamical structure identification

In this subsection an attempt is made to identify a load which apparently cannotbe represented by constant impedance. Again, the voltage source is the same,

3For conciseness of the exposition, no emphasis is put on the equality of geometric andalgebraic multiplicity of eigenvalues of A


Zs

White noise

Vs

Vld

+- v

VMeas

Vl

To WSpace1

II

To WSpaceStep

x’ = Ax+Bu y = Cx+Du

State-Space

ω100sin( t)+

Load

+i -Imeas

I

Π

V

+

-

VSource

11.05

Figure 5.6: Set-up for nonlinear load identification

represented by a constant impedance behind an ideal voltage source. The loadis comprised of a constant impedance connected in parallel with a simplifiedSVC which is installed to keep the load voltage at a constant level of VLoad =10√2 sin(100πt). The block diagram of the set-up is shown in Fig. 5.6 and

Fig. 5.7Fig. 5.8 shows the rms waveshape of the system voltage Vs and load voltage

VLoad. Here one can see that the load voltage was indeed approximately constanteven when the source voltage was experiencing an increase of 5%. One shouldnot expect exact identification of the load in this case because of nonlineari-ties produced by the SVC and noise. On the other hand, all the identificationmethods described in Sections (5.1)–(5.4) provide linear, time-invariant models.Nevertheless, one can try to incorporate the load nonlinearities into a linearmodel of the system. Application of ARMA gives the following estimate of theprocess

A = −427.9193, B = 1.6457e + 005C = 0.0023, D = 0.

(5.39)

In the form of a transfer function, the models is

I(s)

V (s)=

CB

s−A =384.5901

s+ 427.9193, (5.40)

which is just a filter having very fast dynamics. Thus, the dynamics of the com-plex load including an SVC could be satisfactory modeled as constant impedanceload, which is not a trivial conclusion.

Fig. 5.9 illustrates the quality of the fit of the estimated and measured wave-shapes. For the reasons mentioned before, the fit is not as good as in the caseof constant impedance load. But still the model (5.39) is a reasonable trade-offbetween the complexity of (5.39), accuracy, and the amount of effort put intothe identification procedure.


1

Vrms

+ -vVmeas

Vrms

To Workspace

Term

SCR

sign

alrm

s

RMS

Load

a

g

k

m

Electronic switch

Gate pulse

1/9

Gain

V >

Pulse generator

Capacitor

1

+

pu

Simple SVC

Figure 5.7: Block “Load” in Fig. 5.6

0 0.1 0.2 0.3 0.4 0.5 0.60.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

Vs

VLoad

RMS volatges

Vs, V

Loa

d, p

.u.

Time, seconds

VLoad

Vs

Figure 5.8: Load and system voltages in the presence of an SVC

5.6. Concluding remarks on system identification 63

0.38 0.4 0.42 0.44 0.46 0.48−10

−5

0

5

10

15

Time, [sec]

i s(t),

[A]

Instantaneous current is(t)

Measured Estimated

0.41 0.415 0.42 0.425 0.43−10

−9

−8

−7

−6

−5

−4

Figure 5.9: Comparison of estimated and measured currents

5.6 Concluding remarks on systemidentification

In this chapter a brief introduction to system identification has been presented.The main reasons for using identification techniques in power systems are theuncertain nature of the power system and ageing of power system components.Of course, it is preferable to explore every possibility of developing models ofthe system based on physical insights in order “not to estimate what we alreadyknow” [75]. Often a combination of modeling based on physical insights andsystem identification yields best results. These notes complete the presentationof basic principles of power system modeling and identification. Having created,analyzed, and verified a model of the system studied, the engineer can proceedwith the main task—design of a controller capable of fulfilling main control ob-jectives. In the next chapter we therefore present basic controller design methodsthat are relevant to the work reported in this thesis.

Chapter 6

Automatic Control SystemsSynthesis

The main focus of the previous two chapters was on various issues of powersystem modeling and identification. This chapter presents an introduction tothe foundations of one of the key tools of this thesis—automatic control systemssynthesis. The exposition of the material in this chapter is rather axiomatic;however, this approach will certainly suffice for the practical purposes of thisproject. We commence by giving a demonstration of linear control systems.

6.1 Linear control systems

Linear systems have already been encountered in the previous chapters, e.g.,equation (5.31). However, little was said about their properties or solutions tothe linear systems. We therefore proceed with an overview of the main propertiesof linear control systems. Consider the linear time-invariant system:

x(t) = Ax(t) +Bu(t)z(t) = Cx(t) +Du(t).

(6.1)

In equation (6.1), x(t) ∈ Rn is the vector of state variables, u(t) ∈ Rm is thevector of external input variables, z(t) ∈ Rp is the vector of output variables; therest of parameters are constant matrices of appropriate dimensions. Equations ofthe type (6.1) are often used for describing various processes in different branchesof modern science, e.g., in electrical circuits and control theory, signal processing,etc. The output z(t) of the system (6.1) is uniquely given by

z(t) = C

exp(At)x0 +

t∫

0

exp(A(t− τ))Bu(t)dτ

+Du(t), (6.2)

65

66 Chapter 6. Automatic Control Systems Synthesis

where x0 is the initial state vector. The expression (6.2) is bounded if all eigen-values of A have negative real part1. In such a case, the state matrix A is saidto be stable or a Hurwitz matrix2.

Stability is a very important concept in its own right, however, for practicalneeds it is desired to quantify the system’s ability to be controlled in an appropri-ate way by manipulating the external input u(t). Two analytical tools that playan important role in control theory and satisfactorily serve the purpose statedabove are: state controllability and state observability concepts [79].

Definition 6.1. An event is a pair (x, t) ∈ X × T . The event (xf , tf ) can bereached from the event (x0, t0) if and only if there is a trajectory of (6.1) on[t0, tf ] whose initial state is x0, and final state is xf , i.e., ∃ u ∈ U [t0,tf ) such thatφ(t0, tf , x0, u) = xf . This is equivalent to the statement: x0 can be controlledto xf . In shorthand, the last statement is the same as (x0, t0) ; (xf , tf ) orx0 ; xf . 2

Definition 6.2. The events (x, σ) and (z, σ) are distinguishable on the interval[σ, τ ], or in time T = τ − σ, if there is some t ∈ [σ, τ ] and control u ∈ U [σ,τ)that distinguishes them. If (x, σ) and (z, σ) are distinguishable on at least oneinterval, then they are called distinguishable. A systemM is observable on theinterval t = [σ, τ ] if for every pair of distinct states x and z the events (x, σ) and(z, σ) are distinguishable on the interval [σ, τ ]. 2

The formal definitions above are mathematically rigorous, but is of little helpwhen determining controllability numerically. Therefore, an alternative test thatis well suited for determining controllability on a digital computer is presented[66].

Definition 6.3. Two matrices C and O defined as

C =[B,AB, . . . , ABn−1] (6.3)

O =

CCA...

CAn−1

. (6.4)

are called the controllability and observability matrices of the system (6.1). Thus,a system is said to be state controllable if C has full rank and the system is stateobservable if O has full rank. 2

1One should make a distinction between external and internal stability when studyingdynamical systems. There is a subtle difference between these two concepts: internal stabilityalso implies external, but the converse is only true when the system is both controllable andobservable [39].

2Sometimes in literature on the subject, the matrix A is simply called “Hurwitz”. We,however, believe that Adolf Hurwitz (1859–1919) was rather a splendid German mathematicianthan a matrix, even one with strictly negative eigenvalues.

6.2. Linear optimal control 67

Simplifying matters, the term “controllable system” implies that for any ini-tial condition, the states of the system can be driven to any point of Rn in finitetime.

Many authors, e.g., [39], [76] question practical relevance of the above defi-nitions; however, it is our belief that these definitions do have certain value atleast for analysis of linear automatic control systems.

6.2 Linear optimal control

Linear optimal control (LOC) is one of the most elegant and widely used toolsin control theory. A full coverage of LOC in a single chapter is barely possible;instead we confine ourselves to brief surveying of the main concepts, definition,and compact closed-form solutions.

The main task of LOC can be formulated as follows:

Given an LTI system, find a state feedback control law u = −Kx that minimizescertain pre-specified cost functional (optimality condition).

Mathematically, this is rephrased as

minuJ(x, t, u)

Subject to x(t) = Ax(t) +Bu(t).(6.5)

Intuitively, it is expected that the greater the control energy applied to thesystem, the faster the system reaches a new steady-state. It is also known thatbecause of the actuator saturation or some other reasons, the admissible inputu(t) is usually bounded. Therefore, a trade-off is required between the controlenergy and performance of the controlled system.

One way to make such a trade-off is to penalize both excursions of the statevariables and control effort energy. In practice, the cost functional J often hasa special form–that of a quadratic function in x(t) and u(t):

J(x, t, u) = xT (tf )Sx(tf ) +

tf∫

t0

xT (t)Qx(t) + uT (t)Ru(t)dt, (6.6)

where the pre-specified matrices Q,S are semi-positive definite, and R is positivedefinite. The matrices Q,S, and R are the main design “knobs” that enable theengineer to penalize some state or input variables. Thus, the LOC problemreduces to the so-called Linear Quadratic Regulator (LQR).

The LQR problem is not new; there is a well-known unique solution to theLQR problem which is given by


Table 6.1: Eigenvalues of the closed-loop system

Without LQR With LQR

λ1 j3.9633 −1.8044 + j4.3542λ2 −j3.9633 −1.8044− j4.3542

Theorem 6.4. For a given pair (t0, tf ) there exists an absolutely continuoussymmetric n× n function P (t) defined for t ∈ [t0, tf ], which satisfies the matrixRiccati differential equation on [t0, tf ]:

P = PBR−1BTP − PA−ATP −Q, P (tf ) = S. (6.7)

Further, the feedback control u(t) = −R−1BTPx(t) is the unique optimal controlfor x0 on the interval [x0, xf ]. 2

A proof of the theorem can be found in [79]. To illustrate the presentedtheory, consider the following

Example 6.5. Assume that the system under consideration is given by the systemof equations

[δω

]

︸︷︷︸x(t)

=

[0 100π

−0.05 0

]

︸︷︷︸A

[δω

]

︸︷︷︸x(t)

+

[00.1

]

︸︷︷︸B

u(t). (6.8)

Equation (6.8) represents a simplified model of a single-machine infinite-bus sys-tem [63]. We shall use full-state linear optimal feedback regulator (LQR) tocontrol this system. Let R = S = I2 be identity matrices of appropriate dimen-sions andQ = diag(0.25, 1). Solving the Riccati equation for P = 0, the followingcontrol law is obtained:

u = −R−1BTP [δ, ω]T

= −I[0 0.1

] [ 0.0812 2.07112.0711 360.8726

] [δω

]

= −0.2071δ − 36.0873ω.

(6.9)

To evaluate the performance of the controller, let us compare the eigenvaluesof the system with and without the LQR controller. As Table 6.1 reveals, themarginally stable unforced system (6.8) is fairly stable when the control loop(6.9) is closed. 2

Remark 6.6. It is noteworthy mentioning that not only does the LOC theoryprovide elegant, closed-form solutions to a wide class of control problems, butit also has the potential to yield neat solutions for a certain class3 of robust

3Actually, this method has already undergone significant development. To the best knowl-edge of the author, the problem of matched and unmatched uncertainties, robust poles place-ment, and uncertainties in the input matrix have been successfully solved for both linear andnonlinear systems.

6.3. Robust control 69

control problems. This theme is treated in every detail in [51]. Therefore, thereare two different approaches to attacking the problem of robust linear controllersynthesis: i) to apply the LOC theory for robust control or ii) to directly pro-ceed with one of the robust control techniques. The former is seemingly morestraightforward, while the latter possesses more flexible and general tools. Thenext section comprises merely an amalgamation of the basics of robust controltheory.

6.3 Robust control

As was stated in Chapter 5, modeling of electric power systems is not a trivialtask even if the engineer has succeeded in collecting all the necessary informa-tion about the power system. Aging of power equipment and uncertain load4

complicate modeling, analysis, and control of electric power systems.This feature of power systems suggests the choice of control strategy–robust

control–if precise and “guaranteed” control of some quantities, e.g., voltage isrequired. It is very likely that the potential of robust and optimal control tech-niques will have commanding influence on power engineering; however, this timeis yet to come. For this and other reasons, the foundations of H∞ control the-ory are presented in detail. We hope that this will help in disseminating thesetechniques among practicing electrical engineers.

The origins of H∞ optimal control theory were established in the early 1980’sby G. Zames, see e.g., [90]. Since then it has been intensively studied by thecontrol community. A brief introduction to this fascinating theory is given below.This sections establishes the relationship between the robustness issue and H∞ -norm minimization. Most of the section makes an extensive use of frequencydomain due to nice geometrical interpretations of the main concepts, however,time-domain solutions to the standard H∞ problem are also presented.

The standard H∞ control problem can now be formulated. Before this, itis instructive to introduce the main configuration of the controlled plant andcontroller itself. Fig. 6.1 shows the standard block diagram of the dynamicsystem which is controlled by an external controller K. It should be stressedthat the weighting functions which are the main tools for fulfilling the controlspecifications are already absorbed into the plant G. At the present time, thiswill not be discussed, as the structure of the plant is different in the setting forthe frequency and time domains.

The main purpose of the linear robust controllers in this thesis is to providetight voltage control at the terminals of a synchronous generator. Furthermore,the model of the generator is assumed be known exactly, however, the loadvariations are treated as unstructured model uncertainty. Now the problem ofH∞ -controller synthesis can be stated

4More specifically, both the level and consist of the load at a given instant of time areuncertain. In system planning studies, power system loads can be forecasted; however, neithertype nor dynamical characteristics of the load are specified.


G

K

y

zw

u

Figure 6.1: Standard diagram of the plant and controller

C P

w

z++

+

r

Figure 6.2: SISO plant

H∞ Controller Problem formulation

Given the generalized plant G, exogenous inputs w, outputs z, and controlspecifications, find all admissible controllers K such that the H∞ -norm of thetransfer matrix from w to z is minimized, subject to the constraint that all K’sstabilize the plant G.

We commence by reviewing frequency-domain methods for H∞ controller syn-thesis.

6.3.1 Frequency domain solutions to H∞ control problem

The system to be controlled is depicted in Fig. 6.2. For simplicity, the systemis assumed to be SISO. In this figure, P (s) is the controlled plant, C(s) is thecontroller, w represents a disturbance, and z is the output. The resultant transferfunction from r to z is given by the expression:

H(s) =C(s)P (s)

1 + C(s)P (s)(6.10)

The ratio (1 + C(s)P (s))−1 is called the sensitivity function of the feedbacksystem and characterizes how sensitive to disturbances the compensated plantis. To be in line with the accepted practice, we denote the sensitivity functionby capital S. In the ideal case, the sensitivity function S should be zero. Inpractice it is, however, unrealistic to require zero sensitivity. Instead an upper


-1

L(jω)L0 (jω)

Im

0 Re

L

L0

Figure 6.3: Nyquist plot

bound on the peak value of the sensitivity function is specified for a certain rangeof frequencies. That is, one sets an upper limit on ‖S‖∞:

‖S‖∞ = supω∈R|S(jω)| (6.11)

The sensitivity function is a function of frequency, i.e., it varies with frequency,assuming different values for different frequencies. As this is undesirable, aweighting function W (jω) is introduced in order to weaken this frequency de-pendence.

‖WS‖∞ = supω∈R|W (jω) · S(jω)| (6.12)

The loop gain L = PC is another quantity which plays an important rolein H∞ optimization and is closely connected to robustness of the system, seeFig. 6.2.

Due to plant uncertainties (in power systems this can be the level of loadingof the system), the actual plant parameters differ from the nominal ones. Todistinguish between them, the actual plant is denoted by L and the nominalplant as L0. Both the plants are stable if the corresponding Nyquist plots donot encircle the point (−1, j · 0). Following in steps of [47], it may be noted thatplant is stable if the following inequality holds:

|L(jω)− L0(jω)| < |L0(jω) + 1| , ∀ω ∈ R. (6.13)

Graphical interpretation of the inequality is given by Fig.6.3. Inequality (6.13)can be rearranged as follows:

|L(jω)− L0(jω)||L0(jω)|

· |L0(jω)||L0(jω) + 1| < 1, ∀ω ∈ R (6.14)

Now the complementary sensitivity function of the closed loop plant T0 is intro-duced:

T0 = 1− S0 = 1− 1

1 + L0=

L01 + L0

(6.15)


In equation (6.15), S0 stands for the sensitivity function of the nominal plant.Equations (6.14)–(6.15) can be combined to produce:

|L(jω)− L0(jω)||L0(jω)|

· |T0(jω)| < 1, ∀ω ∈ R. (6.16)

The multiplier |L(jω)− L0(jω)|/|L0(jω)| is called the relative size of the per-turbation of the gain loop L from its nominal value L0. If the relative size of theperturbation is bounded, we can write:

|L(jω)− L0(jω)||L0(jω)|

6 |W (jω)| , ∀ω ∈ R, (6.17)

where W (jω) is the aforementioned weighting (given) function. The followingexpression can be obtained after simple manipulations:

|L(jω)− L0(jω)||L0(jω)|

1

|W (jω)| |T0(jω)W (jω)| < |W (jω) · T0(jω)| , (6.18)

|W (jω) · T0(jω)| < 1, ∀ω ∈ R. (6.19)

The last result can be interpreted as follows: for any disturbance that is boundedbyW (jω) (inequality (6.19) holds) the closed-loop plant remains stable (inequal-ity (6.18) holds). Since H∞ is essentially norm minimization, the last equationshould be written in terms of norm notation:

‖W (jω) · T0(jω)‖∞ < 1. (6.20)

Thus, the open loop plant remains stable for any disturbance that is boundedby equation (6.17), if inequality (6.20) holds. However, it must be explicitlystated that stability alone is not the ultimate goal. For most practical systemsthere are two competing requirements: stability and performance. Not all stablesystems perform well, though all systems performing well must be stable, exceptexplosive devices, of course. This consideration in some sense forbids the use ofinequality (6.20) for an H∞ loop shaping and requires a new minimization objec-tive. The so-called mixed sensitivity problem was developed [47] to incorporatethe performance specification into the H∞ controller design. In mathematicalterms it is usually expressed by

∥∥∥∥W1SVW2UV

∥∥∥∥∞, (6.21)

whereW1 andW2 are weighting functions that are the “knobs” of the H∞ -normminimization, S and U are the sensitivity function and input sensitivity function,respectively. The new function V is used to increase the design flexibility.


There is a nonunique solution to the standardH∞ -optimal regulator problem[34],[47] [

XY

]= Z−1

λ

[AB

], (6.22)

where the optimal controller is K = Y X−1, normally A = I, B = 0, and Zλ isdetermined from equation (6.23)

[0 I

−G∗12 −G∗

22

] [λ2I −G11G

∗11 −G11G

∗21

−G21G∗11 −G21G

∗21

]−1 [0 −G12

I −G22

]= Z

∗λJZλ, (6.23)

where (·)∗ operates as X∗(s) = XT (−s) and the plant transfer matrix G isdefined as

G =

[G11 G12G21 G22

]=

W1V W1P0 W2−V −P

. (6.24)

6.3.2 Time domain solutions to H∞ control problem

Time-domain solutions to standard H∞ control problem have been in the focusof attention of control society for quite a long time and resulted in neat, compact,and relatively “simple” expressions. Below we replicate the ones that will shortlybe used in the second part of this manuscript.

Some mathematical preliminaries which will facilitate the further treatmentopen this subsection.

Lemma 6.7. Let matrix H be defined as

H ,

[A RQ −AT

],

and suppose H ∈ dom(Ric) and X = Ric(H). A,Q and R ∈ Rn×n with Q andR symmetric. Then:

1 X is symmetric.

2 λi(A+RX) < 0,∀ i.

3 X satisfies the algebraic Riccati equation ATX +XA+XRX −Q = 0.

2

A proof of the lemma and details on the notation used can be found in [21] and[92]. For now, the most important detail is that X satisfies the associated Riccatiequation. This fact will be used when tackling with suboptimal H∞ controllers.

Unlike the situation with LQR, in the present case close-form solutions to theoptimal H∞ control problem cannot be obtained as the optimal control requiresan iterative search over the set of all admissible controllers.


Let the transfer matrix of the plant G be partitioned as

G =

A B1 B2C1 0 D12C2 D21 0

(6.25)

and the following assumptions hold true

a (A,B1) and (A,B2) are stabilizable.

b (C12, A) and (C21, A) are detectable5.

c DT12 = [ 0 I ].

d

[B1D21

]DT21 =

[0I

].

We now present closed-form suboptimal H∞ controllers which are given by

Theorem 6.8. Let the following two matrices be defined as

H∞ ,

[A γ−2B1BT

1 −B2BT2

−CT1 C1 −AT

]

J∞ ,

[AT γ−2CT

1 C1 − CT2 C2

−B1BT1 A

],

where γ is a given real number.There exists an admissible controller s.t. ‖Tzw‖∞ < γ if and only if the followingconditions hold:

a H∞ ∈ dom(Ric) and X∞ , Ric(H∞) > 0.

b J∞ ∈ dom(Ric) and Y∞ , Ric(J∞) > 0.

c ρ(X∞Y∞) < γ2.

In addition, if these conditions hold, one such controller is given by

Ksub(s) =

[A∞ −Z∞L∞F∞ 0

], (6.26)

where the matrices A∞, F∞, L∞ and Z∞ are defined as

A∞ , A+ γ−2B1BT1 X∞ +B2F∞ + Z∞L∞C2

F∞ , −BT2 X∞

L∞ , −Y∞CT2

Z∞ ,[I − γ−2Y∞X∞

]−1.

2

5The pair (C,A) is said to be detectable if for some L the matrix A + LC is a Hurwitzmatrix.


M

Q

y u

∞

Figure 6.4: Parameterization of all suboptimal H∞ controllers

Again, a proof of the theorem can be found in [21]. It is possible to param-eterize all the suboptimal controllers Ksub that are given by Theorem 6.8. Weonly show the final result. For technicalities and a proof the reader can consult[21] or [92].

Theorem 6.9. Suppose the conditions of Theorem 6.8 are satisfied. Then, forany choice of Q : ‖Q‖∞ < γ and Q ∈ RH∞, the set of all admissible suboptimalcontrollers with ‖Tzw‖∞ < γ is

M∞(s) =

A∞ −Z∞L∞ Z∞B2F∞ 0 I−C2 I 0

.

2

Figure 6.4 shows a schematic block-diagram of the parameterized suboptimalH∞ controllers. To gain momentum, let us design anH∞ -controller for a systemequivalent to that of Example 6.5 and compare the results.

Example 6.10. Consider the system

[δω

]=

[0 100π

−0.05 −0.01

] [δω

]+

[00.1

]u(t), (6.27)

C = [1, 0], D = 0.

The main difference between this system and the system of Example 6.5 is smalldamping added to the system in order to avoid having eigenvalues on the imag-inary axis. As a result, the element (2, 2) of matrix A is different. The followingcontrol specifications are set6:

Closed-loop bandwidth: ωB = 12 rad/sec.

Peak of sensitivity function: Smax = 1.1.

Steady state: Approximate integral action at low frequencies.

6These control specifications are set arbitrarily for demonstration purposes only; however,these values are quite realistic.


Table 6.2: Comparison of eigenvalues of the systemcontrolled by the LQR and H∞ controllers

LQR H∞λ1,2 −1.8044± j4.3542 −4.0925± j4.4791λ3 — −4443

Obj. function: For this example, the objective functions is chosen as

J(S,K) = argminC

∥∥∥∥wPSCS

∥∥∥∥∞, wP =

0.67s+ 10

s+ ε,

where ε = 10−6.

