Modeling and Control of VSC-HVDC

Links Connected to Weak AC Systems

Lidong Zhang

ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING

ELECTRICAL MACHINES AND POWER ELECTRONICS

Stockholm 2010

Submitted to the School of Electrical Engineering in partial fulfillment of the

requirements for the degree of Doctor of Philosophy.

Stockholm 2010

TRITA–EE 2010:022

ISSN 1653-5146

ISRN KTH-EE–10/22–SE

ISBN 978-91-7415-640-9

This document was prepared using LATEX.

To Yibin, Karin, Vivianne

and my parents

and my sister Lixia

iv

Abstract

For high-voltage direct-current (HVDC) transmission, the strength of the ac system is

important for normal operation. An ac system can be considered as weak either because its

impedance is high or its inertia is low. A typical high-impedance system is when an HVDC

link is terminated at a weak point of a large ac system where the short-circuit capacity

of the ac system is low. Low-inertia systems are considered to have limited number of

rotating machines, or no machines at all. Examples of such applications can be found

when an HVDC link is powering an island system, or if it is connected to a windfarm.

One of the advantages of applying a voltage-source converter (VSC) based HVDC system

is its potential to be connected to very weak ac systems where the conventional line-

commutated converter (LCC) based HVDC system has difficulties.

In this thesis, the modeling and control issues for VSC-HVDC links connected

to weak ac systems are investigated. In order to fully utilize the potential of the VSC-

HVDC system for weak-ac-system connections, a novel control method, i.e., power-

synchronization control, is proposed. By using power-synchronization control, the VSC

resembles the dynamic behavior of a synchronous machine. Several additional functions,

such as high-pass current control, current limitation, etc. are proposed to deal with various

practical issues during operation.

For modeling of ac/dc systems, the Jacobian transfer matrix is proposed as a uni-

fied modeling approach. With the ac Jacobian transfer matrix concept, a synchronous ac

system is viewed upon as one multivariable feedback system. In the thesis, it is shown

that the transmission zeros and poles of the Jacobian transfer matrix are closely related to

several power-system stability phenomena. The similar modeling concept is extended to

model a dc system with multiple VSCs. It is mathematically proven that the dc system is

an inherently unstable process, which requires feedback controllers to be stabilized.

For VSC-HVDC links using power-synchronization control, the short-circuit ratio

(SCR) of the ac system is no longer a limiting factor, but rather the load angles. The right-

half plane (RHP) transmission zero of the ac Jacobian transfer matrix moves closer to the

origin with larger load angles, which imposes a fundamental limitation on the achievable

bandwidth of the VSC. As an example, it is shown that a VSC-HVDC link using power-

synchronization control enables a power transmission of 0.86 p.u. from a system with an

SCR of 1.2 to a system with an SCR of 1.0. For low-inertia system connections, simulation

studies show that power-