The H∞ -norm minimization was performed with the help of the µ-toolbox inMatlab [13], which yielded the value of γ = 1.08. The resultant suboptimalcontroller C(s) has a more complex structure as compared to the LQR design,however, the performance of the H∞ controller is better.

C(s) =4617s2 + 46.17s+ 7.254e4

s3 + 4451s2 + 3.64e4s+ 0.01348.

Table 6.2 lists the eigenvalues of the system controlled by the LQR and H∞ con-

trollers. The input to the H∞ optimal controller is the angle deviation of thegenerator’s shaft and the output is the control signal u(t). 2

Note 6.11. It is worth mentioning that selection of weights for the design of bothLQR and H∞ plays an essential role. As yet, selection of weights in robust andoptimal control is more art than science, since from case to case, different ad hocapproaches have to be tried in order to obtain “the best design.”

This example concludes the introduction to basics of LOC and H∞ controllerdesign.

6.4 Control Lyapunov function

Both LQR and H∞ control techniques that were described earlier in this chap-ter are essentially linear control tools. As is known, linear systems are alwaysidealizations of real-life systems. For many practical purposes, the linear systemtheory provides a wide spectrum of excellent techniques for deign and analysiswith solutions marrying simplicity and effectiveness. However, in general thesituation is simple only when the system studied is initially linear or “almostlinear”, i.e., the system nonlinearities do not qualitatively affect the solution.When a linear system is obtained by linearization of a nonlinear model, an ex-plicit assumption is made on the magnitude of deviations of the system’s states

6.4. Control Lyapunov function 77

from an equilibrium point–the deviations should be small for all times. If thisassumption is violated, the results from the linearized model may not be reli-able. If this is the case, the use of linear tools becomes a choice of questionablevalue and application of nonlinear techniques should be considered as a viablealternative.

When studying transient stability issues in electric power systems, nonlinearmodels are commonly used [63]. The necessity for modeling of power systemwith nonlinear differential equations originated from the following fact: When apower system is subjected to a sudden and severe disturbance, its states (gener-ators’ shaft angular frequencies, exciter voltages, etc.) may deviate significantlyfrom the pre-fault steady state. Moreover, a new post-fault equilibrium may bedifferent from the pre-fault, which implies that linear models for pre- and post-fault are also different. All these considerations suggest that nonlinear systemanalysis should be applied to a certain class of power system studies.

There are many nonlinear analysis tools that can be used in power systemapplications. One of the most commonly used techniques is the Lyapunov7 directmethod and its modifications, see [69], [40], and [29]. Below we briefly discussthe so-called Control Lyapunov Function (CLF) and its use in electric powersystem applications.

Stability in the sense of Lyapunov

Let the unforced system be described by a set of differential equations:

x = f(x), (6.28)

where x(t) ∈ Rn is a vector of state variables and f ∈ Rn is a given vector-valuedfunction. Without loss of generality, assume that the system has an equilibriumpoint at the origin. Then the system (6.28) is said to be asymptotically stableif and only if there exists a C1 function V (x) such that [85]

1. V (0) = 0,

2. V (x) is positive definite ∀x\0,

3. LfV (x) < 0, ∀x,

where LfV (x) stands for the derivative of V (x) along f .

7Alexandr M. Lyapunov (1857–1918), an excellent Russian mathematician whose scientificwork had and still has a significant impact on the theory of differential equations, potentialtheory, stability of systems, and probability theory. Both A. M. Lyapunov and A. A. Markovwere students of P. L. Chebyshev.


Control Lyapunov Function

Having found a Lyapunov function V (x) for the unforced system, the functioncan be used for robust controller synthesis. We now briefly illustrate this state-ment. Suppose there exists a Lyapunov function V (x) for the unforced system(6.28). Then LfV (x) is negative

LfV =∂V

∂x

dx

dt=∂V

∂xf (x) < 0, (6.29)

since V (x) is a Lyapunov function. This implies that for the original system(6.28) with controls added,

x = f(x) + g(x)u (6.30)

the derivative along f + gu is

Lf+guV = LfV + (LgV )u (6.31)

Recalling that LfV < 0, one easily establishes the sufficient condition for stabil-ity of the controlled system as

(LgV )u =

(∂V

∂xg

)u < 0. (6.32)

Choosing the function u as

u = −LgV

= −∂V∂x

g (6.33)

yields a control law which, at least, locally stabilizes the system under consider-ation, since

Lf+guV = LfV + LguV

= LfV −(∂V∂x g

)2< 0,∀x.

(6.34)

This result finalizes the brief introduction to CLF theory. This theory will beutilized in the second part of the thesis.

Note 6.12. It should be stressed that in spite of the fact that CLF’s have been inthe focus of attention of engineers for a long time, there are still some obstaclesto utilizing CLF’s. The main reason is the complexity of Lyapunov functions forhigh-dimensional systems. Normally, a series of simplifications is made in orderto obtain manageable CLF’s [69].

6.5 Noncasual control

Non-causal controllers are sometimes used in applications when a controlled sys-tem has some properties that make traditional control techniques ineffective. For

6.5. Noncasual control 79

Plant

FeedforwardController

− delaysTe+

-

r u yFeedbackController

++

Figure 6.5: Linear system controlled by a noncasual controller

example, if perfect tracking is the primary objective of the controller synthesis,one may need to compensate for non-minimum phase zeros of the system. Non-casual controllers have proven to be efficient in solving this type of problems, see[56] and references therein.

The basic idea behind the theory of noncasual control is as follows: If cer-tain information about the system is known beforehand, this knowledge can beutilized in order to synthesize such a manipulated input that control objectivesare best fulfilled. For instance, if the reference signal is known in advance, thiscan be used to generate a control signal which would force the system to followa pre-specified trajectory.

In a simplified form, a linear system controlled by a noncasual controller isshown in Fig. 6.5 In Fig. 6.5, two regulators—feedback and feedforward—areused to control the system. Also a delay function exp(−sTdelay) is introducedto achieve the noncasual effect. The feedback controller is utilized to attenuatenoise and/or disturbances, while the main aim of the feedforward is to improvethe transient of the overall system.

An overview of basics of noncasual control concludes this chapter which wasentirely devoted to reviewing the theoretical foundations of the control strategiesused in the case studies. In the following part of the thesis, most of the theoreticalresults will be applied to two case studies.

Part II

Applications

81

Chapter 7

Oxelosund Case Study

This chapter opens the second part of the thesis work which is chiefly devoted topractical applications of the theory presented in the first part of the manuscript.Two case studies are treated in every detail. For both case studies, several solu-tions motivated by Part I are synthesized and evaluated by means of computer-aided simulations. The simulations are complemented by a discussion on thequality of the solutions and possible extensions of the work.

7.1 Background

The present case study originates from a steel plant located in Oxelosund. Asimplified model of the electrical network of the steel plant and its neighborhoodis shown in Fig. 7.1. When reduced to a reasonably compact size, a model ofthe network consists of the following main components: a synchronous generatordenoted here as G1, a high voltage grid, two transformers T4, T7, a large syn-chronous motor BFF7, and a high priority load that is considered to be sensitiveto variations of the voltage level. Nowadays, generator G1 is connected to a highvoltage grid via a step-up transformer T4, see the left part of Fig. 7.1. In thepresent configuration, the motor BFF7 is located in an electrical vicinity to theload; both the load and motor are fed via the step-down transformer T7.

The most simple and inexpensive method to start a synchronous motor is tounload it and connect directly to the electrical grid. When the shaft of the motorreached the synchronous speed, the motor is loaded. This method is practisedat the Oxelosund steel mill. As a result, at the start-up procedure large inrushcurrents give rise to voltage sags that may have detrimental influence on thesensitive load [86]. Furthermore, the voltage sags caused by the motor affectboth the load and the motor itself. In the most unfortunate circumstancessome parts of the load (this is an aggregated load) may be disconnected by theundervoltage protection and/or the motor may cease to accelerate. In any case,both the motor and the load are strongly affected by the voltage reduction.

83

84 Chapter 7. Oxelosund Case Study

G1 M

T7

Load

PCC

OT20-10

BFF7

T4

M

Current limiting reactor

T7

Load

PCC

OT20-10

BFF7

G1

Figure 7.1: Simplified model of the Oxelosund steel plant. The left plot depictsthe present connection, the right one shows a possible future scenario

At the present time, the problem of voltage sags is solved as follows: prior tostarting the motor, the tap-changer of BFF7 is manually adjusted such that thenetwork voltage exceeds the nominal by approximately 7%. When the motor isconnected to the grid, a voltage sag occurs, which brings the voltage of the motorterminals down to normal. When the motor reaches the synchronous speed, itis loaded and then the tap-changer of BFF7 is returned to its previous positionthat corresponds to the nominal voltage. This method of voltage sags mitigationis clearly slow and relatively unreliable as it requires human intervention in theprocess of voltage control.

There are several alternative solutions to the problem of voltage sags. In ourparticular case, in addition to the voltage sag mitigation methods described inSection 3.3, one can also consider some other alternatives. They include:

• Soft motor starting. A couple of thyristors connected in anti-parallel areused to gradually increase the feeding voltage thus avoiding exceedinglyhigh start-up currents. This solution is nearly ideal, but it still is quiteexpensive–the motor rating is high and the starting currents are corre-spondingly high ∼ 2 kA/phase at a voltage level of 10.5 kV.Autotransformer starters can also be utilized in order to alleviate the volt-age sags caused by a motor start. In this case, the tap controlled in asuitable mode provides a soft start up, increasing the applied voltage inseveral steps, e.g., 50, 65, or 80%. This solution has a significant drawback–the transformer rating must be approximately 4 times that of the motor,which is quite costly since the transformer is in effective use for a fewseconds only.

• Static VAr compensators (SVC). The main idea behind the use of staticVAr systems for voltage sag mitigation is as follows. When a voltage sagwas detected, the control circuits of the SVC respond in such a manner

7.1. Background 85

that the SVC generates more reactive power and thus maintains the pre-specified voltage level. Normally, SVC’s are capable of very fast and precisevoltage control. Appropriately designed control circuits may provide robustoperation and tight voltage control for a wide range of operating conditions,see [24], [60], and [61]. However, it should be noticed that the price of anSVC-based solution may be very high.

• Ferroresonant transformer (FT) can also be utilized for mitigation of volt-age sags caused by a motor start. However, among the others, this ap-proach is less attractive in terms of both technical performance and ex-penses, since the rating of a ferroresonant transformer should be 5 timesthe motor rating. Moreover, FT’s are lossy, see Section 3.3.6.

It can therefore be deduced that the voltage sags caused by a motor start can bemitigated by various solutions which however inherit one common peculiarity–high installation and/or maintenance costs.

A different solution to the problem was reported in a previous work [86],[87]. In [87], the authors suggested to re-design the network’s topology in such away that generator is disconnected from the high voltage grid and re-connecteddirectly to the busbar OT20-10, see the right part of Fig.7.1. A series of sim-ulations was run which revealed that the direct connection increases the shortcircuit capacity of the low voltage busbar reducing thereby the magnitude of thevoltage sag caused by start-up of the motor [86].

One serious obstacle to implementation of the proposed solutions is the factthat after the re-design of the network, the short circuit currents do exceed thetolerable limits. In other words, the busbar is incapable to withstand such highcurrents which may lead to mechanical damage to the busbar. This technicalchallenge was recognized in [87] and a remedy was suggested–semiconductorcurrent limiting devices (CLD). It was shown that a properly sized and designedCLD may limit the fault currents to acceptable limits.

In this case study, we consider the same system, but propose a differentmethod for eliminating the problem of exceedingly high fault currents. It wasconjectured that there could exist an alternative solution that is efficient but lessexpensive–this was the main motivation for seeking another solution.

A literature survey has indicated that one of the most simple, effective, andinexpensive methods of fault current reduction is to use a choke. In our case,it suffices to connect the generator to the busbar OT20-10 through a choke inorder to limit fault currents to acceptable limits. The choke may be sized suchthat the short circuit capacity of OT20-10 is high enough to alleviate the voltagesags; on the other hand, the fault currents remain reasonably low. In addition, toenhance the voltage control, an auxiliary voltage control circuits can be designed.

In the subsequent sections, we demonstrate the proposed method and illus-trate it through computer-assisted simulations. First, the short circuit currentsare calculated and the current limiting choke is sized.


Z 1

E G1 Z 2

Z 3

E m o t .

E syst .

OT20-10

Figure 7.2: Equivalent scheme of the studied part of the Oxelosund plant

Table 7.1: Equivalent circuit parameters

E′′g E′′

s E′′m

1.05pu = 6.3653kV 1pu = 6.0622kV 0.9pu = 5.45596kV

Z1 Z2 Z30.0024 + j0.1748Ω 0.0117 + j0.2655Ω 0.002 + j6.615Ω

7.1.1 An Example of Simplified Short-Circuit Calculation

If a short circuit occurs on the busbar OT20-10, there are three main contributorsto the fault current: generator G1 itself, the high voltage grid PCC, and motorM1. The load also injects some current during the subtransient, though here thiseffect has been neglected. The theory behind the calculations is quite obviousand can be found in most of the textbooks on power system analysis. In thissubsection, the calculation procedure given in [15] was adopted. An equivalentscheme of the Oxelosund plant with its surrounding is depicted in Fig. 7.2. Itwas assumed1 that the voltage drop over the choke must not be greater than2% and the current from the generator during a three-phase short circuit via aresistor of 0.001Ω must not exceed 46 kA. The values of Eg, Es, Em, Z1, Z2,and Z3 are given in Table 7.1.

The generator, system , and motor have the following characteristics.

Generator: Sg = 82 MVA, x′′d,g = 0.17 p.u., ra,g = 0.0014Ω, Vbase =10.5√3kV .

System: The system is represented by a Thevenin equivalent: Zs = rs + jxs =0.0097 + j0.065Ω.

Motor: x′′

d,m = 0.24 pu = j6.615Ω. The subtransient EMF of the motor is foundin Table 7.1.

The resistance of the circuit breakers denoted as rc was included in the cal-culation as well in order to obtain a more accurate result. The value of the

1This numerical values were chosen arbitrarily, but it will be shown later how to estimatethe inductance of the choke based on the allowable voltage drop and vice versa.

7.1. Background 87

resistance is rc = 0.001Ω. Thus, the formulae below represent the equivalentimpedances Z1, Z2, andZ3:

Z1 = x′′d,g + ra,g + rc

Z2 = xs + xt + 2rc

Z3 = x′′d,m + 2rc

The numerical values of Z ’s are listed in Table 7.1. For the circuit of Fig. 7.2,the fault current following a symmetric three-phase short circuit is given by theexpression:

i(t) =√2Iper(t) sin

(ωt− π

2

)+√2I ′′e−

tτ = iper + iaper,

where Iper(t) and iper denote the rms and momentary value of the periodiccomponent, I ′′ is the initial value of the aperiodic component, and iaper stands forthe aperiodic component of the short circuit current. For the sake of simplicity,it is assumed here that Iper(t) does not vary in time and is equal to I ′′. ω isthe power frequency and τ is the time constant which equals the ratio L/r. Thevariables L, x = 2πL, and r are the inductance, reactance, and resistance ofthe circuit, respectively. To estimate the choke’s parameters, it is necessary toevaluate the maximal possible short circuit current, busbar OT20-10 should beable to withstand. It can be proven that the maximum current

imax =√2I ′′

(1 + e−

πrx

)

occurs at the time t ≈ 1/(2f), where f = ω/(2π) is the frequency of the network.This statement can be formulated as a claim:ClaimThe short circuit current given by the equation

i(t) =√2Iper(t) sin

(ωt− π

2

)+√2I ′′e−

tτ ≈√2I ′′

(− cos(ωt) + e−

tτ

)(7.1)

following a symmetric three-phase fault attains a maximal value

imax =√2I ′′

(1 + e−

πrx

)(7.2)

at time t ≈ 12f . 2

Proof. For an extremum to exist, the time derivative of equation (7.1) must alsoexist and be equal zero. The time derivative of equation (7.1) is:

di(t)

dt=

√2I ′′

(ω τ sin(ω t)− e−

tτ

)

τ= 0 (7.3)

The second order derivative of equation (7.1) should be found to assure that theextremum is a maximum:

di2(t)

dt2=

√2I ′′

(ω2τ cos(ω t) + e−

tτ τ−1

)

τ(7.4)


Table 7.2: Comparison of the short circuit currents

Manual calculation EMTDC simulations

ISC , Generator 77.2320 kA 77.4532 kAISC , System 60.0182 kA 54.2125 kAISC , Motor 2.3317 kA 2.3423 kA

Sum: 139.5819 kA 134.008 kA

As is seen from (7.4), if exists, the global maximum is located in the intervalωt ∈ [π/2, 3π/2] ∀ω, τ > 0. Equation (7.3) can be further simplified, taking intoaccount the inequality ω À 1 and recalling that for small t the following holds:e−t/τ ≈ 1− t/τ :

ω sin(ωt)−(1− t

τ

)1

τ≈ 0,

sin(ωt)− 1

ωτ+

t

ωτ2≈ 0,

sin(ωt) ≈ 0,

⇒ ωt ≈ πk, k ∈ Z. (7.5)

Since ωt ∈ [π/2, 3π/2], k must be equal to 1. This observation constitutes theproof of the second part of the claim, i.e., t ≈ π·1

ω = 12f .

Inserting the time that corresponds to maximum current into equation (7.1)yields equation (7.2) and thus completes the proof.

Note 7.1. It should be noted that equation (7.2) was obtained due to somesimplifications, otherwise the solution of transcendental equation (7.3) could notbe expressed in closed form.

Using equation (7.2), the short circuit currents flowing from all the threebranches were calculated. The values obtained were verified by means of EMTDCtime simulations. The comparison of the simplified and EMTDC currents ispresented in Table 7.2. A simple calculation gave the value of the current limitingreactor which in this case equals L = 5 · 10−4H = 0.5mH. Thus, the three-phase short circuit current at node OT20-10 flowing from the generator willnot exceed 46 kA and the total three-phase fault current will be limited by46 + 60.0182 + 2.3317 = 108.3 kA. Finally, the steady-state voltage drop acrossthe choke is estimated to be 2.71% of the phase voltage.

7.1.2 Selecting the size of the choke

In the previous subsection, the upper bound on the fault was chosen based purelyon engineering judgement–the allowable voltage drop was set arbitrarily. Thegoal was to deliver an example which would show how to perform estimation of

7.1. Background 89

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

x 10−3

2

3

4

5

6

7Statistical data

Vol

tage

dro

p ov

er th

e ch

oke

Choke inductance, H

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

x 10−3

20

40

60

80

100

120

Max

imal

pha

se fa

ult c

urre

nts,

kA

Choke inductance, H

Steady state voltage drop, % of the phase voltage

Transient fault current (peak)

1−sec. current (peak)

Figure 7.3: Short circuit data vs. the choke’s inductance

short circuit current and voltage drops. Now, the selection of actual size of thechoke for Oxelosund case study is discussed.

There are two options: i) to specify an allowable voltage drop across the chokeor ii) to specify an allowable short circuit current. Then based on the specifica-tion the inductance of the choke can be estimated. In both cases, Fig.7.3 maybe useful. This figure relates the voltage drop over the choke to the choke’sinductance. Moreover, in the lower part of the figure, fault current is shown as afunction of the choke’s inductance. Therefore, selection of the choke size involvesa trade-off between the allowable voltage drop and maximal fault current. Al-though this figure could have been obtained using Matlab calculations, it wascreated on the grounds of EMTDC simulations.

The present design of the busbar is unable to withstand fault currents oflarge magnitudes e.g., a fault current of 100 kA can cause mechanical damageto the busbar. It was decided to limit the short circuit current to 85 kA. AsFig.7.3 indicates, the choke’s inductance corresponding to this current level is 1mH and the voltage drop across the choke equals approximately 5%.


7.2 Controller Synthesis

In the previous chapter, the use of a choke for fault current limitation was dis-cussed. On the one hand, the choke does limit the short circuit currents; on theother hand, if the choke is over-dimensioned, the steady state voltage drop overthe choke is too high, which might make voltage control ineffective and therebylimit the achievable performance. In order to improve the voltage control, it wasdecided to seek an auxiliary voltage controller that would be able to guaranteevoltage control of acceptable quality.

7.2.1 Design strategy

In this chapter different controller designs are explored. The main objective ofthe controllers is to assure precise and robust voltage control in the presence ofmodel (load) uncertainties. The motor state (ON, OFF) is treated here as themain source of model uncertainty.

A preliminary analysis has shown that alteration of the parameters of G1voltage control circuits does not improve the voltage regulation. Thus, an exter-nal voltage controller is required to fulfill the control objectives.

Before proceeding with the controller design one important observation shouldbe made:

The instants of time when the motor is connected to the grid are known in ad-vance. Moreover, the voltage sags due to the motor start-ups can be accuratelyestimated based on the knowledge of the parameters of the grid and motor.

The above observation forms the basis of a new, external voltage controller thatwill be incorporated in the voltage regulator circuits of generator G1. We there-fore propose a new controller design that is capable of enhancing the process ofvoltage regulation. The key steps of the design are further explained as follows.

1 First, an appropriate model of the system has to be developed. For the systemunder consideration, an analytically derived model could have been set up;however, missing and unreliable data complicated the modeling. Because ofthis reason, a model of the power system was created in PSCAD/EMTDCsimulation program [1]. Our model is analogous to that described in [87].It should be noted that some parameters of the generator and motor wereunknown and the model was tuned by trial-and-error method by A. Wik-strom of ABB Automation Products AB. The tuning procedure includedalteration of some parameters of the model until a good match betweenthe model and field measurements was obtained.

2 Controller design normally requires having a model of the process to be con-trolled. In some case, this is not a necessary condition, but this situation isbeyond the scope of this chapter2. In this case study, two linear models of

2By employing subspace techniques it is possible to design an LQG without having a linearmodel of the process. It suffices to have field measurements for the controller design.

7.2. Controller Synthesis 91

the power system were obtained using system identification and lineariza-tion. Nonlinear time-domain simulations performed in EMTDC indicatedthat energization of the motor BFF7 does not result in significant devia-tions of the state variables from the equilibrium or nonlinear effects. Thisobservation justified the use of linear models in the study.

3 Having obtained a linear model of the system, dynamical properties of thelinear model have to be carefully inspected. In our study, the most impor-tant task is to quantify the capability of generator G1 to mitigate voltagesags due to starts of the motor.

4 To improve the voltage control, an extra controller is designed based on thei) model of the system, ii) knowledge of the magnitude of voltage sags,and iii) scheduling of the motor starts. Technically, there are variouscontroller configurations which can fulfill the control objectives. Below wediscuss some of them.

The remainder of the chapter is therefore devoted to demonstration of the abovesteps with emphasis put on the last step, for it contains the most interestingfinding of this thesis work.

Note 7.2. The need for a linear model of the system under consideration isdictated by the choice of the controller. That is, the design of H∞ -(sub)optimalcontrollers requires a linear model of the system. Despite the fact that H∞ -control [partially] has already been extended to nonlinear systems [33], this areastill remains the subject of ongoing research.

7.2.2 System modeling

EMTDC modeling

The power system consisting of the components listed earlier in this chapter wasfirst modeled in EMTDC, see Fig. 7.4. Although Fig. 7.4 does not indicate this,but the synchronous generator was modeled with seven differential equations:three for the d-axis, two for the q-axis, and two differential equations modelingthe mechanical dynamics of the generator. The models of the exciter and gov-ernor associated with the generator included the main sources of nonlinearities.For more details on the models used, the reader can consult the EMTDC user’smanual [1]. The load was represented by constant impedance and the motorBFF7 was modeled as a synchronous generator with a negative torque appliedto the shaft, see Fig. 7.4.

In EMTDC, the so-called “trapezoidal” method (with a fixed step-size) isimplemented for numerical integration of the equations. The minimal allowablestep-size of the integration is 50 microseconds, but in our computations largerstep-sizes were chosen–0.1 millisecond for short circuit current calculation and 1millisecond for studying dynamical behavior of the system when energizing themotor.

92Chapter

7.

Oxelo

sundCase

Study

T2

A

B

C

A

B

C

Tm

va =

55 13

5 10

.5

#1

#2

1000

0.00

1

0.00

1

0.00

1

3 Phase RMS

A

B

C

Iane

t

Ibne

t

Icne

t

0.001

0.001

0.001

Ib

Ic

Ua

Ub

Uc

Ia

Isa

Isb

Isc

0.00

1

0.00

1

0.00

1

5.51

25

5.51

25

5.51

25

0.02

3395

8

0.02

3395

8

0.02

3395

8 Iloa

da

Iloa

db

Iloa

dc

Ifau

lta

0.00

1

Ifau

ltb

0.00

1

0.00

1 If

aultc

Faul

t

A

B

C

FAU

LTS Hydro

Governor

Tm w Tm0

SP

HG

Tm w

Ef If

Te

Tm

A

B

C

Vabc If Ef

Vref

Exciter (SCRX)

1.0

EF

WO

UT

MT

M

Generator

81.25 MVA, 10.5 kV

A

B

C

3 Phase RMS

GT

M

0.001 Iga

0.001

0.001

Igb

Igc

Blm

7 Tm

w

Ef

If

w

Te

Tm

A

B

C

A

B

C

0.00

1

0.00

1

POUT

QOUT

PANG

GTM

WOUT

EF

Generator

Pnet

Qnet

Vnet

PRO

T

0

Busbar

Ua

Ub

Uc

Control

of System Breaker

0 - Closed

1 - Open

SYST 0

ENABBlm 1

win

Blm

Measurments on

Motor Blm 7

PBlm

QBlm

PANGBl

TMBlm

WOUTBlm

winBlm

TIM

E

Release Generator

ENAB

TMBlm

WOUTBlm X 2

* 3.183e-3

ompu

* TMBlm

* *

1

k -1

Synch motor Blm 7

3.6 MW, 10.5 kV

Torque

-1

Fault Currents

Ifb

Ifa

Ifc

*

1.0

Events

Faul

t

Fault Blmsw

Blm

sw

Torque

Vcanet Vbc

net

Vab

net

Vca

bus

Vbcbus

Vabbus

Vabbus

Vbcbus

Vcabus

Vbcnet

Vcanet

Vabnet Net

Iga

Igb

Igc

SYST

A

B

C

RR

L

RR

L

RR

L

A

B

C

PROT

A

B

C

3 Phase RMS

A

B

C

3 Phase RMS

VlR

0.00

1

Vger

Vger

IBlm

a

IBlm

b

IBlm

c Blm

brea

A

B

C

Load

1e-3

1e-3

1e-3

Vrefer

BT

25

VlR

e -sT

Blm

sw

e -sT

Blmbrea

Blm

sw

* 0.02

contr

A

B

Com

par-

at

or

TIM

E

40

R/S

Fl

ipfl

op

S R

Q

e -sT

I

P

1.

B -

E

+

F

+

cont

r

VlR

1 D

+

F

-

Vrefer

* 0.

05

Figu

re7.4:

EMTDC

model

ofelectrical

netw

orkof

theOxelosu

ndsteel

plan

t


Table 7.3: Coefficients of the numerator in descending order

0.0021 −0.302 52.83 1.504e3 1.84e4 1.57e59.52e5 2.73e6 9.74e6 8.13e6 1.32e6

Table 7.4: Coefficients of the denominator in descending order

1 58.87 1.323e3 1.723e4 1.572e5 1.035e64.646e6 1.445e7 3.2e7 2.21e7 3.313e6

Identified linear model of the system

Two linear models of the power system were obtained using system identificationand linearization of the equations describing the system. First, the identifiedlinear model is presented.

When system identification is used for obtaining linear models of a process,appropriate inputs and outputs have to be chosen. In the case under investi-gation, several signals can be used as an input for the voltage control at busOT20-10. For example, the reactive power drawn from the generator, the cur-rent, or the rms voltage at bus OT20-10. The last option was selected as themost natural. Thus, generator G1 will keep the voltage at the nominal levelduring starting of the motor, the voltage deviations due to the dead-band of thetransformer tap-changer, distant faults on the high voltage side of the network,and variation of the other loads. The reference voltage of the exciter Vexciter, ref.was chosen to be the input and the VRMS at bus OT20-10 was chosen to bethe controlled output. Thus the transfer function to be identified is given by theexpression:

G(s) =∆VRMS, OT20−10(s)

∆Vexciter, ref.(s)(7.6)

ARMAX and a Prony method based identification techniques were used to obtainlinear models of the system. In ARMAX, an initial model was found with amodified N4SID method3. A comparison (not shown here) revealed that theProny method yielded a little more accurate models which are used for controllersynthesis.

The excitation signal ∆Vexciter, ref. was chosen to be an impulse with a du-ration of 3 seconds and a magnitude of 0.05 pu, i.e., 5%. The numerator anddenominator of the identified transfer function are listed in Table 7.3 and Ta-ble 7.4.

The coefficients of the transfer function must be read row-wise, that is, thesummands of the numerator and denominator of the transfer function are givenas follows:Numerator → 0.0021s10 − 0.302s9 + 52.83s8 + · · ·+ 1315000,

3All the identification work except the Prony analysis was done in Matlab, [7], [53]. TheProny method based identification was performed with the help of DSI toolbox [4].


0 1 2 3 4 5 6 7 8 9 10−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03System Identification

Time, seconds

Vol

tage

incr

emen

t at p

oint

OT

20−

10Actual plant Identified plant

Figure 7.5: Step response of the actual and the identified plants. A 0.05 p.u.step was applied to Vexciter ref.

Denominator → s10 + 58.87s9 + 1323s8 + · · ·+ 3313000.The responses of the identified and actual systems are depicted in Fig.7.5. Thematch is quite good making it difficult to distinguish between the two responsesin the time range between the 3rd and 10th seconds.

Note 7.3. The order of the model was selected based on the following consid-erations: Firstly, it is desirable to have an accurate model of the system, thus,the dynamics of the generator, exciter, governor, and motor must present in themodel. Simple arithmetics gives: 3 + 2 + 2 + 3 = 10, therefore one should have10 state variables to accurately model the system. Secondly, the order selectionprocedure described in [43] was utilized. In this procedure, the number of statesis determined by increasing the order of the system until the signal-to-noise ratioexceeds some threshold.

Tables 7.3 and 7.4 represent the nominal plant; to guarantee robust perfor-mance of the system, the controller has to robustly regulate the voltage despitethe state of the motor. We therefore identify three more plants according toTable 7.5.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6Step responses of the identified plants

Vol

atge

at b

us O

T20

−10

, p.u

.

Time, seconds

Plant APlant BPlant CPlant D

Figure 7.6: Step responses of the four identified plants corresponding to thefour cases from Table 7.5

Table 7.5: Four operational conditions

Case Vref.,gen.,p.u. Motor state

A 1.0 ONB 1.0 OFFC 1.05 OND 1.05 OFF

Linearized model of the system

In fact, a linear model of the studied power system of Oxelosund has alreadybeen derived in Example 4.2. The only remaining task is to insert the numericalvalues of the system’s components given in Appendix B.

Thus, the following linear model is developed:

Table 7.6: Coefficients K1, . . . ,K6, K7δ, K7E′q

K1 K2 K3 K4 K5 K6 K7δ K7E′q0.7068 0.8350 0.2277 1.6943 0.1643 0.6531 0.0665 0.2915


0 1 2 3 4 5 6 7 8 9 100.97

0.98

0.99

1

1.01

1.02

1.03

1.04

Gen

erat

or G

1te

rmin

al v

olta

ge, p

.u.

! !

" $# % & '

() # %*,+-./0

Figure 7.7: Comparison of step responses: Linear model vs. EMTDC

Ac =

0 −0.0677 −0.079979 0 0 0314.16 0 0 0 0 0

0 −0.18825 −0.48791 0 0.11111 00 −493 −1959.3 −50 0 −10000 0 0 50 −50 00 −0.16435 −0.65308 0 0 −1

(7.7)

B = [0, 0, 0, 3000, 0, 1]T,

C = [0, 0.1643, 0.6531, 0, 0, 0] ,D = 0.

(7.8)

The transfer function of the plant equivalent to the state space description(7.7)–(7.8) is given by

H(s) =10880s3 + 7256s2 + 1.627e5s+ 1.085e5

s6 + 101.5s5 + 2671s4 + 16810s3 + 64350s2 + 2.306e5s+ 1.226e5(7.9)

It is customary to evaluate the quality of linearized models by comparing timeresponses of the linearized and nonlinear plants. Such a comparison of stepresponses of the full-order EMTDC and reduced-order model (7.7)–(7.8) is de-picted in Fig. 7.7. In the figure, it can be seen that the main dynamical phe-nomena were captured, resulting in a reasonably good correspondence betweenthe curves. The ripple-shaped curve corresponding to the full model is due tothe numerical noise caused by the calculation of the rms value of the signal.


Note 7.4. The time constant of the voltage regulator of generator G1 is 0.02seconds. This implies that without an auxiliary voltage controller, the ideal re-sponse time is limited to the aforementioned figure. However, this ideal time ishardly achievable in practice, as there are various delays caused by the trans-ducers and communication system.

Quantifying the voltage sag

The control strategy being developed in this chapter requires the knowledge ofthe magnitude of the voltage sag caused by start of the motor. The details of thecontrol strategy will be presented in the next subsection; now we demonstrate amethod for estimation of the voltage sag magnitude.

To simplify the calculation of the magnitude, the influence of the active powerchange on the voltage level at bus OT20-10 is neglected. At the very moment themotor is connected to the net, the reactive power Q drawn from the net increasesby certain amount ∆Qm which may be estimated from the motor rating. Theincrease ∆Qm causes a reduction of the motor terminal voltage. For simplicity,it is also assumed here that the voltage remains constant during the start of themotor. The magnitude of this constant voltage may be determined by a standardload flow routine. Then, the following equation can be utilized to calculate thevalue of ∆VG1:

∆Qm =VG1,new (VG1,new − VOT20−10)

xchoke

(7.10)

The equation above can be solved for VG1,new:

VG1, new =VOT20−10

2+

√VOT20−10

2 + 4∆Qmxchoke

2(7.11)

Substituting the values for ∆Qm = 3.2 MVAr and VOT20−10 = 0.9821 p.u., thenew voltage VG1, new = 1.0092 p.u. is obtained. Thus, the desired quantity ∆VG1is found to be ∆VG1 = VG1, new − VG1, old = 1.0092 − 0.9881 = 0.0211 p.u., thatis, approximately 2.11%.

It was not mentioned explicitly, but it was also assumed that the increasein the reactive power consumed by the motor is completely compensated bygenerator G1.

Note 7.5. The voltage sag due to a motor start can in the first approximationbe estimated by using the equation

∆V = 1− ZmotorZmotor + Zsystem

or in terms of rated power of the system and motor

∆V = 1− SsystemSmotor + Ssystem

.


In the two equations above, Zmotor is the equivalent motor impedance, Zsystemand V denote the Thevenin impedance and voltage of the system measured atthe motor terminals, respectively.

7.2.3 Disturbance scheduling–Coordinated controllerdesign

Since the controls of G1 are slow and it is infeasible to speed up its response (seeNote 7.4), a new approach had to be sought. This subsection reports main resultson the use of coordinated control to enhance the process of voltage regulation.

Normally, the voltage in the power systems is controlled by the genera-tors, synchronous condensers, Static VAr Systems, transformer tap-changers,and some other auxiliary devices. In normal conditions, the voltage levels arekept within some pre-specified limits; however, certain activities may cause devi-ation of the voltage from its nominal value. Among these activities are changesin the load levels, motor starts, etc.

The modern high power electronic devices allow reasonably fast and pre-cise voltage control, see Section 3.3 and [23], but this can involve considerableinvestments. On the other hand, voltage quality problem can be reduced byintroducing coordinated voltage control. Here, this method is understood in thesense of the following

Definition 7.6. Coordinated control is a complex procedure of combining avail-able controls of power system components that possess appropriate control ca-pabilities. Coordinated control is aimed at achieving enhanced regulation of aprocess. 2

The above definition only suggests that control efforts of several componentsshould be combined if the operation of a system in a decentralized control moderesults in unsatisfactory performance. In practice, however, there are a numberof issues to be addressed applying the concept of coordinated control. These are:

• What is the configuration of the auxiliary (master) control?

• What signals and communication links should be used?

• What measures should be taken to suppress undesirable interactions be-tween different control channels?

Clearly, no general recommendation can be given for using coordinated control–engineering judgement is needed to better utilize the potential benefits of thismethod in each particular study. Perhaps, the only exception would be the ruleof thumb: “in order to achieve fast control, fast circuits should be employed”.That is, fundamental properties of the system have to be studied and then basedon the results of the study an appropriate input-output couple should be selected.It should be noticed that in many cases the use coordinated control yields verygood results [49].


Let us consider a different scenario. If a preliminary analysis shows that thereare fundamental or some technical limitations on the achievable performance,then another solution has to be sought. One solution would be to incorporateinformation of certain kind into an appropriate control scheme. This method ishere referred to as disturbance scheduling and is defined in more precise termsby

Definition 7.7. Let a system consist of a process and a disturbance. Assumethat the characteristics of the disturbance are known or they can be estimatedwith desired precision. Then, an auxiliary controller having the characteristicsof the disturbance incorporated in its operation is termed disturbance schedulingcontroller. That is, a disturbance scheduling controller operates such that pre-ventive measures are taken prior the occurrence of the disturbance. This controlconcept is therefore called disturbance scheduling since the disturbance is indeedscheduled. 2

This “noncasual” mode of operation may be very effective in solving a numberof practical task, where other controller designs fail to meet control specifications.For instance, if perfect tracking is to be achieved by a non-minimum phase sys-tem, the use of such noncasual controllers have proven to be a sound alternativeto any other controller design [56].

Remark 7.8. The concept of disturbance scheduling is very similar to coordi-nated control method; however, one important difference must be recognized:while coordinated control normally results in a hierarchial control structure, thedisturbance scheduling in general does not require having a centralized (subor-dinated) control scheme.

Since an example is the best explanation, the use of disturbance schedulingfor improved voltage regulation is demonstrated below.

The controls of the components can be coordinated with known or estimatedcauses of voltage quality problems, for example, motor starts. In case the con-ventional voltage regulator is either slow or imprecise or the specifications onvoltage quality are quite tight, the utilization of information known in advancemight significantly simplify the process of voltage regulation. For example, inthe present case study, it was possible to assign the functions of voltage controlto a local generator. The voltage regulator of the generator was unable to bringthe voltage quickly enough back to its nominal value due to the presence of thereactor (Fig.7.5 clearly indicates that a 5% increase in the generator referencevoltage yields approximately a 2.5% voltage increase at bus OT20-10.) The ideabehind the method described in this study is illustrated in Fig.7.8. We introducetwo modes of operation which are shown in Fig. 7.8b and Fig. 7.8c (these twomodes will be referred to as ‘mode A’ and ‘mode B’, respectively.) In mode A,the reference voltage of G1 gradually increases to 1.05 p.u. (from time t1 to t2.)Due to this, the voltage at bus OT20-10 increases to 1.02%. At time t2 themotor is connected to bus OT20-10. The connection of the motor reduces thevoltage, but this only brings the voltage back to 1 per unit.


Time

.1, . .V

ref Gpu∆

1

1t

2t

3t

4t

1.05

1

Time

20 10, . .

OTV p u−∆

1t

2t

3t

4t

1.02

Time

20 10, . .

OTV pu−∆

1t 2

t3t

4t

1.01

1

0.99

a

b

c

Mode A

Mode B

Figure 7.8: Voltage Control


Exciter & extraController

Load

G1 Motor

T7

OT20-10

Figure 7.9: Information flow for the control scheme used

In mode B, the voltage at OT20-10 is boosted to 1.01 p.u. and then themotor is energized. Because of the motor connection, the voltage decreases to0.99 p.u. and then stabilizes at 1 per unit. It seemed very attractive to design acontroller which could exactly compensate for the voltage reduction during thestarting of the motor. However, in the present case it was impossible becauseof the time constants of the processes of voltage increase and decrease, whichwere substantially different. The time constant of the voltage regulator wassubstantially greater than that of the motor start up.

In principle, the time constant of the voltage regulator is dependent on theslope of the voltage increase, (in Fig. 7.8a, between time t1 and t2.) The greaterthe slope, the faster the response. Although, in the limit case t1 = t2 = t, theslope reduces to a step change, which will correspond to some time constant τ ofthe voltage regulator and any further increase in the speed of the response willbe impossible. In this case study the voltage was increased step-wise; however,still the response was too slow (this difficulty was overcome by the use of anH∞ controller.) We also need a feedback controller to eliminate steady statecontrol error and to ensure robust operation. Fig. 7.9 schematically shows theinformation flow for the control scheme used. In the figure, the dashed lines showthe information necessary for the proper operation of the controller, namely, thestate of the motor (ON or OFF), the rms voltage at the busbar OT20-10, andpossibly the setting of the tap-changer of transformer T7. However, the latterinformation was not used in this case study.

Most probably, the installation of a more advanced controller could haveimproved this aspect of the voltage regulation (this conjecture will be checkedlater in this chapter), but first it was decided to investigate the appropriatenessof PID controllers for voltage control.

It is expected that such a coordination of the motor and generator will keepthe voltage within some pre-specified limits. In the present case, it was decided


to limit the voltage from 0.99 p.u. to 1.01 per unit. It should be noted thatapart from the economic benefit (no auxiliary equipment needs to be installed),the presence of a generator brings the possibility to mitigate voltage sags causedby motor starts or distant ground faults.

We therefore add a PID controller to the system. The voltage at OT20-10 issensed and the error signal Vref., gen. − VOT20−10 is sent to the PID controller.The need for the PID controller is dictated by the inability of the internal voltageregulator installed at G1 to maintain the nominal voltage level. Details of thePID controller are supplied in the subsequent section.

Proportional Control

Proportional-Integral-Derivative (PID) controllers have become classical sincethey were invented. There are many advantages with PID controllers: theirorder is low, their theory is well developed, and the tuning rules are alreadycreated [66]. However, there are also several drawbacks associated with thistype of controller: some difficulties while tuning, the performance is not alwaysvery good, i.e., oscillatory output of the system which is controlled by a PIDcontroller. Also it is difficult to account for model uncertainties, which meansthat the controller may not be robust under changing operation conditions.

In this study only fixed-parameter PID controllers are considered. The con-troller used has the following configuration [66]:

Gc(s) = Kp

(1 +

1

Tis+ Tds

), (7.12)

where Kp, Ti, and Td are the controller parameters to be found. Since the struc-ture of the PID controller and the plant are fixed, the only remaining goal is toselect such a set of the controller parameters that certain performance specifica-tions are fulfilled. In Appendix B a tuning rule is proposed which incorporatessome estimated uncertainties. Applying the optimization procedure the tandem‘controller-controlled plant’ is tuned so that the performance specifications onthe response (rise time, peak response, steady-state, settling time, etc.) are notviolated under varying operational conditions.

When the uncertainties are identified and time domain specifications are set,an optimization procedure can be applied.

When tuning the PID controller, there are two possibilities: to use the identi-fied model of the plant or the reduced-order linear model provided by analyticallinearization. Accounting for the simplicity and reasonable accuracy the Pronymethod based identification provides, we will tackle only with the identifiedmodels.

The task is to design a PID controller capable of performing well being inoperation with the plant given by equation (7.9) under all four operational con-ditions in Table 7.5. The initial guess Td = 0, Kp = 5, and Ti = 0.5 sec. wasobtained with the help of the Genetic Algorithm [35].


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.2

0.4

0.6

0.8

1

Time, seconds

Vol

tage

at O

T20

−10

, P.U

.

Robust PID. Step responses of the identified plants

Specs Plant APlant BPlant CPlant D

Figure 7.10: Step responses of the compensated plants

Remark 7.9. In this study, the genetic algorithm is used as follows: a coarsegrid is set on the admissible space of parameters. Then, the GA is initiated andgenerations evolve. The input variables are the sum of squared errors

J =N∑

t=0

(Vref (t)− VOT20−10(t))2

= (Vref − VOT20−10)T (Vref − VOT20−10),

where N is the length of the data array and the output variables are the un-known parameters. In this case, the unknown parameters are the PID controller:Td, Kp, and Ti.

The values of Td, Kp, and Ti were then determined more precisely usingthe Matlab Optimization Toolbox [18]. The final values of the PID controllerparameters are Td = 0, Kp = 0.9213, and Ti = 0.2342 sec. The responses of thecompensated plants are shown in Fig.7.10. One may notice here, that all thecompensated plants are stable and the characteristics of the responses are closeto the specification which is represented by a thick line in Fig.7.10.

The bus OT20-10 voltage profile at motor BFF7 connection is shown inFig. 7.11. In the figure, the dashed line corresponds to the case when no controlleris installed, dotted line depicts the voltage which is controlled by the PID withTi, Td, and Kp found earlier in this section. Finally, the solid line shows thesystem response with the same PID controller as above and voltage coordinationas presented in Section 7.2.1.


12.7 12.8 12.9 13 13.1 13.2 13.3 13.40.975

0.98

0.985

0.99

0.995

1

1.005

1.01

1.015

1.02Compensated plant. Motor start

Vol

tage

at O

T20

−10

, P.U

.

Time, seconds

PID+coordination No controller PID only H∞+coordination

Figure 7.11: Comparison of the voltage profiles at a motor start

The use of the PID controller alone improves the quality of the voltage;however, the magnitude and duration of the voltage sag are greater than in thelast case, i.e., case 3. It can be observed that, in general, the response of the‘coordinated-compensated’ plant (case 3) is in a reasonable agreement with thevoltage waveshape in 7.8c. Furthermore, in case 3 the voltage does not sag below99% of its nominal value. In addition, a small reduction of the magnitude of thesag was achieved due to dynamical compensation of the voltage reduction. Ifthe speed of the voltage controller had been greater, this reduction could havebeen substantially greater. The motor disconnection is not shown in the figurebecause the voltage deviation caused by the disconnection was negligibly small.

H∞ controller design

The design of an H∞ controller is based on the theory presented in Section 6.3.More details on this subject are provided in [21] and [47]. Similarly to thePID controller design, time domain data were used to obtain transfer functionsof the nominal plant and for uncertainty modeling. As in the case with PIDcontroller, an attempt was made to capture the range of the uncertainty byidentifying several plant models, including the most and least stressed modelsas the boundary cases. The optimization was performed with the µ-Analysisand Synthesis Toolbox in Matlab [13]. Adopting equation (76) in [47], the


Table 7.7: Summands of the numerator (H∞ controller, descending order)

107.3s11 1.121e4s10 4.213e5 s9 9.158e6s8 1.456e8s7 1.715e9s6

1.582e10s5 9.925e10s4 5.711e11s3 1.511e12s2 5.037e12s 2.754e12

Table 7.8: Summands of the denominator (H∞ controller, descending order)

1s11 261.9s10 2.693e4s9 1.005e6s8 1.048e7s7 2.172e8s6

1.006e9s5 1.167e10s4 2.376e10s3 1.297e11s2 1.258e11s 3.136e10

uncertainty was estimated using the expression:∣∣∣∣Di(s)−D0(s)

D0(s)

∣∣∣∣ 6 |VW1| , (7.13)

where i = 1, . . . , 4 stands for the number of the transfer function denominator.The weighting function W1 was chosen such that equation (7.13) is satisfied forall i.

The resultant transfer function of the controller was of 11th order (see Tables7.7–7.8) and was reduced to a 6th order controller (given by Tables 7.9–7.10.)

In this study we used the “Schur model reduction method” for reducing thefull-order controller. The chief instrument behind this model reduction techniqueis to obtain a model Gm(jω) = Cm(sI −Am)−1Bm +Dm of the full-order plantG(jω) = C(sI −A)−1B +D such that the following inequality holds:

‖G(jω)−Gm(jω)‖∞ 6 2

n∑

i=k+1

σHankel(i), (7.14)

where σHankel denotes the Hankel singular values4 of the stable and unstableprojections of G(jω).

The order reduction was needed because of the high gains (∼ 1012) of thefull-order controller, which complicate implementation of the controller. Beingreduced to a lower order, the gains also reduce (∼ 105). As a result of thecontroller order reduction, the response of the compensated plant is slightly moreoscillatory, but still its performance is very good. Analyzing the time responsesand Bode plots of the full-order and reduced-order controllers, one can judgewhether the reduction is good enough.

The Bode plots of the reduced and full-order plants are shown in Fig. 7.13.It can be seen in the plot that the elimination of five modes did not qualitativelychange the behavior of the controller.

The evaluation of the H∞ controller performance is shown in Fig. 7.11. Inthe figure, it can be noticed with the H∞ controller installed the voltage sag

4If a system is given by the quadruplet A,B,C, and D, then the Hankel singular valuesof the system are: σi =

√λi(PQ), where P and Q are solutions of the matrix equations

AP + PAH +BBH = 0, AHQ+QA+ CHC = 0.


0 0.5 1 1.5 2 2.5 30.7

0.75

0.8

0.85

0.9

0.95

1

Time, seconds

RM

S v

olta

ge a

t bus

OT

20−

10, p

.u. HStep response of the ∞ compensated plant

Figure 7.12: EMTDC simulation: Step response of the linear model controlledby the H∞ regulator

Bode Diagrams

10

15

20

25

30

35

40

45

50

Mag

nitu

de (

dB)

100

101

102

−100

−50

0

50

100

Frequency (rad/sec)

Pha

se (

deg)

Dashed line − 11th order controllerSolid line − 6th order controller

Figure 7.13: Bode diagrams of the full-order (dashed line) and reduced-order(solid line) controllers


Table 7.9: Summands of the numerator (Reduced-order H∞ controller,descending order)

1.0728e+2s6 4.7840e+3s5 8.5222e+4s4 7.1922e+5s3

2.4062e+6s2 9.3078e+6s 8.0668e+6

Table 7.10: Summands of the denominator (Reduced-order H∞ controller,descending order)

1s6 2.0182e+002s5 1.4430e+004s4 3.0221e+004s3

2.2720e+005s2 2.9996e+005s 9.3889e+004

associated with the starting has smallest magnitude than that in all other cases.Physically this improvement is due to the high gains of the controller which forcethe system to “act” faster and more precisely.

Fig. 7.14 depicts a comparison of the system voltages when i) the PID and ii)H∞ controllers are used to enhance the voltage control. In this test, the generatorvoltage reference point was altered step-wise. The figure clearly indicates thesuperiority of the H∞ controller over the PID controller.

7.2.4 Conclusions

In this chapter, the theoretical foundations of the first part of the thesis wereapplied to a case study. An attempt was made to improve the quality of electricpower at a steel mill by use of information known beforehand and an auxiliarycontroller.

Main emphasis was on the investigation of the possibility to take preventivemeasures and schedule the known disturbance in such a way that the system’svoltage magnitude is maintained at a pre-specified level.

Two controllers—the classical PID and more sophisticated H∞ —were de-signed and implemented in EMTDC. The performance of the proposed “distur-bance scheduling/coordinated control” scheme was verified by means of nonlin-ear simulations. The results of the verification indicate that the main controlobjective—tight voltage control—was met.

In summary, the following was done in this case study:

• The generator voltage control and motor starting were coordinated.

• Power system identification has been applied to H∞ -controller design.

• A numeric technique for PID tuning has been developed and applied to thecase study. The tuning technique successively employs a Matlab-basedGenetic Algorithm toolbox and the Optimization toolbox.

Furthermore, it is essential to note that the use of the H∞ controller resulted inimproved performance of the plant. That is, the dynamical voltage compensation


0 1 2 3 4 5 6 7 8 9 10−0.03

−0.02

−0.01

0

0.01

0.02

0.03dV

RM

S, a

t OT

20−

10

Voltage control: H∞ vs. PID controller

Time, seconds

PID compensated plant

Hinf

compensated plant

Figure 7.14: EMTDC simulation: Voltage control. H∞ versus PID controller

reduced the voltage sag caused by a motor start. For the sake of comparison, thePID controller could not reduce the sag depth which was 2% (however, the overallperformance was improved); on the contrary, the H∞ controller did reduce thesag depth to 1.2%.

As an intermediate result, the three-phase short circuit currents were cal-culated to ensure the ability of busbar OT20-10 to withstand such a systemfault.

The main result is an improved voltage waveshape at motor starting–thevoltage fluctuations are within ±1% at energization of the motor. With theproposed voltage control algorithm, the duration of the voltage sag does notexceed 100 ms. It is noteworthy mentioning that in the present days starting ofthe motor causes a voltage sag lasting approximately 12 seconds and having amagnitude of approximately 5%, see Fig. 3.1.

Chapter 8

Gruvon Case Study

8.1 Background

8.1.1 Economical Motivation

Deregulation of electricity markets is taking place in many countries across Eu-rope and the US. The deregulation is primarily aimed at the introduction ofcompetition, and consequent reduction of electricity prices. It is also expectedthat privatization will simplify the market entering for small and medium sizebusinesses. The introduction of distributed generation makes the distributionnetworks more active than in the past. This brings the necessity for new marketregulations and puts more responsibility on the new generators. That is, alongwith the merits of distributed generation, there will be new tasks for utilities,such as maintenance of system reliability and improvement of power quality.

8.1.2 System studied

The system under consideration represents a model of the power grid of a papermill located in the city of Grums, Sweden. The electrical network of the papermill is shown in Fig. 8.1. Since only a part of the network is relevant to thepresent investigation, the network is reduced as is depicted in Fig. 8.2(a).

The reduced network consists of high priority load (LD1, LD2) and commonload (LD3), two local generators (G1, G3), two transformers (T02, T05), and astrong grid denoted here as NET, see Fig. 8.2(a). In today’s configuration, theloads are fed by the local generators and the network NET over the year, exceptfor one hundred hours, when the sensitive load is disconnected from the maingrid and is fed by the local generators, as an island, see Fig. 8.2(b). During thesehundred hours the risk of voltage sags is considered to be high. Thus, voltagesags caused by faults on the distribution or transmission network are the majorfactor that jeopardizes the proper operation of the steel plant.

109

110 Chapter 8. Gruvon Case Study

T01 T02 T03 T04 T05 T06

NET

G1 G3

LD1 LD2

894 MVA, 30kV

LD3 LD4

B30

A30

S01B10 S02B10

S01A10 S02A10

CLD

GS GS

Figure 8.1: Electrical network of the paper mill of “Billerud”

8.1. Background 111

G3 T05

∆∆∆∆LD

GS

G1 T02

GS

LD1

LD2

(b)

GS

CLD

G3 T05 LD3

NET

LD2

GS

G1 T02

LD1

(a)

Figure 8.2: System studied: (a) normal connection, (b) connection in the“islanded mode”

The disconnection of the busbar “NET” involves operation of a part of thesystem in the so-called “island mode” which is normally avoided due to sev-eral considerations such as personnel safety and possible problems with powerquality (voltage or frequency level deviations.) However, in the case under inves-tigation, the system is designed so that it may be operated in the island mode.In normal conditions, the active power produced by the local units is normally12 MW greater than the local load (19 MW consumption, 31 MW production).Currently, during the hours with high risk of thunderstorm, the power producedby the generators is manually reduced until balance between production andconsumption is achieved. Then the generators are connected in parallel, and thecurrent limiting device CLD1 disconnects the generators and their loads from therest of the net. Thus, the generators G1 and G3 run in the “islanded mode”, seeFig. 8.2. When the thunderstorm is over, the generators will be re-connected tothe net in the reverse order, i.e., they will get connected to the net and then theactive power set-points will be restored. It is worth remarking that the systemis protected from voltage sags only during the time when it is operated in the“islanded mode”. If a voltage sag occurs when the system is in the normal state,important technological processes may shut down causing loss of production.

1At the present time, the current limiting device is a mechanical switch, but it is called“CLD” anyway, since now we are only discussing a possible future scenario for which having afast CLD is essential.


Table 8.1: Characteristics of commonly usedcircuit breakers [86]

Disconnection time

Mechanical breaker 1− 10 cyclesSolid-state breaker < 0.5 cycleHybrid switches > 1 ms

8.1.3 Fast Switches and Current Limiting Devices

If there are several feeders in a local electrical network, then one of the simplestand most reliable methods to mitigate a voltage sag is to transfer the load fromthe faulted to backup feeder. Here we implicitly assume that the voltage sag wascaused by a fault on one of the feeders. In the present case, no backup feederis available; however, a fast switch can be used to disconnect the faulted feeder.After the disconnection, the control equipment of the generators has to rapidlybring the system to a new acceptable equilibrium.

There is a wide spectrum of current breaking devices commercially available,ranging from mechanical breakers to semi-conductor current limiting devices [86].

Table 8.1 reveals technical characteristics of some circuit breakers. As Ta-ble 8.1 indicates, modern power limiting (breaking) devices are capable of veryfast operation and have reasonable rated power. Thus, it will be further assumedthat technically one can almost instantaneously disconnect currents up to 1 kA,which is sufficient for this case study.

8.1.4 New mode of operation

As was already mentioned, deregulation of electricity markets stimulates moreeffective use of generator capacities. Evidently, if there were no risk of powerdisruption (mainly voltage sags) propagating to the system from the networkNET, it would be feasible to run the generators most of the time based only onthe economical considerations, e.g., estimation of the profit.

However, this is not a very realistic situation and this fact supposes a searchfor a technical solution which would be capable of fulfilling two solid goals:

• Allow the personnel of the paper mill to run the generators permanently.

• Effectively mitigate the voltage sags due to faults on the network outsidethe paper mill.

8.2 Coordinated Control

We conjecture that the combined use of the knowledge of the power surplusproduced by the generators and advanced controllers is a viable solution of tasks

8.3. System modeling and controller design 113

presented in the previous subsection. In what follows, we focuse on the technicalaspects of our solution.

Let us consider a typical scenario of a sudden disconnection of the circuitbreaker CLD, see Fig. 8.2. If there is no supplementary control, the disconnec-tion of the generators will cause power imbalance and as a consequence frequencyexcursions from 50 Hz2. Unless the output power of the generators is manuallyadjusted, the frequency will be different from 50 Hz and will take a value deter-mined by the power imbalance and speed-droop characteristics of the generators.On the other hand, in this case study one can at any time instant easily observethe surplus/shortage of power by monitoring and analyzing the currents flow-ing through the transformers T02 and T05. This implies that it is possible todesign a simple logical circuit (controller) which could acquire and analyze therelevant data and control the circuit breaker CLD. If a voltage sag having an un-acceptable magnitude is detected, this controller should fix and hold the powerexported/imported from/to NET and then disconnect CLD. Next, the powerset-points of the generators G1 and G3 must be adjusted in such a way thatpower balance is restored. It should be noted that a very good alternative is toinsrease/decrease the set-points of the generators proportionally to their sharesbefore the disconnection. For example, suppose before disconnection PG1 = 13MW, PG3 = 18 MW, PLD1 + PLD2 = 19 MW. Thus the power surplus was 12MW. Therefore, after the opening of CLD, the total output power of G1 hasto be reduced by 13 · 12/(13 + 18) = 5 MW and the output power of G3 by18 · 12/(13 + 18) = 7 MW. If this is done, the power oscillations between thegenerators will be minimized.

8.3 System modeling and controller design

8.3.1 System modeling

Models of power systems are usually of a large order, far beyond that acceptablefor controller design purposes and for analysis. To overcome this difficulty, somesimplifying assumptions have to be postulated.

• The dynamics of the turbines and governors are neglected, Pmech.,i = con-stant.

• The load is represented by constant impedances.

• It is postulated that damping of the generators is uniform.

• Transfer conductances of the network are neglected.

2In this case study, simplified models are used which do not include frequency controlcircuits which normally present in real power systems. It is, however, instructive to performanalysis similar to that of this chapter.


Also expressions for Lyapunov functions are greatly simplified if the modelused is simple. For this reason, the system studied in this paper is modeled bya third order model as shown below [68].

dδidt = ωi

Midωidt +Dωi = Pm,i −

2∑j=1

BijEiEj sin (δij) + ui,

T ′do,i

dEi

dt = Efd,i − Ei +(xd,i − x′d,i

) 2∑j=1

BijEj cos (δij) + σi

i = 1, 2.

(8.1)

In equation (8.1), the number of machines is 2; δi, ωi, Ei, Efd,i, Mi, Di arerespectively the rotor angle, rotor angular frequency deviation, quadrature-axiscomponent of the electro-motive force (EMF), excitation voltage, inertia anddamping coefficients of the machine i. The manipulated inputs are ui and σi.Tdo,i is the open-circuit transient time constant; xd,i and x

′d,i are the synchronous

and transient reactances of the respective generator. Bij is the transfer admit-tance of the equivalent internal-node network [68].

The set of equations (8.1) describes the system operating in the islandedmode. The equilibrium point of this mode of operation can be translated to theorigin, resulting in the system [40]:

dδidt = ωi

Midωidt = −Dωi −

2∑j=1

Bij(EsiE

sj sin

(δsij)− EiEj sin (δij)

),

dEi

dt = −αi (Ei − Esi )− βi

2∑j=1j 6=i

Bij(Esj cos (δij)− Ej cos (δij)

)

αi =1−(xd,i−x′d,i)Bi,j

T ′do,i

, βi =xd,i−x′d,iT ′do,i

i = 1, 2.

(8.2)

Note that the superscript s is used to denominate the steady-state quantityof the system after transition, be it to or from the island mode. Again, thosesteady-state quantities are well-defined, since the knowledge of the pre-transitionpower flow can be used for calculation of the post-transition quantities such asthe rotor relative angles δsi or Es

i , etc.

Control strategy

The fact that after the disconnection the power balance between the consumptionand production is maintained does not guarantee that the system will not fallout of synchronism. One may encounter a situation when the system becomesunstable if the power imbalance before disconnection was very large. Moreover,the load LD1 and LD2 are not constant and may vary substantially in time(modeled by ∆LD in Fig. 8.2(b)). These considerations suggest the use of a


supplementary control loop which would stabilize the system when it is sub-jected to such internal disturbances. Since the disconnection of NET changesthe equilibrium point of the system and causes significant deviation of the sys-tem’s states from the equilibrium, the use of linear controller becomes a choiceof questionable value. Instead, we propose the use of a Lyapunov method basedrobust controller.

In this case study, the framework of the controller design rests on two funda-mental concepts: i) Lyapunov’s direct method and ii) Control Lyapunov Func-tion.

8.3.2 Controller Design

In the literature on this subject, the use of energy functions as the first candidatefor a Lyapunov function is advised. Indeed, the total energy of a system isalways positive and for unforced realistic systems it always dissipates, i.e., itstime derivative is negative. Consider the energy function

V (x) =1

2

2∑

i=1

Miωi2 +

2∑

i=1

αiβi

(Ei − Esi )2

−2∑

i=1

δi∫

δsi

(Pm,i − Pe,i) dδi +1

2

2∑

i=1

αiβi

(Ei − Esi ). (8.3)

As was shown in [55], V (x) is indeed a Lyapunov function for (8.2). In [55] itwas also demonstrated how an improved excitation controller given by (8.4)

σi = ∆Efd,i(t) = −Ki (Ei(t)− Esi ) , i = 1, 2 (8.4)

can be derived based on (8.3).In the present paper we complement the decentralized excitation controller (8.4)by a controller designed based on the Control Lyapunov Function theory. Thecontroller is of the form (6.33) with gi ≡ 1:

ui = −ki∂V (x)

∂ωi

= −kiMiωi (8.5)

The supplementary controller (8.5) is decentralized, i.e., it uses only local signalsof the respective generator. The inputs to this controller installed on generatori are

• The angular frequency deviation ωi of Gi.

• The deviation of the EMF Ei of Gi from the steady-state Esi .


1 1k M

2 2k M

1K−

2K−

1ω∆

2ω∆

1( )E t∆

2( )E t∆

1u

2u

1σ

2σ

Figure 8.3: Structure of the controller based on CLF

GS

CLD

G3 T05 LD3

NET

LD2

GS

G1 T02

LD1

Disturbancescheduling and CLF-

based controller

3 3 3( ), ,E t ω σ∆ ∆

busV

1 1 1( ), ,E t ω σ∆ ∆CLDSw

LDSw

Figure 8.4: System controlled by the CLF regulator

The outputs of the controller are

ui : A signal which is to be added to the Gi active power set point, i.e., Pm,i.

σi : An auxiliary signal added to the set point Efd,i of the ith generator.

The controller structure and the controlled system are schematically shown inFig. 8.3 and Fig. 8.4.

The constants ki may be chosen arbitrarily, since any set of Lyapunov func-tions for a given system is invariant under the operations of summation andmultiplication by a positive scalar. Theoretically, the greater ki the more damp-ing is added to the system. However, nonlinearities of the governors will limitthe achievable damping when ki are increased above certain threshold.

Remark 8.1. The auxiliary controllers designed in this section are optimal for thisparticular choice of the Lyapunov function. Had we found a different Lyapunovfunction, the “optimal” CLF-controller (8.4) would have been different.


0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19

1

1.005

1.01

1.015

1.02

1.025

1.03

System voltage as a function of time

Time, sec.

G3 v

olta

ge w

ith a

nd w

ithou

t ext

ra c

ontr

ol, p

.u.

With extra controllers

Without extra controllers

Figure 8.5: System voltages with and without extra controllers

Remark 8.2. One intrinsic feature of CLF-controllers is their inability to directlyassess the “quality of the transient”. That is, the CLF-controller is designedbased on the system topology after a transient, when the system states arealready set in motion. Thus, nothing is done during the transient. While usinglinear robust analysis tools, one does have the flexibility to control the systemstates before, during, and after the transient. Of course, in spite of this fact, ifthe linearity assumptions are violated, the Lyapunov theory should be utilizedfor analysis and controller design. In addition, the quality of the transient canbe indirectly controlled by choosing appropriate damping.

Remark 8.3. Despite the fact that the CLF-based controller was designed usinga simplified models3 of the power system, the simulations are performed ondetailed, nonlinear models.

It should be noted that in this case study, the voltage controllers of thegenerators are capable of supporting the nominal voltage level. Therefore, thereis no need in auxiliary voltage controllers. Fig. 8.5 shows the voltage at theterminals of G3. As the figure reveals, there is an insignificant voltage swell ofapproximately 3% which, however, rapidly declines in a time less than 2.5 cycles.

3Indeed, in general case it is a challenging task to find a Lyapunov function for a sys-tem having 3 state variables. For larger systems, finding a Lyapunov function is even morecomplicated exercise.


0 1 2 3 4 5 6 7

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Time, sec.

Freq

uenc

y de

viat

ion,

p.u

.

System’s frequency deviation ∆f, p.u.

Coordination and extra controllerOnly CoordinationNo extra control

Figure 8.6: Frequency deviation caused by islanding of the system

8.4 Simulations

The performance of the system with and without controllers is evaluated bymeans of computer simulations. A detailed nonlinear model of the system underconsideration is built and tested in Power System Blockset [10]. Fig. 8.6 andFig. 8.7 present the main results of the study. As can be noticed in Fig. 8.6, inthe presence of coordinated control and supplementary controls (8.4) and (8.5),the system exhibits the best performance. The power balance is maintained,thus the steady-state frequency is 50 Hz. During the transient which followedthe opening of CLD, the maximum frequency deviation was 0.016 p.u., then ω3settled below 0.01 p.u. in a time span less than 0.9 sec. As is evident from Fig. 8.6,the coordination alone does bring the frequency deviation to zero, though it takeslonger time. In the case of no auxiliary control, the system’s frequency does notreturn to 50 Hz and remains above 0.012 p.u. Moreover, the maximum excursionof the frequency is slightly above 0.035 which is approximately 2 times greaterthan that of the case with 2 auxiliary controllers.

Fig. 8.7 depicts the phase portrait of the state variables of G3. Again, onecan observe that in the case of full control, the state variables quickly approachthe new equilibrium.

8.5. Conclusions 119

−2 0 2 4 6 8 10 12 14

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

δ3, deg.

ω3, p

.u.

No Extra Control

Coordination only

Full Control

Simulation time T = 3 sec.

Figure 8.7: Phase portraits for the case of i) No extra control, ii) Coordinationonly, iii) Full Control. In equations (8.4) and (8.5) K1 = K2 = 10, k1 = k2 = 40

8.5 Conclusions

This chapter presented some results of the study which aims to investigate thepossibility of employing disturbance scheduling for power system stabilizationand electric power quality improvement. Unlikely the first case study, we nowsolve a different problem–that of stabilization of a power system. There are, how-ever, some similarities which bridge the two applications: in both cases certaininformation known in advance was utilized; the two case studies are concernedwith power quality phenomena, namely, voltage sag and frequency quality.

In this case study, we applied the concept of disturbance scheduling methodand the Lyapunov Control Function theory for the design of a power systemcontroller. The controller designed has the following properties:

• The controller is simple, linear, and decentralized.

• The controller uses local input signals that are available and measurable.That is, no observer and communication system are needed.

• Damping of a power system with multiple generators is additive [55].

• The control is robust and assures local asymptotic stability of the system.

The aforementioned properties of controller presented in this chapter make itapplicable to distribution networks with local generators which is especially ap-propriate in the deregulated electricity market environment.


The design of the controllers is based on a simplified model of the system.However, the validation of the model with controllers is performed on a detailedmodel in Matlab. The simulations performed clearly show that the controllerfulfils the objective stated in the subsection 8.1.4 allowing a more reliable modeof operation, which also brings economical merits.

Chapter 9

Conclusions and FutureWork

9.1 Conclusions

“Behind the public eye a quiet revolution is taking place, one that will perma-nently alter our relationship with energy–the building block of our industrial,digital society” [17]. In this quotation, Distributed Generation (DG) is impliedas the main driver of the revolution. The introduction of DG and electric powersystem restructuring create a new reality for both utilities and customers. Thenew reality will create more competitive electricity markets and bring the cus-tomer service to a qualitatively new level.

In the light of these facts, it becomes more important to operate the powersystem such that the quality of service and reliability of power delivery are ableto meet more stringent specifications imposed by the new electricity market.

Power quality is one of the most important quantifiers of the quality of cus-tomer service. The issue of electric power quality has been in the focus of re-search of both power industry and academia for the last decade. The researchhas resulted in a number of techniques aimed at enhancing the quality of electricpower.

This thesis work presents a new contribution to the existing methods ofpower quality improvement. More specifically, this work is concerned with theinvestigation of the possibility to mitigate voltage sags by using coordination ofpower system equipment that has voltage control capabilities.

The voltage sag mitigation method presented in the thesis is essentially basedon the conjecture that utilization of certain information which is known in ad-vance facilitates the deign of auxiliary controllers and eventually leads to en-hancement of power quality.

Information that is known beforehand is employed for taking preventive mea-sures prior to the occurrence of intentional disturbances on the system. This

121

122 Chapter 9. Conclusions and Future Work

process is here termed “disturbance scheduling”. Here, the disturbance can becaused by motor starting or some other topological change of the system. Thecombination of advanced controllers and disturbance scheduling comprise themain tool for achieving enhanced regulation of a process of interest.

Two case studies have been presented which illustrated the use of the pro-posed power quality improvement methods. The first of the two case studiespresented in the manuscript is concerned with the issue of voltage sag mitigationat a steel mill. The sensitive load of the mill is subjected to voltage sags causedby a start up of a large synchronous motor. With the aid of computer simula-tions, it was shown that the problem of voltage sags can be effectively alleviatedby applying disturbance scheduling and an advanced voltage controller.

In the second case study, another power quality phenomenon—quality offrequency—was the chief subject of the investigation. The local power systemof a paper mill has been studied. The main objectives of the second case studywere to improve the reliability of the system and formulate a method for fullerutilization of the system’s technical and economical potentials. In more preciseterms, the latter objective was to apply a new control algorithm which wouldallow the operation of the backup generators such that they not only providebackup power, but also market signals are incorporated in their operation.

The core idea of the control algorithm proposed is the conjecture that dis-turbance scheduling and coordinated control can assure proper, economicallybeneficial, and reliable operation of the local system, which is especially impor-tant when the electrical systems are becoming deregulated and restructured.

It is also worth mentioning that the new control in addition to providing a newmode of operation also improves the frequency control of the system, which is es-pecially important since at the present time—according to field measurements—the frequency control might be ineffective.

The work that was reported in this manuscript is summarized below:

• An overview of voltage sag mitigation equipment and methods has beendone.

• The so-called disturbance scheduling-coordinated control method has beenapplied to confront two common power quality phenomena, namely, voltagesags and frequency variations.

• A brief survey of linear and one of the nonlinear control techniques relevantto the case studies has been presented.

• A short overview of some identification techniques has complemented thethesis.

• Finally, the issue of eigenvalue cluster identification has been partially ad-dressed in the thesis, however, this complicated question requires furtherinvestigation.

9.2. Future work 123

9.2 Future work

The work presented in this thesis has the covered main aspects of applicationof disturbance scheduling for power quality improvement. Despite the fact thatnumerous simulations were run and case studies were carried out, there is stillspace for future work which could shed more light on some interesting issuesthat have not been treated herein. The following topics can be addressed in thefuture:

• Use of noncasual control in combination with robust linear and nonlinearcontrollers. It is especially interesting to study the use of µ-analysis, sincethe combination of noncasual and robust control can effectively guaranteethe robust performance and robust stability of the system in both steadystate and transient. The ability of robust control techniques to deal withsystem uncertainties is another significant advantage that is especially use-ful for applications in power systems. However, certain care should beexercised when applying robust control, since robust controllers in powersystem applications are very likely to have a very large order. Therefore,the use of model reduction techniques will be necessary.

• In special cases, when nonlinear control techniques are to be used, distur-bance rejection and reference tracking should be tested. That is, one couldmodel the system and the disturbance and then try to find a feedback con-troller such that the disturbance is captured in the zero dynamics of thesystem. If such a controller exists and it is realizable, then the disturbancewould have no impact on the controlled output (voltage or frequency orboth).

• Analysis of the properties of the above mentioned controller structuresshould be performed. Here, such properties as stability and performanceare essential.

• It is a well-known fact that having an accurate model of the controlledobject is one of the key components of successful controller design. It is alsoknown that some power system loads do possess nonlinear characteristics.Therefore, in some cases, nonlinear models of power system loads may berequired for accurate system modeling. This implies that nonlinear loadidentification may have to be needed for load identification. Partially, thisis done in Appendix E; however, nonlinear load identification might beworth a further investigation in the future.

• The eigenvalue cluster identification technique was proposed in a heuristicmanner, which implies that statistical analysis of the proposed identifica-tion method is required.

• The proposed disturbance scheduling technique can be implemented uti-lizing neural networks, which can result in new “smart control systems”.

Appendix A

Identification of EigenvalueClusters in Large PowerSystems

In this appendix the issue of eigenvalue cluster identification and assessment ofcomputational difficulties associated with the identification are addressed. A newidentification algorithm is introduced which is conceived as a complement to theProny analysis-based identification.

In order to perform system studies and analysis, a model the process of inter-est is required. All real-world processes are nonlinear in their nature; however,many of them can be satisfactorily approximated by linear models. If a physicalprocess can be well described by a model which is linear in some neighborhoodof an equilibrium point, two fundamentally different techniques are usually em-ployed for linear model obtaining. These are:

• Analytical linearization of the nonlinear equations describing the process(power system in our case) is the most valuable in the sense of gaininginsights in the dynamical properties of the system. In power system appli-cations there is however a substantial obstacle to using this linearizationmethod: it will most likely lead to an oversized model which will requiremodel reduction in order to obtain a manageable equivalent of the system.In addition, this high-order model will contain full information about thesystem, despite the fact that not all states of the system are normally ex-cited by an event1. That is, if the relative amplitude of the oscillationsdue to an event is small, these modes may be regarded as “not excited”.Besides this, the full model will contain states which correspond to physical

1Here the word “event” only denominates some kind of perturbation of the system. Dis-connection or re-closure of a line are typical events.

125

126 Appendix A. Identification of Eigenvalue Clusters in Large Power Systems

phenomena that might be irrelevant to a particular study, e.g., transientsdue to capacitor bank energizing and the like.

• Identification techniques are often involved when a model of a process iscreated based on field measurements. Identification techniques are oftenused if explicit modeling of the process in question is not feasible, for detailssee the discussion in Subsection 4.4.2. More rarely, model reduction of thepower system (power system equivalencing) employs identification tech-niques (Maximum Likelihood Identification, Least-Squares-Error Identifi-cation) for obtaining reduced-order power system equivalents [63]. Thereare a large number of system identification methods available, however,many of them inherit one common feature–difficulties when dealing withidentification of closely spaced transfer function poles which in this thesisare synonymously called “eigenvalue clusters”. On the other hand, thepresence of eigenvalue clusters may have an adverse impact on the systemstability and performance. This consideration is, to a certain extent, thedriving force behind the work presented in this appendix.

Now the main objective of the study can be formulated as

Given time-domain data (measurements), find a linear time-invariant modelof the process that generated the data. If the process contains an eigenvaluecluster, its presence should be reflected in the model. That is, the clustershould not be approximated by a composite eigenvalue.

The objective stated above clearly requires use of an identification technique.The Prony analysis that has been briefly reviewed in Chapter 5 is best suited forfulfilling the aforementioned objective. This is so because the Prony analysis is:

• A classical technique that has been used by researchers for a substantialtime span.

• Numerically robust and inexpensive.

• A flexible identification tool which allows the analyst to separately treatthe poles of a transfer function and then the zeros. This is feature isinstrumental in the algorithm developed in this Appendix.

We will return to the Prony analysis a bit later; first, the issue of eigenvalueclusters is discussed in more detail.

A.0.1 Multiple eigenvalue issue

The fact that power systems are never at rest substantially complicates thesystem analysis, since this volatility implies that models of the power systemsare time-variant. Moreover, it is well known that power systems are nonlinear.As an example, saturation (ceiling voltages) in the exciter circuits or rate limiters(gate velocity) in the governors can be mentioned.

127

Nonetheless, this problem is usually “overcome” by assuming that the rel-ative movement of the system around an equilibrium point is relatively small.This assumption justifies the use of the aforementioned linear analysis. That is,the system under consideration is treated as a stationary system. While in manycases this is acceptable, the situation may become somewhat more complicatedif a phenomenon called “modal resonance” occurs. In simple terms, modal reso-nance is a phenomenon that takes place when two or more eigenvalues interactwith each other. Physically this may be explained as follows.Gradually changing its state2, a power system moves from one equilibrium pointto another. In general, the eigenvalues of the Jacobian matrix evaluated at a newequilibrium differ from the previous ones. Furthermore, some of the eigenvaluesmay start approaching each other. As is well known from perturbation theory,the eigenvalues of a matrix depend continuously on the parameters of the matrix,e.g., the following inequality holds

∣∣∣∣∂λi∂pj

∣∣∣∣ = a <∞, (A.1)

where a is a real number, λi and pj are the ith eigenvalue and jth parameterof the matrix in question, i.e., A = A(p). However, when several eigenvaluesare concentrated in a relatively small domain of the complex s-plane, the mag-nitude of a may experience significant growth. That is, small variations of theparameter pj may induce substantial change of the eigenstructure of the system.In particular, the eigenvalue λi can become quite mobile as the parameter pjis varied. In the worst case, the eigenvalues can eventually cross the imaginaryaxis and assume values in the right half-plane causing violent oscillations on thepower system. Therefore, an initially stable power system may enter a poten-tially dangerous operating region if an eigenvalue cluster is present in the system.As a result, operation in such a region can compromise stability of the systemor its performance or both.

A mathematical description of modal resonance in power systems is quitecomplicated and is not as yet fully understood. Nevertheless, this issue hasalready received some attention [50], [65], and [38]. Modal resonance is closelyrelated to the issue of eigenvalue clusters, implying that a better understanding ofthe mechanisms leading to the formation of eigenvalue clusters may also elucidatethe issue of modal resonance.

We therefore briefly review some basic fact about closely spaced poles. Let usassume that a power system can be linearized3, i.e., the four matrices A, B, C,andD exist and A is nonsingular, see (4.18). Suppose next that the power systemwas linearized at an equilibrium point x0. In addition, all the eigenvalues of Aare distinct, i.e., there are n simple eigenvalues. Then, in principle, the matrix Acan be used to obtain the Jordan decomposition Y HAX = J , ([Y H ]ij = [Y ∗]ji),where Y , X, and J are respectively the matrix of right eigenvectors, matrix of

2This motion is due to variations of some parameters of the system.3We confine ourselves to considering small deviations of the system states.


SystemA, B, C, D

Feedbackgain

1x

u

Figure A.1: State Feedback Controller

left eigenvectors, and the Jordan block. Some elementary considerations lead toan interesting conclusion [30]:

∣∣∣∣∂λi∂p

(x0)

∣∣∣∣ 61

|yHx| =1

S(λ) , (A.2)

where p is some parameter, y and x are the columns of matrices Y and Xassociated with the corresponding eigenvalue λi.

This shows that under certain conditions the eigenvalue may be sensitive toa parameter change if the angle between corresponding eigenvectors approachesπ/2 radian. The final conclusion is [30]: if the matrix A has a defective eigen-value [the case when r repeated eigenvalues share k : k < r eigenvectors], O(ε)perturbations in A may induce O(ε1/n) perturbations in the eigenvalue λi.

It should also be noticed that the presence of repeated eigenvalues makes thecorresponding eigenvectors troublesome as well. The main points discussed inthis section are exemplified below.

Example A.1. Let matrices A, B, C, and D be the reduced-order model of ahypothetical ‘power system’.

A =

−1.1 0 0 1−5.3482 1.7 −4.39 5.872

0 1 −0.1 00 0 1 −0.5

, (A.3)

B = [0, 1, 0, 0]T, (A.4)

C = [1, 0, 0, 0] , D = 0. (A.5)

Let us also assume that there is a path for partial static state-feedback F ∈ R1×4.F1 ∈ [0, 0.02], the remaining components of F are zeros:

F = [interval(0, 0.02), 0, 0, 0],

see Fig. A.1. The feedback gain is intentionally assigned a very small value. Inreal world, this can be some neglected dynamics or an “insignificant” structuraluncertainty of the system. Now we investigate the impact of the feedback gain

129

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.051

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

Real Axis

Imag

Axi

s

Parameter−cahnge inducedeigenvalue movement

The feedback gain wasonly varied within [0, 0.02] p.u. in uniformsteps of 0.001!

Figure A.2: Collision of the upper half-plane eigenvalues

F1 on the system eigenvalues. The root locus of the controlled system is shownin Fig. A.2. The main conclusion which can be drawn from Fig. A.2 is: be-ing marginally stable, the controlled system becomes unstable exposing a jumpbehavior. Clearly, the movement of the system eigenvalues is nonlinear in F1.

It can be noted that introduction of the feedback is equivalent to altering theterm A2,1 by the amount of F1. It is also interesting to inspect the eigenvectorsof the system as F1 is altered between 0 and 0.02. Figure A.3 depicts the systemeigenvalues. One may notice that despite the fact that the perturbation was‘negligibly’ small the eigenvectors are displaced by large angles. The numericalvalues of the eigenvectors for 2 cases—F1 = 0.01 and F1 = 0.012—are given inTable A.1. 2

A.0.2 Transfer function identification

In this subsection we recapitulate and elaborate the Prony identification tech-nique which has been previously discussed in Chapter 5.

It is worth reminding that the classical Prony analysis is a signal identificationmethod rather than transfer function identification. However, if the input signalhas a special form, than an extension of the Prony analysis can be used not onlyfor signal, but also for transfer function identification.

We now propose an algorithm that can be applied to measured data in orderto identify a transfer function of the process that generated these data.


0.2

0.4

0.6

30

210

60

240

90

270

120

300

150

330

180 0

Figure A.3: Change of eigenvectors. The thin line represents the components ofthe eigenvectors at F1 = 0.01 and the thick line marks the same components

for F2 = 0.012

Table A.1: Eigenvectors for 2 different feedback gains: F1 = 0.01 andF2 = 0.012

E(F=0.01) =

0.0427 -0.3081i 0.0427 +0.3081i -0.3188 -0.0529i -0.3188 +0.0529i

-0.2205 +0.5591i -0.2205 -0.5591i 0.5441 +0.2135i 0.5441 -0.2135i

0.5006 +0.2537i 0.5006 -0.2537i 0.2544 -0.5007i 0.2544 +0.5007i

0.3755 -0.2934i 0.3755 +0.2934i -0.2959 -0.3884i -0.2959 +0.3884i

E(F=0.012) =

0.2061 +0.2269i 0.2061 -0.2269i 0.3135 +0.0915i 0.3135 -0.0915i

-0.3431 -0.4920i -0.3431 +0.4920i -0.4825 -0.3355i -0.4825 +0.3355i

-0.5030 +0.2568i -0.5030 -0.2568i -0.3419 +0.4407i -0.3419 -0.4407i

-0.0028 +0.4766i -0.0028 -0.4766i 0.2345 +0.4264i 0.2345 -0.4264i

131

Algorithm A.2. Prior to performing system identification, accurate field datashould be collected. The choice of sampling interval and signals to be measuredis critical. When possible, quantities in all three phases should be acquiredin order to have some redundancy. Provided the field data are available, thefollowing can be done to obtain a linear time-invariant transfer function of aprocess

1. Identify an initial model of the process. When identifying an initial model,it is irrelevant which identification method is used; the most importantpart is the quality of the estimate. In very noisy environment the Pronyanalysis may not be the best choice.

2. Determine a region in the s-plane which can contain closely-spaced poles,i.e., eigenvalue clusters. It can be noted that it is relatively easy to find aregion which might contain eigenvalue clusters since we are only interestedin clusters that are located in a neighborhood of the imaginary axis. Theother eigenvalue clusters can be left lumped together, since well dampedeigenvalues should not cause any severe problem even if they are repeated.

3. Apply an optimization technique to obtain a more accurate estimates of theclosely-spaced transfer function poles. The Newton optimization techniquehas proven to be a very effective tool for performing various minimizationtasks, provided a reasonable initial guess is available. In our case, an initialguess can be easily estimated.

4. Incorporate the improved poles in the model and apply the Prony analysisto the model. At this stage, the zeros of the transfer function are estimated.

2

In brief, the core idea is to obtain an initial transfer function of the model thatpossibly has eigenvalue clusters. Next, the closely spaced poles of the transferfunction are specialized. Finally, having obtained a set of correct poles, the zerosof the transfer function are determined based on optimal curve fitting in a leastsquares sense. These steps are explained below in more detail.

The identification of the poles of a transfer function is adopted from earlywork reported in [32] and [71]. For the sake of consistency, some of the equationsfrom the article [71] are reproduced here. This also seems pedagogical since theseequations are used for obtaining an initial guess for the Newton method.

Remark A.3. So far, nothing has been said about how the data for the identi-fication was obtained. In practice, time domain data may be obtained in twodifferent ways, namely, by recording field measurements or by performing com-puter simulations. In both cases, the system must be set in motion by applyinga pre-specified perturbation/excitation, then the system’s response should berecorded and analyzed.

The following simplifying assumptions are made in order to facilitate theexposition:


• All poles of the transfer function are distinct.

• Prior to the disturbance, the system was at rest, i.e., zero initial conditionmodel is assumed.

• The feedforward term is zero.

• The system has no transport delay.

It should however be stressed that these assumptions are only made to simplifythe presentation; in the original work these assumptions were relaxed. Followingin the lines of [82], the transfer function to be identified is given by

H(s) =

n∑

k=1

Rks− λk

, (A.6)

where Rk and λk are the unknown parameters which are the residues and polesof H(s). The pre-specified order of the system is denoted by n. The perturbationused in this study is a train of impulses

Uinput(s) =

L∑

j=0

cjexp(−sDj)− exp(−sDj+1)

s− λn+1, (A.7)

where Dj ∈ R is the delay of jth impulse, D0 ≡ 0, L is the number of impulses inthe train, and cj ∈ R is the impulse amplitude. As usual, s = α+ jβ is reservedfor the Laplace variable.

The system’s (A.6) response to the perturbation (A.7) may be determined as

Y (s) = U(s)G(s) =

[L∑

j=0

cj

(e−sDj − e

−sDj+1

)][ n∑

k=1

Rk

(s− λk) (s− λn+1)

]. (A.8)

After some algebraic manupulations4, the transfer function residues becomeavailable, keeping in mind zero initial condition:

Ri =Bi(λi − λn+1)

c0eλi(DL+1−D0) − cL +L∑j=1

(cj − cj−1) eλi(DL+1−Dj)

, i = 1, n, (A.9)

where B’s are defined as

Bi = Qi

L∑

k=0

ck

eλi(DL+1−Dk) − eλi(DL+1−Dk+1)

,

Qi =

Ri

λi−λn+1, i = 1, 2, . . . , n

Qn+1 = −n∑

k=1

Qk.

(A.10)

4The derivation is not very complicated but is bulky and fairly lengthy: first the summandsin (A.8) are regrouped, then partial fraction expansion is done, finally the inverse Laplacetransform is applied to the equation and then the time variable is shifted.

133

0 1 2 3 4 5 6 7−1

−0.5

0

0.5

1

1.5

2

Tra

in o

f im

puls

es

Time, seconds

c1

c2

c3

D1

D2

D3

D4

Figure A.4: Train of impulses

Now, if λ’s were known, the residues Ri would be readily available by solvingequation (A.9) for Ri in a least squares sense. Since λ’s can be calculated aswas explained in Section 5.1, all components of the transfer function can bedetermined.

A.0.3 Newton’s method for signal identification

Let a measured signal x(t) and its estimator x(t) be given by

x(n) =K∑

i=0

Cizni + w(n) (A.11)

x(n) =

K∑

i=0

Cizni , (A.12)

where n = 0, 1, 2, . . . , N − 1, N is the number of data samples, K stands for theorder of the model, i.e., the number of superimposed damped sinusiods, w(n) isthe noise contamination having the probability density function PDF: N (0,W ),respectively. Ci is the complex amplitude of the mode zi, see equation (5.3).


According to equation (5.5) we have

1 1 · · · 1z1 z2 · · · zn...

.... . .

...

zN−11 zN−1

2 · · · zN−1n

C1C2...CN

=

x(0 · Ts)x(1 · Ts)

...x((N − 1) · Ts)

.

The sampling interval is denoted Ts. The task is now to estimate the unknowneigenvalues zi. The last equation can be rewritten in the matrix form:

x = ZC.

In [82] it was shown that the following equation also holds

x(K)x(K + 1)

...x(N − 1)

=

x(0) x(1) · · · x(n− 1)x(1) x(2) · · · x(n)...

......

...x(N −K − 1) x(N −K) · · · x(N − 2)

aKaK−1

...a1

,

(A.13)

where the vector a consists of the coefficients of the characteristic polynomial ofthe system:

zKi − (a1zK−1i + a2z

K−2i + . . .+ aK) = 0, i = 1, 2, . . . ,K. (A.14)

Equation (A.13) may be written in a more compact form

x = Xa.

When performing signal estimation, one of the most common approaches isto use the so-called maximum likelihood criterion. That is, one attempts tomaximize the probability density function p(x; θ)

p(x; θ) =1

(2π)N/2 det1/2(W )exp

[−1

2[x− x]T W−1 [x− x]

]

=1

(2π)N/2 det1/2(W )exp

[−1

2[x−Xa]T W−1 [x−Xa]

](A.15)

where

θ = a (A.16)

is the parameter vector to be estimated.

Since the exponential function exp(−x) is a continuous and non-increasingfunction, it is maximized by minimizing its argument. Moreover, since the ex-ponential function is continuous, it is minimized if the natural logarithm of itattains a minimum. In other words, one has to find such a set of the parameters

135

θ that (A.15) is maximized, which will give the best correspondence between theobserved data set and the model of (A.11).

∂ ln(x, θ)

∂θi=

1

2

∂ (Xa)T

∂θiW−1 (x−Xa) . (A.17)

The last expression forms the gradient:

∂ ln(x, θ)

∂θ=

1

2

∂ (Xa)T

∂θW−1 (x−Xa) (A.18)

= −XTW−1(X −Xa). (A.19)

Setting the gradient ∂ ln(x, θ)/∂θ = 0 and solving (A.19) for a, the minimumvariance estimate of a is obtained as

a =(XTW−1X

)−1XTW−1X. (A.20)

An initial estimate of the transfer function poles is made available through fac-torization of equation (A.14).

Assume that z1 is the pole closest to the unit circle. We now can initializeadditional pair of poles zK+1, zK+2 in a neighborhood of z1. To reflect this inthe signal model, we convolve the characteristic polynomial with the initializedpoles

p(z) = (z − zK+1)(z − zK+2)K∏

i=1

(z − zi) (A.21)

= zK+2k − (a1zK+1k + a2z

Kk + . . .+ aK+1) = 0 (A.22)

to obtain a new characteristic polynomial.

The aim of this Appendix is now formulated

Given time-domain measurements and the modified characteristic polynomial(A.22) apply an optimization technique to obtain an estimate of the poles ofthe system that generated the measurements.

We propose a new objective function

J = aT

[N∑

k=1

Y T (k)(AAT

)−1Y (k)

]a, (A.23)

where

AT =

aK+2 aK+1 · · · a1 1 0

. . .. . .

. . .

0 aK+2 aK+1 · · · a1 1


Y (k) =

xK+2(k) · · · x1(k) x0(k)xK+3(k) · · · x2(k) x1(k)

......

......

xN−1(k) · · · xN−K−4(k) xN−K−3(k)

.

There are a number of optimization algorithms developed for such kind ofminimization problems. Among them are: Newton’s method (with various mod-ifications), the Secant method, and many other [27]. It is well-known that havinga good initial guess, the Newton method provides fast convergence. In fact, itmay be superlinear, or more precisely, quadratic. In our case, “a good initialguess” can be obtained relatively easy by using the Prony analysis or equa-tion (A.20) or Discrete Fourier Transform. This makes the Newton method areasonable alternative for the task at hand. The basic idea behind the Newtonmethod is shown below:

θk+1 = θk −H−1k gk, (A.24)

where Hk denotes the Hessian matrix of J evaluated at θk, k is the number ofiteration, and the vector gk is the gradient of J .

A.0.4 Gradient and Hessian matrix

In this subsection explicit expression for the gradient (g) is given. The gradientis formed by differentiation of J with respect to the vector of parameters a:

g =∂J

∂aT=

N∑

k=1

[vT (k)Y (k) + aTY T (k)

∂v(k)

∂aT

], (A.25)

where

v(k) =(AAT

)−1Y (k)a (A.26)

and

∂J

∂aT=

[AA

T]−1

[Y (k)−

[∂(ATA)

∂a1

,∂(ATA)

∂a2

, . . . ,∂(ATA)

∂aK+1

][IK+1 ⊗ v(k)]

], (A.27)

where ⊗ denotes tensor product5

Writing the Hessian matrix explicitly would involve substantially more com-plicated analysis, which can be found in e.g., [80].

5For two matrices A ∈ Rm×n and B ∈ Rp×q , the tensor product is defined as:

A⊗B =

a11B a12B · · · a1nB

a21B a22B · · · a2nB

.

.

....

. . ....

am1B am2B · · · amnB

.

It is easy to see that A⊗B ∈ Rmp×nq .

137

Initial guess

A good choice of the initial conditions is the key to the successful optimization. Ifthe initial guess selected was not close enough to the real values of the parameters,then the Newton optimization may not have superlinear convergence. In theworst case it can diverge, in spite of fact that the order of the model ‘K’ wasproperly chosen.

Let us consider two possible ways of obtaining an initial guess for the Newtonoptimization–the Discrete Fourier Transform (DFT) and the Prony analysis.

DFT is a well-established method which is numerically robust and inexpen-sive. However, the DFT cannot provide an initial guess of pure real transferfunction poles, i.e., λi ∈ R. In addition, the DFT is very likely to lump closelyspaced poles to one. The spectral leakage phenomenon makes it a difficult taskto separate them.

The Prony method is a sound alternative to the DFT for the initial guessobtaining. It is relatively inexpensive numerically and in noise-free environmentresults in a reasonably accurate set of transfer function poles. In noisy environ-ment, like in the case with the DFT, the Prony method can lump closely spacedpoles; however, this drawback can be eliminated by the proposed Newton opti-mization. Moreover, for the Prony method can identify any number of pure realand/or complex poles.

Accounting for all the above, the Prony analysis was used to obtain an initialguess for the Newton optimization in this case study6.

Selection of the Order of the Model

The major assumption of this subsection is that the function J = J(K) is convex.This may or may not be so; however, one can set a coarse grid of K’s and thenevaluate the objective function J on the grid. By doing this, a convex region ofJ(K) may be located and the corresponding minimum be found.

The process of the signal identification begins with setting the order of themodel KR ← Kmax to a maximal value and KL ← Kmin to a minimal value,unless the user has an educated guess of the model’s order. Next the minimiza-tion is performed for the current value of KR, then the same must be done forKL. Let us define Km ← E((KR + KL)/2). ‘E(·)’ is used here to denote thenearest integer towards plus or minus infinity. Then the values of the objec-tive function J(KR) and J(KL) should be compared. If J(KR) > J(KL) thenKR ← Km, otherwise KL ← Km. The new iteration begins with updatingKm ← E((KR + KL)/2). The iteration is repeated until the global minimumis attained. Graphical interpretation of the proposed algorithm is depicted inFig.A.5.

It is expected that such a binary search will reduce the number of runs ofthe minimization. For example, let Kmax = 40 and K∗ = Kopt = 7. Then the

6The initial guess obtained with the Prony analysis is equivalent to that given by equa-tion (A.20) if W = IK .


Y e s

N o

N o

Y e s

max. min .: , :

: , : 1.2

R L

R Lm

K K K K

K KK E i

= =+ = =

R LK K=

( ) ( )R LJ K J K>

R mK K←

.

.

opt L R

Stop iterating

K K K= =

: 1,

:2

R Lm

i i

K KK E

= ++ =

L mK K←

( )L

Evaluate

J K ( )R

Evaluate

J K

: 1,

:2

R Lm

i i

K KK E

= ++ =

Figure A.5: Flowchart of the selection of system’s order

A.1. Application examples 139

optimal order can be attained (in the best case) after only 5 runs. In addition,one can also follow in steps of [64] and split the data into two sets: training andtest sets. This gives the opportunity to use a) the training set for adjusting themodel’s order and b) the test set for evaluating the “quality of the model.”

A.1 Application examples

This section illustrates the use of the optimization routine proposed. Identifi-cation is performed for two different cases–noisy and noise-free environments.Each case is considered from two perspectives–close and well-isolated eigenval-ues (called here ‘distinct’) are identified. The only identification performed inthis section covers the identification of the transfer function poles, since only thispart of the transfer function identification differs from the existing extensions ofthe Prony method. The identification of transfer function residues does not differfrom the procedure described in subsection A.0.2. In all cases the sampling inter-val was 10−2 seconds and the number of samples was fixed to 700. The numberof samples as well as the sampling time were chosen arbitrarily (the time stepmust be smaller than the Nyquist period.)

A.1.1 Distinct poles (Large separation)

This section presents identification of transfer function poles that are well sepa-rated. In this work, two poles are said to be well-isolated (|λi−λk| > ε, ∀ i 6= k)if the value of S(λ) in (A.2) is large, which is different from case to case.

The example considered earlier in this chapter on page 128 could have beena good case study for applying the theory developed in Section A.0.3; however,in such a case we would not be able to assess the ability of the routine to findthe damping of poles7. Therefore, a new example should be created.

In this section, the signal model is of the form

y(n) =K∑

i=1

Ci exp(σin∆t) sin(ωin∆t+ ψi), (A.28)

where y(n) is the measurement at time n, σi and ωi are the damping and thefrequency of ith pole λi = σi + jωi, ∆t is the sampling interval; Ci and ψi arethe amplitude and the phase of ith residue.

Note A.4. The primary goal of this section is demonstrate the use of the Newtonoptimization for identification of the poles of a transfer function. That is, only σiand ωi are of interest here. However, the residues of the signal are also identified.

7In that example, all the poles were located on the imaginary axis.


Table A.2: Convergence illustration. Distinct poles, noise-free caseIter. No. 1 2 3 4 5 6‖J∗ − Jn‖2 7.21 2.17 7.5998e−1 1.2114e−1 2.2753e−3 2.2756e−6

Summarizing all the above, the following model is selected for validation ofthe algorithm

C1 = 1, σ1 = −1, ω1 = 2π, ψ1 = π/5,

C2 = 1.5, σ2 = −1.5, ω2 = 1.5 · 2π, ψ2 = 1.5 · π/5. (A.29)

Noise free signal

According to (A.16), the parameters of the model can be put in a vector form:

θ∗ = [1, 1.5, 2π, 3π, π/5, 1.5π/5,−1,−1.5]T .

The Prony method was not used for an initial guess obtaining since in theabsence of noise and absence of close poles the Prony method produces nearlyideal transfer function. Thus, an arbitrary initial guess was chosen which ishowever far enough from the true values:

θ0 = [0.8, 1.36, 5.969, 10.1473, 0.754, 1.2818,−0.9,−1.53]T .

A reasonable solution to the problem was obtained after 6 iterations. The errornorm in the parameter vector was ‖θ∗−θ6‖2 6 2.2756 ·10−6. Fig.A.6 graphicallyillustrates the rate of convergence of the optimization. To be able to trace downthe convergence, the values of ‖J∗ − Jn‖2 are listed in Table A.2 which revealsa superlinear convergence of the optimization. The estimates could have beenobtained with a pre-specified accuracy which would only take more iterations.Also the Prony analysis gives excellent results whose accuracy is only limited bythe capabilities of the software used in the case study (Matlab.)

Noisy signal

This section presents results on the identification of well separated poles in noisyenvironment. The main argument for performing identification in noisy envi-ronment is the conjecture that even if the time domain data were generated bymeans of a computer simulation and consequently supposed to have no noise,the data contains modes that can be seen as noise. Since the order of the powersystem is high, there is no reasonable way of inclusion of all the modes into amodel. So, neglected modes will certainly produce some equivalent of noise. Itcan be shown that as the number of modes neglected increases, the probabilitydensity function of the sum of the truncated modes approaches the Gaussianwhite noise. Fig. A.7 illustrates this statement. In Fig. A.7A and Fig. A.7Bone can see a sinus wave and its probability density function. Obviously, the


1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

Iteration number

|| J(

θ* ) −

J(θ

6)||2

Convergence of the Newton opimization

Figure A.6: Convergence illustration. Distinct poles, noise-free case

−10 −5 0 5 10−0.5

0

0.5

1

1.5

2

2.5D: PDF of a sum of 50 sinus waves

0 0.2 0.4 0.6 0.8 1−10

−5

0

5

10

15

20C: Sum of 50 sinus waves

−2 −1 0 1 20

0.1

0.2

0.3

0.4

0.5B: PDF of a sinus wave

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1A: Sinus wave

Figure A.7: PDF of a sum of superimposed sinus waves


400 450 500 550−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Sample number

Noi

sy s

igna

l

Noisy data

Figure A.8: Distinct poles. Noisy data

PDF of a sinus wave has the rectangular shape. As the number of superimposedwaves increases (in this example the sin waves have different amplitudes andphases), the PDF of the resulting signal will approach the Gausian distribution,see Fig. A.7D.

All this means that the problem of ‘noisy signals’ not only arises when dealingwith field measurements, but also in the case of simulations of large power system.

The noise added to the ‘clean’ signal had the following characteristics: meanvalue µ = 0, standard deviation w = 1. To be able to obtain reasonable esti-mates, the Signal-to-Noise Ratio (SNR) was set to 41 dB. The effectiveness of theidentification algorithm was evaluated by comparison to the correct values. Theidentified parameters were also compared with the those obtained with the Pronyanalysis. A fragment of the signal embedded into noise is shown in Fig.A.8. Ta-ble A.3 gives the comparison of the two methods. One may notice here that theproposed algorithm performed better while identifying all the parameters exceptfor σ2.

A.1.2 Closely spaced poles

The following model was used for the test with noisy data

C1 = 1, σ1 = −1, ω1 = 2π, ψ1 = π/5,

C2 = 1.07, σ2 = −1.07, ω2 = 1.07 · 2π, ψ2 = 1.07 · π/5. (A.30)

The difference between each pair of the parameters is only 0.07.


Table A.3: Distinct poles, noisy data. Comparison between The Newtonoptimization and Prony analysis

Parameter C1 C2 ω1/(2π) ω2/(2π) −σ1 −σ2Newton Opt. 1.015 1.477 1.001 1.499 1.012 1.499Prony analysis 1.021 1.464 0.995 1.503 1.045 1.503

True 1 1.5 1 1.5 1 1.5

Table A.4: Convergence illustration. Closely spaced poles, noise-free caseIter. # 1 2 3 4 5 6 7

‖J∗ − Jn‖2 3.29 1.19 2.6e−1 4.12e−2 1.74e−2 6.35e−4 5.91e−6

Noise-free signal

The parameters may be put in the vector form according to (A.16):

θ∗ = [1, 1.07, 2π, 2.14π, π/5, 1.07π/5,−1,−1.07]T .The initial guess is

θ0 = [0.8, 0.88, 5.969, 6.5659, 0.754, 0.8294,−0.9,−0.99]T .The final result was obtained after 8 iterations. The error was estimated tobe ‖J(θ∗) − J(θ8)‖2 6 10−5. The presence of closely spaced poles normallycomplicates the optimization; however, in spite of the poles proximity, they wereidentified correctly. Table A.4 gives the numerical values of the error in theoptimization process.

Noisy signal

Noise used in this run had the same shape like in the case with distinct poles.First, the Prony analysis was run to obtain an initial guess. The Prony initialguess for 1 composite pole was:

C1 = C2 = 2.076309/2 = 1.0382

ω1 = ω2 = 1.035152 · 2πψ1 = ψ2 = 0.8999 · π/5σ1 = σ2 = −1.0732.

To initiate the Newton optimization, a second damped sinus wave [whose pa-rameters are indexed with the subscript 2] was added to the vector of parameters.It is noteworthy that the initial estimated are close to the actual parameters,the only drawback is the impossibility to distinguish the lumped poles by us-ing the Prony analysis alone. Table A.6 presents the rate of convergence of theNewton algorithm for the case of closely-spaced poles. Again, one can recognizequadratic convergence.


Table A.5: Comparison of the identified parameters. Closely spaced poles,noisy signal

Parameter C1 C2 ω1/(2π) ω2/(2π) −σ1 −σ2Newton opt. 1.0000 1.0700 1.0000 1.0700 1.0000 1.0700Prony analysis 2.0763 — 1.0352 — 1.0732 —

Actual values 1 1.07 1 1.07 1 1.07

Table A.6: Rate of convergence. Closely spaced poles. Noisy data

Iter. # 1 2 3 4 5 6J∗ − Jn 2.85 1.45 4.13e−1 0.47e−1 2.98e−3 9.54e−7

A.2 Discussion on the results

In this chapter some results on eigenvalue cluster identification are presented.An optimization technique is developed that is capable of identifying the closely-spaced poles of a transfer function in the presence of noise.

The main tool for the identification method proposed is the well-known New-ton algorithm. In order to obtain an initial guess for the Newton optimization,the Prony analysis is used. As the simulations show, in the noise-free environ-ment the poles (closely spaced and distinct) are perfectly identified with bothmethods. As the signal used for identification is contaminated by noise, differ-ences in performance of the two methods emerge.

Analyzing noisy signals, neither close, nor distinct poles can be preciselyidentified by the Prony method. The proposed method, however, is capable of‘distinguishing’ closely spaced poles, despite heavy numerical computations. Inall cases the algorithm developed in this chapter performed at least as good asthe Prony method. It should come as no surprise, since our method was usedto improve the poles identified by the Prony method. Generally speaking, theproposed algorithm was not conceived as a Prony method substitute, but ratheras a complement of it.

It may be noticed that as closely spaced poles approach each other, it becomesmore difficult to separate them. In the case they collide, the Newton optimizationwill become unstable and prone to numerical failures. However, it is not verylikely that a real power system will have exactly the same poles. Furthermore, ifthere are repeated poles, the Newton optimization still can be applied; however,in such a case the objective function must be modified accordingly.

Another potential danger for the proposed method is low SNR of the signalat hand. As certain threshold is passed, it will be impossible to identify theparameters of a model with desirable precision. If this is the case, higher orderspectral methods should be tested to find the repeated or closely spaced poles. Itshould however be realized that, in general, as certain threshold of SNR is passed,

A.2. Discussion on the results 145

identification of a transfer function by any method becomes more complex anderror prone.

A.2.1 Future work

Statistical properties

There are many improvements that can be done to the algorithm developed inthis chapter. Some of the possible continuations of the study are listed anddescribed below.

• A method for repeated poles identification should be sought. So far, onlyclosely spaced poles were identified. It seems relatively simple to modify(A.11) such that the model explicitly contains repeated poles.

• Mathematically rigor analysis of the convergence of the algorithm proposedshould be performed.

• Monte-Carlo simulations should be run in order to evaluate the statisticalproperties of the new algorithm. Also it is necessary to estimate the lowerbound on the signal-to-noise ratio at which identification is still reliable.

Appendix B

Main data for theOxelosund and Gruvon casestudies

The main data used in the present case study is listed in the two tables below.Table B.1 shows the data used for the transformer T7 in Fig. 7.1 while Table B.2gives that for the synchronous generator and motor. All the values are given inper unit of the devices.

Table B.1: Case study. Main data for the transformer

ST , MVA f , Hz V1, kV V2, kV xT , p.u. xT ,Ω

55 50 135 10.5 0.1 0.20045

In the steady state P0 = 18.7MW , Q0 = 2.7878 MVAr, IG1,phase = 1.48

kA, ILoad,phase = 0.916 kA, ISystem,phase = 1 kA, IBFF7,phase = 1.48 kA,

VLoad,phase = 10.31 kA (= 0.9821 p.u.), SSystem = −9.9 + j7.86 MVA, andSLoad = 6.945 + j9.26 MVA.

147

148 Appendix B. Main data for the Oxelosund and Gruvon case studies

Table B.2: Case study. Main data for the synchronous generator and motor

Parameter Synchronous generator Synchronous motor

Xp, p.u. 0.06 (0.081426 Ω) 0.06 (1.654070 Ω)Xd, p.u. 2.18 (2.958484 Ω) 1.78 (49.07083 Ω)X ′d, p.u. 0.17 (0.230707 Ω) 0.35 (9.648796 Ω)

X ′′d , p.u 0.13 (0.176423 Ω) 0.24 (6.616290 Ω)

Xkf , p.u. 0 0Xq, p.u. 2.09 2.09X ′′q , p.u. 0.23 0.23

Ra, p.u. 0.0014 (0.0026895 Ω)Ta, s 0 0T ′do, s 9 9T ′′do, s 0.05 0.05T ′′qo, s 0.05 0.05

Air gap factor 1 1

H, s 5.22 4ω, rad/s 314.159 314.159D, p.u. 0.05 0.05Vbase, kV 6.06218 6.06218Ibase, kA 4.467 0.2199Vt, p.u. 1 0ψ0 0 0

P0, MW 19.6 0Q0, MVAr 0 0

Exciter

T1, s 0.02

Ta, s 1.5Tb, s 1TE , s 0.02K 100 No exciter

Emax, p.u. 5Emax, p.u. −5Vbase, kV 6.06218Ibase, kA 4.467Rreverse,Ω 15000

149

Royal Institute of Technology

The Oxelosund case study

Created:Last Modified:Printed On:

February 3, 1999June 27, 2000 (es97-52)July 04, 2000 (es97-52)

SS 1

T2

ABC

ABC

Tm

va =

5513

510

.5

#1#2

1000

0.00

1

0.00

1

0.00

1

3 PhaseRMS

A B CIa

net

Ibne

t

Icne

t

0.001

0.001

0.001

Ib

Ic

Ua

Ub

Uc

Ia

Isa

Isb

Isc

0.00

1

0.00

1

0.00

1

5.51

25

5.51

25

5.51

25

0.02

3395

8

0.02

3395

8

0.02

3395

8Iloa

da

Iloa

db

Iloa

dc

Ifau

lta0.

001

Ifau

ltb0.

001

0.00

1If

aultc

Faul

t

A B CFA

UL

TSHydro

Governor

TmwTm0

SP

HG

Tmw

Ef If

Te

Tm

A

B

C

VabcIfEf

Vref

Exciter (SCRX)

1.0

EF

WO

UT

MT

M

Generator

81.25 MVA, 10.5 kV

A

B

C

3 PhaseRMS

GT

M

0.001Iga

0.001

0.001

Igb

Igc

Blm

7 Tm

wEf

Ifw Te

Tm

A B C

A B C

0.00

1

0.00

1

POUT

QOUT

PANG

GTM

WOUT

EF

Generator

Pnet

Qnet

Vnet

PRO

T

0

Busbar

Ua

Ub

Uc

Control

of System Breaker

0 - Closed

1 - Open

SYST0

ENABBlm1

win

Blm

Measurments on

Motor Blm 7

PBlm

QBlm

PANGBl

TMBlm

WOUTBlm

winBlm

TIM

E

Release Generator

ENAB

TMBlm

WOUTBlm X2

*3.183e-3 ompu

*TMBlm

**

1

k

-1

Synch motor Blm 7

3.6 MW, 10.5 kV

Torque

-1

Fault Currents

Ifb

Ifa

Ifc

*

1.0

Events

Faul

t

Fault Blmsw

Blm

sw

Torque

Vcanet Vbc

net

Vab

net

Vca

bus

Vbcbus

Vabbus

Vabbus

Vbcbus

Vcabus

Vbcnet

Vcanet

Vabnet Net

Iga

Igb

Igc

SYST A B C

RR

L

RR

L

RR

L

A B C

PROT

A

B

C

3 PhaseRMS

A

B

C

3 PhaseRMS

VlR

0.00

1

Vger

Vger

IBlm

a

IBlm

b

IBlm

cBlm

brea

A B C

Load

1e-3

1e-3

1e-3

Vrefer

BT

25

VlR

AB

Com

par-

ator

e-sT

Blm

sw

e-sTBlmbrea

Blm

sw

*0.02

contr

A B

Com

par-

ator

TIM

E

40

R/S

Flip

flop

SR

Q

e-sT

ompu

1.0

N(s)D(s)

Order = 7

I

P

1.

B-

D+

E

+

F

+

cont

r

VlR

1

B

-

D+

F

+ Vrefer

*0.

05

N(s)D(s)

Order = 2 WOUT

D-

F

+

N

D

N/D

314.

1592

6

Figu

reB.1:

EMTDC

schem

aticdiagram

ofthecase

study

Appendix C

Comparison of the softwareused in the thesis work

In this work, several computer simulation and analysis programs were used tocarry out power system simulations. Among those are Matlab and its tool-box Power System Blockset [Simulink], SIMPOW, and EMTDC. Each of theaforementioned programs has advantages as well as drawbacks. Table C.1 givesa comparative analysis of the programs’ capabilities. In general, it can be saidthat for best results one should use several programs. The combined use ofthe programs offers reasonable speed of the simulation, additional functional-ity, and ensures the correctness of the simulations. While dealing with a largepower system the best option is to use SIMPOW which provides fast simulations;conversely, if the system at hands is of a small size, the best option undoubt-edly is the Power System Blockset. A well-developed components library makesEMTDC also a good alternative for medium-size power systems which, however,lacks a linear analysis tool.

This study has shown that the direct use of Matlab—that is, without PowerSystem Blockset—for power system simulation is neither convenient nor fast.However, setting up the algebraic and differential equations provides best insightsinto the dynamics of the system studied.

151

152Appen

dix

C.

Compariso

nofthesoftw

are

used

inthethesis

work

Tab

leC.1:

Com

parison

ofthesoftw

arepackages

used

! " #

$ % % #

% %

& '

( # ) $ %

%

* ( +

" %

" ,

& # + ' +

%

( - ./0

& # + % ,

, "

) ' % )

1 ( #

"

2$ % "

( $ % )3 % ( #

& ( #

( #

#

2$ % "

" " " % " )4 ( # )

4 % 5 #

#

'

4

# #

' 6 ' +

7 " )

7

" #

# %

' % "

' " "

# ( + ) ) +8 2 # % + " " + ) 9

' #

& &

Appendix D

PID Controller Tuning

The goal of the optimization is to determine such a set of the PID parametersthat the time (frequency) domain specifications on the response are fulfilled.The tuning procedure described in this section heavily relied upon numericaloptimization. As the objective function in general is not convex, the optimizationmay fail. If the optimization procedure does so, another set of initial conditionsshould be considered. If the change of the initial guess does not lead to anacceptable solution, another controller type must be adopted.We start with a formal description of the task:

minimizep

|F (ti)− F ∗ (ti)| ,subject to pi,lower 6 pi 6 pi,upper

(D.1)

where t is a vector representing time, F (t) is a vector-valued function which is theactual step response of the compensated plants, F ∗ (t) is the desirable responseof the compensated plants, p = [p1, p2, p3]

T is a vector of the PID controllerparameters (for instance, p1 = Kp, p2 = Ti, and p3 = Td), pi,lower and pi,upperdenote the lower and upper bounds of the admissible parameter space. F ∗ (t)is specified by the engineer and incorporates the desired characteristics of theresponse, i.e., the overshoot, rise, steady-state, and settling time. To performthe optimization task, the MATLAB Optimization Toolbox was used [18]. Letus consider an example of optimization of a convex function. Fig.D.1 shows thefunction to be minimized. Each point on the surface represents the absolute valueof the difference between the desired response and the response produced by thecompensated plant. Changing parameters p1 and p2, the optimization procedureresults in the minimum of the function, provided the function is convex. If thefunction is not convex, there is a risk to hit a local minimum; however, havingselected a good initial guess, optimization is likely to lead to an acceptablesolution. In this case study the Genetic Algorithm [35] was used which offereda good initial guess and substantially reduced the time needed for obtaining thefinal set of the PID parameters.

153

154 Appendix D. PID Controller Tuning

05

1015

2025

0

5

10

15

20

250

1

2

3

4

5

6

7

8

Parameter P1

Optimization of a convex function

Parameter P2

Figure D.1: Optimization of a convex function

Table D.1: Main data for the case study

xd, Ω x′d, Ω T ′do H, s.

G1 12.1895 2.0947 4.48 2.3G2 6.7837 1.1658 4.6 1.65

D.1 Main data for the Gruvon case study

Table D.1 lists the main data for the system studied in this paper. The impedanceconnecting G1 and G3 is equal z = x = 0.1 Ω. The active power PG1,0 = 13MW, PG3,0 = 18 MW. The total load of LD1 and LD3 equals 19 MW. Theterminal voltage of both generators is 10.5 kV. The power exported to NETbefore disconnection of CLD is 12 MW. All power losses are neglected.

Appendix E

Power System Identification

E.1 Introduction

This appendix reports on the measurements which were performed on May 15th

between 14.30 and 17.00 hours at the paper mill “Billerud” located in Grums,Sweden. A one-line diagram of the electrical network of “Billerud” is depictedin Fig. 8.1 on page 109.

The objective of the study was to acquire measurements which could be usedfor obtaining linear models of the local generators and some prioritized load.

To the best knowledge of the author, accurate linear dynamic models of thegenerators and load are not yet available. It is therefore important to obtainthese models in order to perform system analysis and enable the analyst todesign auxiliary controllers such as, for example, power system stabilizers.

Depending on the nature of the study, linear or nonlinear analysis can becarried out. While nonlinear analysis has proven to be essential in stabilitystudies, it has not as yet received much attention from practicing engineers.Furthermore, the complexity of nonlinear analysis and lack of generic results,e.g., in the form of unified general theorems, are often serious obstacles to usingnonlinear theory.

On the contrary, linear analysis has proven to be an invaluable tool possessinggreat flexibility and a number of generic results [46]. For instance, if stable, alinear time-invariant system is globally exponentially stable everywhere in thestate space. Also, controller design techniques are very lucid, e.g., using µ-analysis the engineer has “knobs” for simultaneous control of both stability andperformance while explicitly specifying main sources of system’s uncertainty.Therefore, if the use of linear models is justified and the analyst is enabled toapply the rich arsenal of tools offered by linear analysis, an LTI model of theprocess of interest should be obtained.

As was discussed in Chapter 4, linear time-invariant models can be obtainedin several ways, one of them involves the so-called linearization of nonlinear

155

156 Appendix E. Power System Identification

Table E.1: Data for system identification

t 0 0.5 1 1.5 2

x(t)× 100 0 0.39346 0.63211 0.77687 0.86466u(t)× 100 1 1 1 1 1

equations. That is, normally LTI models are derived from nonlinear differential-algebraic equations describing the physical nature of the system studied. Thismethod is exemplified below.

Example E.1. Consider the nonlinear affine system

x(t) = − sin(x(t)) + u(t). (E.1)

Expanding the sin function in the Taylor series sinx = x− (1/2)x2 +O(x3) andneglecting all nonlinear terms, one obtains a linearized version of the originalsystem:

∆x = −1 ·∆x+ 1 · u = −a1∆x+ b1u. (E.2)

Often for conciseness the symbol ∆ is omitted but small deviations from theequilibrium are assumed. 2

When linearization is not a feasible option because of complexity of the sys-tem or uncertain system’s parameters, identification (curve fitting) can be con-sidered as a sound alternative to obtaining LTI’s of the system. That is, thesystem being studied is subjected to some perturbation and then the relation-ships between the input and output are analyzed in order to extract an LTImodel of the system. That is, in this case field measurements are used to obtainlinear models of the system.

Example E.2. Consider now a different scenario: one has only a data set consist-ing of 5 points1 shown in Table E.1. For simplicity, suppose that the structure ofthe model—first order DE—is known, c.f. equation (E.2). The only unknowns arethe parameters of the model. The task is to develop a linear model based on thedata available. Let us assume zero initial conditions and apply the ARX methodfor extracting a linear model corresponding to the data given by Table E.1. Inthis particular case, there is a closed form solution to the identification task.This solution is given by:

θ =

[a1b1

]=

[1

N

N∑

t=1

φ(t)φT (t)

]−11

N

N∑

t=1

φ(t)x(t), (E.3)

where the vectors φ(t) are defined as φT (t) = [−x(t− 1), u(t− 1)]. Solving equa-

tion (E.3), one finds the coefficients [a1, b1]T = [−0.60654, 0.39346]/100. In the

1The data were obtained from a numerical simulation of Eq. E.1. The system was subjectedto small perturbations in the control input u in order to get a response which could be usedfor system identification, see Table E.1.

E.1. Introduction 157

continuous time domain, this almost exactly corresponds to equation (E.2), i.e.a1 = −1, b1 = 1. Thus, the model obtained with the help of system identificationprovided an excellent approximation of the original nonlinear model. 2

The main purpose of Example E.2 was to illustrate the concept of systemidentification. However, the identification method mentioned above (ARX) willnot be applied to the identification of actual systems in this chapter. The mainreason for rejecting ARX is its inability to assist in determining the order of themodels to be identified and the potential risk to hit a local minimum when asearch algorithm is used for numerical minimization of the loss function [36].

To overcome the technical difficulties with ARX, a subspace system identifi-cation technique is employed for identification of the models in this Appendix.

E.1.1 Technical details

The field tests, which are reported here, consisted of 4 major experiments, aimedat identification of the following power system components:

1 Turbine and governor of G1.

2 Turbine and governor of G3.

3 Voltage regulator and exciter of G1.

4 Voltage regulator and exciter of G3.

In all tests and for all signals, the sampling frequency was constant and equal to2 kHz. Owing to the relatively high sampling rate, it was possible to measurethe instantaneous values of quantities of interest. The following signals weremeasured:

1 Time in seconds.

2 Current of T05 in Amperes.

3 Current of G1 in Amperes.

4 Field voltage (DC) of G3 in Volts.

5 Voltage of the system in kV.

6 Field voltage (DC) of G1 in Volts

7 Current of G3 in Amperes.

8 Rotor angular frequency of G1 in revolutions per minute (rpm).

9 Rotor angular frequency of G3 in rpm.

10 Current flowing through the cable connecting the buses S01A10 and S02A10in Amperes.


When active (P) or reactive (Q) powers were needed instead of currents, thewell-known relations

P = V × I × cosφ (E.4)

Q = V × I × sinφ (E.5)

were used, where φ was the angle between the voltage and current. In thisAppendix, the term “rms” is defined as

S(t)rms∆=

√1

T

∫ t0+T

t0

S(t)2dt, (E.6)

where S(t) is an arbitrary ac signal (voltage or current), t0 is an initial time,and T is one or a half of the period of S(t), i.e., 20 or 10 ms.

In some tests, several (slow-varying) signals were re-sampled off-line at alower sampling rate of 1000 Hz. Thus, no information loss was encountered, butsignal processing was significantly faster.

E.1.2 What models can be obtained?

Before performing system identification, the fundamental question to be askedis “what kind of models can be obtained from these measured data?” Implicitlythis question has already been answered–the use of subspace system identificationhas predetermined the type of the dynamical models which could be obtained–linear, time-invariant (LTI) models. Some of the advantages of LTI models havebeen mentioned above; the reader is advised to refer to any book on moderncontrol theory2 for more details on this fascinating branch of modern engineering.Although the usage of LTI’s may seem oversimplified, LTI models are found to beappropriate for the purposes pursued in the present research, i.e., fixed-parameterLTI controller design and analysis.

As was already written, when the use of linearization is questionable, nonlin-ear analysis is to be applied. Thus, it is essential to have nonlinear models of thesystem components such as generators or loads. In addition, it is informative tovalidate the performance of the linear models by running a computer simulationof a detailed nonlinear model.

In the remainder of the Appendix we therefore first present the main resultsand discusses the most interesting findings from linear system identification andthen–their nonlinear conterparts.

2See for example [66].

E.2. Preprocessing of the data and qualitative analysis 159

40 40.5 41 41.5 42 42.5 43 43.5 442400

2600

2800

3000

3200

3400

3600

3800

Rotor mechanical speed of ω1 and ω

3, rpm

Mec

hani

cal s

peed

of ω

1 and

ω3, r

pm

Time, sec.

ω1

ω3

Figure E.1: Rotor mechanical speed of generators G1 and G3

E.2 Preprocessing of the data and qualitativeanalysis

In this section we briefly overview the data measured at the field tests and performqualitative and quantitative analyzes. When relevant, comparisons with existingstandards/typical values are done.

E.2.1 Rotor angular speed of G1 and G3

First, the angular velocity ω of both generators is examined. Since each of thesynchronous generators G1 and G3 has one pair of poles, it is expected thatω = f · 60 · 1 = 50 · 60 = 3000 revolutions per minute (rpm). Figure E.1shows a fragment of the measured signals ω1 and ω3 in rpm in steady state(no perturbation applied to either generator). As can be noted in Figure E.1,the mechanical speed (angular frequency) of the generators differs from 3000rpm. For G1 and G3 the mean angular frequencies are 2855.7 and 2934.7 rpm,respectively. The maximum (minimum) deviations from the nominal speed are+22.2(−19.3)% for G1 and −2(−3.2)% for G3. That is, on the average thespeed of the generators is below the synchronous speed. This observation is quiteinteresting since under the test the generators were not run in the islanded mode,excluding the possibility of asynchronous speed. Moreover, while performing themeasurement, it was noticed by the author3 that the fundamental frequency was50 Hz, i.e., it was equal to the nominal frequency. As is stated in [16], “95% of

3There was another analog meter mounted in the control board and connected to the grid.


the 10 second averages shall not be outside the range 49.5−50.5 Hz”. That is, theallowable frequency deviation should not exceed ∆f = ±0.5/50 · 100 = ±1%. Inthe present case, the average frequency deviations are −4.7186% and −2.1697%,respectively. Thus, if the measurement is correct, being operated in the islandedmode, the system frequency deviations will exceed the value specified by thestandard at least 2 times.

One can also note that the inaccuracy of the measurement of G1 is muchgreater than that of G3. More precisely, the standard deviation of the measure-ment of ω1 is equal to 196.6851, while for ω3 it is only 4.2412.

The deviation from the synchronous speed can be partially explained by asystematic error of the measuring device or a scaling factor error. The latteris however less likely. To investigate the nature of this discrepancy, a discreteFourier transform (DFT) of the current of the generator G1 was calculated.The spectral contents of the current, however, indicated no deviation from thefundamental frequency, see Fig. E.6, but the presence of an unusually largeDC component in the current. The inspection of the spectrum of the currentalso reveals an interesting observation: the 2d harmonic magnitude is negligiblysmall. This observation lands in certain controversy, since such a high DC offsetshould have caused saturation of the transformers and subsequent given riseto the second harmonic. It is noteworthy mentioning that a sampling rate of2000 Hz should reliably represent all harmonics up to the 20th. As there wasalmost no second harmonic, we should conclude that either the DC offset wasdue to a measurement error or the DC current produced by the generator wasabsorbed by the nearest load since the DC current of the cable has negligiblysmall magnitude, see Fig. E.5.

Leaving the current for now, the main conclusion to be drawn is: As the fre-quency of power is an essential power quality characteristic, this question shouldbe resolved by more accurate field measurements. Since the measuring deviceused in this test is most likely to produce erroneous results, another meter hasto be used for the measurement. It should be emphasized that it is important toclarify this uncertainty with the frequency as ω’s are used by the governors of G1and G3 for speed regulation. Moreover, if new pieces of equipment e.g., powersystem stabilizers (PSS) are to be installed, the precise and reliable frequencymeasurements have to be readily available.

E.2.2 System voltage

In this test, both generators were set in the constant reactive power control mode(QG1, QG3 = const). This means that the field voltages were held constant dur-ing the test. Since the voltage regulators had been disabled, the system voltageshould not have varied very much. Fig. E.2 depicts the rms (calculated with awindow of 1 cycle) system voltage and the spectral contents of the measured mo-mentary line-to-line voltage. Indeed, as one can see from the figure, the voltagedoes not vary significantly and the voltage spectrum contains the 5th and 7th

harmonics due to some load consisting of power electronics equipment.


10 20 30 40 50 60 70 8010

10.2

10.4

10.6

10.8

Time, seconds

VS

yste

m, R

MS

0 50 100 150 200 250 300 350 400 450 500

10−4

10−2

100

Frequency, Hz

Spe

ctru

m o

f VS

yste

m

Figure E.2: System voltage at test 1. The upper subplot shows the rms voltagein kV and the lower subplot depicts the spectrum of this voltage

E.2.3 System currents

The system currents are shown in Fig. E.3. It is instructive to examine thecurrents by a closer consideration of Fig. E.3. As the figure indicates, the gen-erators’ currents are 180 degrees out of phase as compared to the currents ofT05 and the cable connecting G1 and G3. Since physically this is unrealistic,we suggest that while connecting the measuring circuits, in some cases the po-larity of the clip-on ampermeters was reversed. To find out which ones werewrongfully connected, the power flow of the whole system has to be examined.Another immediate conclusion: the current of G3 is exceedingly large. Again,the power flow analysis can shed some light on this issue. Firstly, let us usethe measurements to roughly estimate the output power of G3. We proceed asfollows: S =

√3 · V · I =

√3 · 10.4 · 18/

√2 = 229.3 MVA. That is, the output

power is approximately 8 times greater than the rated power (28.75 MVA) ofG3. Before proceeding with the power flow study, we state the following facts:

• The system voltage was measured correctly. This is rather a postulate thana fact, but we needed it for obvious reasons. The other facts are based onlaws of physics.

• The loads of the system were consuming both active (P ) and reactive (Q)power during all tests. Here we implicitly assume that the load is mainlycomprised of asynchronous machines and power inverters which in steadystate always consume P and Q.

• Both generators were producing active power during all tests.


26.2419 26.2519 26.2619 26.2719 26.2819

−1.5

−1

−0.5

0

0.5

1

1.5

x 104 System currents, A

Time, seconds

Cur

rent

, A

IT05

IG1

IG3

ICable

Figure E.3: System currents at test 1

• The power produced by each of the generators could not exceed their ratedpower, which are 16 and 28.75 MVA, respectively.

A rudimentary analysis (not shown here) reveals that the aforementioned factshold true when:

1. The sign of the data arrays corresponding to IT05 and IG1 is reversed, i.e.,multiplied by −1.

2. The numerical values of the data corresponding to the current of G3 arereduced by a factor of 10.

3. The load closest to G3 consumes 16.7 MVA, which is 2.7 MVA greater thanwas expected.

4. The load closest to G1 consumes 5 MVA, as expected.

These discrepancies are corrected for by performing corrections 1 and 2 for alltests, since the measurement set-up was unchanged during all tests.

Fig. E.4 shows the active power produced by G1 and G3, the power importedfrom the net and transferred via the cable. Since this test was concerned withthe identification of the dynamics of turbine and governor of G1, the governor ofG3 was disabled and the output power remains constant, which is clearly seenin Fig. E.4.

The current which was carried by the cable connecting the generators G1and G3 is now considered in more detail. While performing this measurement, itwas noticed by the author of the manuscript that the current ICable waveshape


0 10 20 30 40 50 60 70 80 90−1

−0.5

0

0.5

1

1.5x 10

7

Time, seconds

Sys

tem

pow

ers,

W

PG3

P

T05

PCable

PG1

Figure E.4: System active powers at test 1

was severely distorted by harmonics. The total harmonic distortion (THD) wasapproximately 17%. We now examine the spectrum of ICable, see Fig. E.5.

As Fig. E.5 indicates, only the 5th harmonic has now noticeable magnitudein spite of the fact that at a sampling frequency of 1000 Hz, there must presentall harmonics up to the 10th. Hence, there took place an unexplained effectof “disappearing” of the 7th and some other harmonics. Alternatively, the dataacquisition device that was used –“Argus” made an erroneous DFT of the currentof the cable. This possibility, however, is very unlikely, since the waveshape of thecurrent was distorted to such a degree that the distortion was visually noticeable.In the measurements, which the author analyzed off-line, the current waveshapereveals almost no distortion.

Now the currents of the generators G1 and G3 are examined. The waveshapeof the currents has already been shown in Fig. E.3; however, the scaling has hid-den an interesting observation–the presence of large DC component, see Fig. E.6and Fig. E.7. On the other hand, the DC components of the cable current andthe current of T05 were negligibly small. This suggests that if no measuring errorwas made, the DC components were generated locally at the busbars the gener-ators are connected to. To obtain a numerical estimate of the DC component,we may calculate the mean value or spectrum of the current. According to thespectrum of IG1, the DC components of the currents were about 10% and 4.14%of the fundamental component, respectively. This is worth comparing with atypical value of 0.1% cited in [22].


29.32 29.34 29.36 29.38 29.4 29.42 29.44 29.46 29.48

−300

−200

−100

0

100

200

300

Time, seconds

Cur

rent

I Cab

le, A

0 50 100 150 200 250 300 350 400 450 500

10−5

100

Frequency, Hz

Spe

ctru

m o

f IC

able

Figure E.5: Current ICable at test 1. Observe the insignificant harmonicpollution of the signal

0 50 100 150

10−4

10−2

100

Frequency, Hz

Spe

ctru

m o

f IG

1

18 18.02 18.04 18.06 18.08 18.1−600

−400

−200

0

200

400

600

800

Time, seconds

Cur

rent

I G1, A

Figure E.6: Test 1: Current IG1, A. Note an asymmetry of the positive andnegative peaks of IG1 in the upper subplot


18 18.01 18.02 18.03 18.04 18.05−2000

−1500

−1000

−500

0

500

1000

1500

2000

Time, sec.

Cur

rent

I G3, A

0 50 100 150 200 250 300 350 400 450 500

10−5

100

Frequency, Hz

Spe

ctru

m o

f IG

3

Figure E.7: Test 1: Current IG3, A. Note an asymmetry of the positive andnegative peaks of IG3 in the upper subplot

E.2.4 Field voltages of G1 and G3

Since the exact type of the exciters installed on the local generators was un-known, it has been assumed that the exciters are of the type DC1A (commonlyknown as ‘Type I excitation system’) [2]. Fig. E.8 shows the field voltages of thegenerators G1 and G3. It is should be noticed that despite the inductive charac-teristics of the field winding, the voltage in addition to the DC components alsocontains a lot of harmonics having large magnitudes. Inspection of Fig. E.9 re-veals that there are some inter-harmonics in the spectrum which is quite unusual.Also, it is seen in Fig. E.8 that at certain time instances the DC voltage acrossthe field winding reverses the polarity. The author does not have a reasonableexplanation for this phenomenon. If there were noncontinuously acting regula-tors such as the type DC3A, discontinuities could have been induced an EMFwhich was greater in magnitude than the DC voltage applied to the winding ofthe exciter. The field voltage of G1 never had negative polarity, though Efd,1was considerably smaller in magnitude than Efd,3. Since the excitation systemof the generator is one of the most important pieces of equipment, which playsthe central role in supplementary stabilizing controller design, more extensiveand precise measurements are required.

Now one immediate conclusion can be drawn: the preliminary analysis per-formed indicates that proceeding with system identification on the available datamay result in inaccurate models, which is unacceptable since having accuratemodels of the exciters and governors is essential for both analysis and successful


0 5 10 15 20 25 30−50

0

50

100

150

200

Time, seconds

Efd

, G1, E

fd, G

3, V

29.95 30−50

50

150

Figure E.8: Test 1: Field voltages of G1 and G3 in Volts. The field voltage ofG3 is shown in blue line, while the green line depicts the Vfd of G1. A smallportion t ∈ [29.95, 30] of the signal is enlarged and overlaid the upper right

corner of the plot.

0 100 200 300 400 500

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Frequency, Hz

Spec

tra

of E

fd,1

and

Efd

,3

Figure E.9: Spectra of Efd,1 and Efd,3. The blue and green lines denote thespectra of Efd,3 and Efd,1, respectively

E.3. Load identification 167

controller design. Therefore, new field measurements must be recorded payingattention to

• Harmonic contents of all signals

• Measuring quantities in all three phases

• DC current offsets

• Field voltages of the synchronous generators

• Precision of the measurements of mechanical speed of the generators’ shafts.

Our experience shows that prior to recording the signals a series of short trialrecords should be made and analyzed off-line. This would give the engineerthe right message about the expected quality of quantities of interest and helphim to spot the measuring equipment which is likely to introduce erroneousmeasurements.

Despite the fact that identification of parameters of the generators is unfea-sible, load identification is still possible, as the load can be identified based onlyon the voltage and current measurements which are reliable and accurate.

E.3 Load identification

In this section two models of the electrical load of the paper mill are estimated.There are two different possibilities: to represent the load as a black box andidentify it or to pre-select a structure of the load based on physical insights andthen estimate the parameters. In our case there exists a reasonable model of theload, so the task of this section reduces to estimating the unknown parameters ofthe model. The main results of the estimation are presented in this section.

In the field tests that have been conducted, neither the load current nor thepower were directly measured. Instead, the load current is calculated off-line asfollows:

ILoad = IG1 + IG3 + IT05. (E.7)

Although in principle it was possible to measure each of the loads connected tothe busbars S01A10 and S02A10, for simplicity the loads LD3 and LD4 weretreated as one aggregate load, see Fig. 8.1.

There exist different models for power system loads. The choice of a partic-ular model depends mainly on the knowledge of the physical characteristics ofthe load in question. The three common static models are the so-called “con-stant admittance, constant current, and constant power” models [3]. In thisstudy, an attempt was made to identify a “dynamical” model of the load, sincea large fraction of the load consists of asynchronous motors and power electronicequipment.


Let us consider the generic model of power system load described in [3].

Tpx = P0

[VV0

]Nps

− PdPd = x+ P0

[VV0

]Npt(E.8)

where Pd is the active power demand of the load, P0 and V0 stand for the nominalpower and voltage, respectively. The parameter Tp denotes the time constant ofthe extra state variable–x. The exponents Nps and Npt are the unknown steadystate and transient voltage indices to be estimated. Normally, these indices havethe following lower and upper bounds:

0 6 Nps 6 3

0.5 6 Npt 6 2.5(E.9)

Note E.3. Although in the equations above we only estimate the parameters thatdescribe the dynamical behavior of the active power, equivalent expressions arevalid for the reactive power.

Obviously, the model given by equation (E.8) is nonlinear unless Nps = Npt =1. Therefore, there are two different ways of obtaining a model based on equa-tion (E.8):

1. Apply a nonlinear estimation technique in order to identify the unknowntime constant Tp and indices Nps and Npt.

2. Linearize the model (E.8) and apply a linear identification to find theunknowns.

The latter alternative is much easier to implement. Moreover, when workingwith a linear or linearized model, even if the nonlinear model of the load wereavailable, it should be linearized before incorporating into the overall systemwhich is linear.

Now the main goal of the Appendix is formulated

Given measurements of the system voltage (input) and currents and the loadmodel (E.8), estimate the model parameters Nps, Npt, and Tp. The outputvariable was not measured and has to be synthesized based on the signalsavailable.

In spite of the fact that analysis and controller design are performed on linearmodels, it is a good practice to validate the performance of the controlled systemon the detailed nonlinear model. Therefore, it having an accurate nonlinearmodel is also important. We will return to nonlinear identification later. First,an attempt is made to obtain a linear model of the load.


E.3.1 Identification of a linear model

Linearizing the original model (E.8) and introducing the notation

θT = [Nps, Npt, Tp],

we thus obtain a linear equivalent of (E.8):

x = − 1Tpx+ 1

TpP0

V0(Nps −Npt)V = A(θ)x+B(θ)V

Pd = x+ P0

V0NptV = Cx+D(θ)V.

(E.10)

Note that in the model (E.10) the matrix C is fixed and known, i.e., it is not afunction of θ.

Several data sets are available for the identification of the load. The setdenoted G3_SP.txt is the most reasonable choice since in this test the systemvoltage was subjected to the largest perturbations. The large voltage variationscould better excite the dynamics of the load. The data set contains measure-ments of all the variables for the time interval t ∈ [0, 86] sec. Since the signalscorresponding to the load power were not recorded, these have to be calculated,see Section E.1.1 for some detail on the calculation of power.

A Matlab routine was written based on the theory presented in [52] toperform the identification work. The routine yielded the following values of theparameters of interest:

Nps = 0.9611 (E.11)

Npt = 2.2708 (E.12)

Tp = 0.3838 sec. (E.13)

It is interesting to note that the indices are within the bounds given by (E.9).A comparison of the response of the estimated model and measured signal areshown in Fig. E.10. As the figure indicates, there is a reasonable agreementbetween the model’s response and the measurement.

E.3.2 Identification of a nonlinear model

We now apply a nonlinear identification technique in order to estimate the un-known parameters in equation (E.8). Before proceeding with technical details ofthe identification algorithm, the “general plan of attack” is outlined below.

1 The structure of the system (E.8) suggests the use of a Nonlinear Final ImpulseResponse (NFIR) algorithm.

2 The continuous nonlinear differential equation (E.8) should be discretized.

3 The nonlinear part of the system should be expanded in a Taylor series andthe (unknown) coefficients of the expansion be estimated.


10 20 30 40 50 60 70 8010

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6

11.8

12

Time, sec.

Measured and simulated model output

Measured load powerFitted load power

Figure E.10: Comparison of the measured active power and that given bymodel (E.10)

4 The nonlinear part is next reconstructed using the previously found coefficientsof the Taylor series.

Now the steps 1− 4 are explained in more detail.

First, the measured voltage is normalized by dividing it by the nominal volt-age: V = V/V0. As the voltage normally does not deviate significantly from thenominal value4, V is approximately 1. Next, the power functions V Nps and V Npt

are expanded in a Taylor series

V Nps ≈ 1 + α1V + α2V2 + . . .+ αnV

n (E.14)

V Npt ≈ 1 + β1V + β2V2 + . . .+ βmV

m, (E.15)

where the indices n andm are pre-selected in order to attain the desired precisionof approximation. The coefficients αi and βi are unknown and have to be found.

The original model of the load (E.8) can be discretized as shown below:

Tpx(k + 1) = P0

[1 + α1V (k) + α2V

2(k) + . . .+ αnVn(k)

]− Pd(k)(E.16)

Pd(k) = x(k) + P0

[1 + β1V (k) + β2V

2(k) + . . .+ βnVn(k)

]. (E.17)

4See Table 3.1 on page 21.


Assuming for simplicity that n = m, a linear regression can now be formed:

Pd(k) = γ1V (k − 1) + γ2V2(k − 1) + . . .+ γnV

n(k − 1) (E.18)

+ P0

[1 + β1V (k) + β2V

2(k) + . . .+ βnVn(k)

], (E.19)

where the new coefficients γi are defined as γi = (αi − βi)P0/Tp. Finally, intro-ducing the parameter vector

θ =[γ1, γ2, . . . , γn β1, β2, . . . , βn

]T

and the modified voltage measurement vector

ϕ(k) =[V (k − 1), V 2(k − 1), . . . , V n(k − 1) V (k), V 2(k), . . . , V n(k)

]

we obtain

Pd(k)− P0 = ϕ(k)θ, k = 2, 3, . . . ` (E.20)

or in a more compact form

Pd,0 = Φθ (E.21)

Φ =

V (1), V 2(1), . . . , V n(1) V (2), V 2(2), . . . , V n(2)

V (2), V 2(2), . . . , V n(2) V (3), V 2(3), . . . , V n(3)...

...

V (`− 1), V 2(`− 1), . . . , V n(`− 1) V (`), V 2(`), . . . , V n(`)

(E.22)

where ` is the total number of data samples; Pd,0 is numerically equal to themeasured vector Pd reduced by the nominal power P0: Pd,0 = Pd − P0.

The last expression is clearly linear in the parameter vector θ which can befound as

θ = Φ†Pd,0. (E.23)

In equation (E.23), the symbol † denotes the pseudo-inverse, for more detail onpseudo-inverses refer to page 54.

Having found the parameters γi and βi, one can recover the unknown param-eters Tp and αi solving the set of equations

P0Tpαi −

P0Tpβi = γi (E.24)

(first Tp is retrieved and then αi) in a least squares sense.


Finally, the least squares technique is applied one more time to reconstructthe unknown indices Nps and Npt. This is done by solving the set of overdeter-mined equations

Nps = (lnV (k))−1 lnn∑

i=0

αiVi(k), α0 ≡ 1 (E.25)

Npt = (lnV (k))−1 lnn∑

i=0

βiVi(k), β0 ≡ 1, (E.26)

In the equations above, the index k assumes values k = 1, 2, . . . , `. Again, solvingthe overdetermined system of equations, the use of singular value decompositionis advised, due to its excellent numerical properties.

Note E.4. The presented identification method is solely based on the assumptionof existence of the Taylor series expansion in (E.14)–(E.15), i.e., the assumptionthat each of the functions V Nps and V Npt can be approximated by a Taylor series.This assumption is indeed valid since it can be easily shown that the functions areregular at V = V0, moreover the interval of convergence is V0 − 1 < V < V0 + 1,which assures the convergence of the series in almost all practical cases.

E.3.3 Conclusions

A series of field measurements was taken at a real-life paper mill. An attempt wasmade to identify models of the synchronous generators, exciters, governors, andload of the paper mill. Unfortunately, due to some discrepancy in the measure-ments, reliable and accurate identification of the parameters of the generatorsand associated equipment was unfeasible. However, here a felicitous quotationshould be cited “In all human affairs there are efforts, and there are results,and the strength of the effort is the measure of the result” [12]. That is, in spiteof the infeasibility to obtain accurate models of some equipment of the mill, anumber of potential problems were pinpointed, e.g., the inaccurate measurementof the mechanical speed of the rotor shaft (generator G1). Also, useful practicalexperience was gained, which would prevent such situations from occurring inthe future. Moreover, the analysis that was carried out in this Appendix mightbe informative for the personnel of the paper mill. Of course, there are openquestions which require further investigation.

The main achievement reported in this Appendix is the estimation of theparameters of the load. Linear and nonlinear models of the load are developedthat could be used in both small-signal analysis and stability studies. The pa-rameters of the linear model are estimated based on the well-known technique–minimization of Prediction Error. A Taylor series expansion lies in the basis ofthe nonlinear estimation. That is, using a Taylor series the original nonlinearfunction is replaced by the truncated series and then the problem is translatedto a linear parameter estimation. The nonlinear model of the load may find an


application in stability analysis or detailed computer-aided simulations aimed atvalidating designs that were based on simplified models, including linear ones.

Bibliography

[1] EMTDC V3. User’s manual. Manitoba HVDC Research Centre, Winnipeg,Manitoba, 1988.

[2] IEEE recommended practice for excitation system models for power systemstability studies. IEEE Std 421.5-1992, August 1992.

[3] Analysis and Control of Power System Oscillations. Task Force 07 of Advi-sory Group 01 of Study Committee 38. Technical report, CIGRE, December1996.

[4] DSI tools, User’s manual. January 1997.

[5] Technical Report 187, Electra, December 1999.

[6] Advanced Power Quality Analysis. University of Wisconsin Madison, Lec-ture Notes, 1999.

[7] Matlab – The Language of Technical Computing. User’s guide. The Math-Works, Inc., 1999.

[8] Power Quality in Electric Power System, Course material. ABB T&D Uni-versity, Lecture Notes, Ludvika, Sweden, 1999.

[9] Power Quality Solutions. A brochure. ABB Power T&D CompanyPower Systems Division, 1999. This document is downloadable fromwww.abb.com/usa/.

[10] Power System Blockset. User’s Guide. TEQSIM International Inc. and TheMathWorks, Inc., 1999.

[11] Using Simulink. The MathWorks, Inc., 1999.

[12] James Allen. As You Thinketh. Putnam Publishing Group, 1959.

[13] Gary J. Balas, John C. Doyle, Keith Glover, Andy Packard, and Roy Smith.“µ-Analysis and Synthesis Toolbox. User’s Guide”. 1998.

175

176 Bibliography

[14] David Berlinski. On Systems Analysis: An Essay Concerning the Limita-tions of Some Mathematical Methods in the Social, Political, and BiologicalSciences. The MIT Press, Cambridge, Massachusetts and London, England,1978.

[15] V. M. Blok, G. K. Obushev, L. B. Paperno, and S. A. Guseva. Guidelines toelectrical engineers for project work designing. High school, Moscow, 1990.In Russian.

[16] M. H. J. Bollen. Understanding Power Quality Problems. Voltage Sags ansInterruptions. IEEE Press, New York, 2000.

[17] Anne-Marie Borbely and Jan F. Kreider. Distributed Generation: ThePower Paradigm for the New Millennium. CRC Press, 2001.

[18] T. Coleman, M. A. Branch, and A. Grace. Optimization Toolbox User’sGuide. 1999.

[19] D. A. Pierre and D. J. Trudnowski and J. R. Smith and T. A. Short. AnApplication of Prony Methods in PSS Design for Multumachine Systems.WN 117-2 PWRS, IEEE/PES Winter Power Meeting, February 1990.

[20] J. C. Das and J. Casey. Effects of excitation controls on operation of syn-chronous motors. Industrial & Commercial Power Systems Technical Con-ference, 1999.

[21] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis. State-spacesolutions to standard H2 and H∞ control problems. IEEE Transactions onAutomatic Control, 34(8):831–847, August 1989.

[22] R. C. Dugan, M. F. McGranaghan, and H. W. Beaty. Electrical PowerSystems Quality. McGraw Hill, New York, 1996.

[23] A. Ekstrom. High Power Electronics. HVDC and SVC. Course material.The Royal Institute of Technology, Stockholm, 1990.

[24] M.M. Farsangi, Y.H. Song, and Y.Z. Sun. Supplementary control designof svc and statcom using H∞ optimal robust control. Proceedings of theInternational Conference on Electric Utility Deregulation and Restructuringand Power Technologies, 2000.

[25] W. Favoreel, B. De Moor, and P. Van Overschee. Model-free subspace-basedLQG-design. Proceedings of the American Control Conference, San Diego,California, pages 3372–3376, June 1999.

[26] M. Feder and E. Weinstein. Parameter Estimation of Superimposed SignalsUsing the EM Algorithm. IEEE Transactions on Acoustics, Speech, andSignal Processing, 36(4):477–489, April 1988.

Bibliography 177

[27] A. Forsgren, P. E. Gill, and M. H. Wright. Numerical Nonlinear Program-ming. Class Notes. Dept. of Mathematics, KTH., Stockholm, Sweden, 1998.

[28] B. A. Francis. Course in H∞ Control Theory. Springer-Verlag, Berlin, 1987.

[29] M. Ghandari. Control Lyapunov Functions: A Control Strategy for Dampingof Power Oscillations in Large Power Systems. PhD thesis, Royal Instituteof Technology, November 2000.

[30] G. H. Golub and C. F. van Loan. Matrix Computations. The Johns HopkinsUniversity Press, London, 3rd edition, 1996.

[31] E. Handschin. Real time control of electric power systems. Proceedingsof the symposium on real-time control of electric power systems, ElsevierPublishing Company, 1971.

[32] J. F. Hauer, C. J. Demeure, and L. L. Schaft. Initial Results in PronyAnalysis of Power System Response Signals. IEEE Transactions on PowerSystems, 5(1):80–89, February 1990.

[33] J. W. Helton. Extending H∞ control to nonlinear systems: control of non-linear systems to achieve performance objectives. Philadelphia, PA: Societyfor Industrial and Applied Mathematics, 1999.

[34] J. W. Helton and O. Merino. Classical Control Using H∞ Methods Theory.Optimization and Design. Society for Industrial and Applied Mathematics,1998.

[35] C. R. Houck, J. A. Joines, and M. G. Kay. A Genetic Algorithm for FunctionOptimization: A Matlab Implementation. 1996.

[36] Jinglu Hu, Kotaro Hirasawa, and Kousuke Kumamaru. A homotopy ap-proach to improving PEM identification of ARMAX models. Automatica,37(9):1323–1334, September 2001.

[37] A. Isidori. Nonlinear Control Systems. Springer, New York, 3rd edition,1995.

[38] L. Jones. On zero dynamics and robust control of large AC and DC powersystems. PhD thesis, Royal Institute of Technology, November 1999.

[39] T. Kailath. Linear Systems. Prentice-Hall, Inc., 1980.

[40] N. Kakimoto, Y. Ohsawa, and M. Hayashi. Transient Stability Analy-sis of Multimachine Power System with Flux decays via Lyapunov’s Di-rect Method. IEEE Transactions on Power Apparatus and Systems, PAS-99(5):1819–1828, Sept/Oct 1980.

[41] S. M. Kay. Fundamentals of Statistical Signal Processing. Estimation theory,volume 1. Prentice Hall, New Jersey, 1993.

178 Bibliography

[42] M. Klein, G. J. Rogers, and P. Kundur. A fundamental study od inter-areaoscillations in power systems. Transactions on Power Systems, 6(3):914 −−921, August 1991.

[43] V. Knyazkin. Power System Identification for Controller Design. Master’sthesis, The Royal Institute of Technology, B-EES-9904, Stockholm, January1999.

[44] V. Knyazkin and L. Soder. The use of coordinated control for mitigationof voltage sags caused by motor start. The proceedings of 9th InternationalConference on Harmonics and Quality of Power, III:804–809, October 2000.

[45] E. Kreyszig. Advanced Engineering Mathematics. John Wiley & Sons Ltd.,7th edition, 1993.

[46] P. Kundur. Power System Stability and Control. EPRI power system engi-neering series. McGraw-Hill, Inc., 1994.

[47] H. Kwakernaak. Robust Control and H∞ –Optimization–Tutorial Paper.Automatica, 29(2):255–273, 1993.

[48] H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. John Wiley& Sons, New York, 1972.

[49] Xianzhang Lei, E.N. Lerch, and D. Povh. Optimization and coordinationof damping controls for improving system dynamic performance. IEEETransactions on Power Systems, 16(3):473–480, August 2001.

[50] Chih-Ming Lin, V. Vittal, W. Kliemann, and A. A. Fouad. Investigation ofmodal interaction and its effects on control performance in stressed powersystems using normal forms of vector fields. Power Systems, IEEE Trans-actions on, 11(2):781–787, May 1996.

[51] F. Lin. Robust Control Design: An Optimal Control Approach. AFI Press,1997.

[52] L. Ljung. System Identification – Theory for the User. Prentice Hall, En-glewood Cliffs, N.J., 1987.

[53] L. Ljung. System Identification Toolbox. User’s Guide. The MathWorks,Inc., 1997.

[54] J. Machowski, J. W. Bialek, and J. R. Bumby. Power System Dynamicsand Stability. John Wiley & Sons, Chichester, 1997.

[55] J. Machowski, S. Robak, J. W. Bialek, J. R. Bumby, and N. Abi-Samra.Decentralized stability-enhancing control of synchronous generator. IEEETransactions on Power Systems, 15(4), November 2000.

Bibliography 179

[56] L. Marconi, G. Marro, and C. Melchiorri. A solution technique for almostperfect tracking of non-minimum-phase, discrete-time linear systems. In-ternational Journal of Control, 74(5):496–506, 2001.

[57] M. W. Marshall. Using series capacitors to mitigate voltage flicker prob-lems. Rural Electric Power Conference. Papers Presented at the 41st AnnualConference, pages B3 –1–5, 1997.

[58] M. F. McGranaghan, D. R. Mueller, and M. J. Samotyj. Voltage Sags inIndustrial Systems. IEEE Trans. Industrial Applications, 29(2):397–403,April 1993.

[59] A. S. Meghani and H. A. Latchman. H∞ vs. classical methods in the designof feedback control systems. IEEE Proceedings Southeastcon ′92, 1:59–62,1992.

[60] J.V. Milanovic. Tuning of svc stabiliser to compensate the influence of volt-age dependent loads. Proceedings of the 36th IEEE Conference on Decisionand Control, 3:2553–2558, 1997.

[61] J.V. Milanovic and I.A. Hiskens. Damping enhancement by robust tuningof svc controllers in the presence of load parameters uncertainty. IEEETransactions on Power Systems, 13(4):1298–1303, November 1998.

[62] I. J. Nagrath and D. P. Kothari. Modern Power System Analysis. TataMcGraw-Hill Company Limited, New Delhi, 1985.

[63] Yao nan Yu. Electric Power System Dynamics. Academic Press, New York,1983.

[64] T. T. Nguyen. Parametric harmonic analysis. IEE Proc.-Gener. Transm.Distr., 144(1):21–25, January 1997.

[65] Y.-X. Ni, V. Vittal, W. Kliemann, and A. A. Fouad. Nonlinear modalinteraction in HVDC/AC power systems with dc power modulation. PowerSystems, IEEE Transactions on, 11(4):2011–2017, November 1996.

[66] K. Ogata. Modern Control Engineering. Simon & Schuster, (Asia), PteLtd., Singapore, 1995.

[67] P. Van Overschee and B. De Moor. Subspace algorithms for the stochasticidentification problem. Proceedings of the 30th Conference on Decision andControl, Brighton, England, pages 1321–1326, Dec. 1991.

[68] M. A. Pai and R. W. Sauer. Power System Steady State and the Load FlowJacobian. IEEE Trans. On Power Systems, 5(4), November 1990.

[69] M.A. Pai. Energy Function Analysis for Power System Stability. KluwerAcademic Publishers, 1989.

180 Bibliography

[70] D. A. Pierre. On the Simultaneous Identification of Transfer Functionsand Initial Conditions. Proceedings: American Control Conference, pages1270–1274, June 1992.

[71] D. A. Pierre, D. J. Trudnowski, and J. F. Hauer. Identifying LinearReduced-Order Models for Systems with Arbitrary Initial Conditions us-ing Prony Signal Analysis. IEEE Trans. Automatic Control, 37(6):831–835,June 1992.

[72] G. Rogers. Power system oscillations. The Kluwer international series inengineering and computer science. Kluwer Academic Publishers, 2000.

[73] A. Sannino, M. G. Miller, and M. H. J. Bollen. Overview of Voltage SagMitigation. IEEE Winter Meeting, Singapore, 2000.

[74] P. W. Sauer and M. A. Pai. Power System Dynamics and Stability. PrenticeHall, New Jersey, 1998.

[75] J. Sjoberg, Q. Zhang, L. Ljung, A.Beneveniste, B. Delyon, P.-Y. Glorennec,H. Hjalmarsson, and A. Juditsky. Nonlinear Black-box Modeling in SystemIdentification: a Unified Overview. Automatica, 31(12):1691–1724, 1995.

[76] S. Skogestad and I. Postlethwaite. Multivariable Feedback Control. Analysisand Design. John Wiley & Sons Ltd., 1996.

[77] J. R. Smith, F. Fatehi, C.S. Woods, J.F. Hauer, and D. J. Trudnowski.Transfer function identification in power system applications. IEEE Trans.Power Systems, 8(3):1282–1290, August 1993.

[78] T. Soderstrom and P. Stoica. System Identification. Prentice Hall, Engle-wood Cliffs, N.J., 1989.

[79] E. D. Sontag. Mathematical control theory: deterministic finite dimensionalsystems. Springer, New York, 2d edition, 1998.

[80] D. Starer and A. Nehorai. Maximum likelihood estimation of exponentialsignals in noise using a Newton algorithm. Fourth Annual ASSP Workshopon Spectrum Estimation and Modeling, pages 240–245, 1988.

[81] D. J. Trudnowski. Order reduction of large-scale linear oscillatory systemmodels. IEEE Trans. Power System, 37(6):451–458, February 1994.

[82] D. J. Trudnowski, J. M. Johnson, and J. F. Hauer. SIMO system Identi-fication from Measured Ringdowns. Proceedings of the American ControlConference, pages 2968–2972, June 1998.

[83] D. J. Trudnowski, J. R. Smith, T. A. Short, and D. A. Pierre. An applicationof prony methods in pss design for multimachine systems. IEEE Trans. onPower Systems, 6(2):118–126, Feb. 1991.

Bibliography 181

[84] M. Vidyasagar. Nonlinear Systems Analysis. Electrical Engineering.Prentice-Hall, 1978.

[85] T. L. Vincent and W. J. Grantham. Nonlinear and Optimal Control Sys-tems. John Wiley & Sons, Jun 1997.

[86] A. Wikstrom. Reduction of Voltage Dips by Redesign of an Industrial Dis-tribution System Using Fast Switches and Current Limiting Devices. Dept.of Electrical Engineering, Royal Institute of Technology, Stockholm, 2001.Licentiate Thesis.

[87] A. Wikstrom, P. Bennich, and G. Andersson. Improvements of the Per-formance of an Industrial Power System by Use of Fast Switches. TheProceedings of the 31st Annual North American Power Symposium, pages183–187, 1999.

[88] Giann-Dong Yang, Hung-Chung Tai, and C. C. Lee. Systematic Approachto Selecting H∞ Weighting Functions for DC Servos. Proceedings of the33rd IEEE Conference on Decision and Control, 2:1080–1085, 1994.

[89] L. A. Zadeh and C. A. Desoer. Linear System Theory. The State SpaceApproach. McGraw-Hill, New York, 1963.

[90] G. Zames. Feedback optimal sensitivity: model preference transformation,multiplicative seminorms and approximate inverses. IEEE Trans. on Auto-matic Control, 26:301–320, 1981.

[91] L. Zhang. Three-phase Unbalance of Voltage Dips. PhD thesis, ChalmersUniversity of Technology, Goteborg, Sweden, November 1999.

[92] K. Zhou, J. G. Doyle, and K. Glover. Robust and Optimal Control. PrenticeHall, Upper Saddle River, 1996.

On the Use of Coordinated Control of Power System ...

Documents

Transcript of On the Use of Coordinated Control of Power System ...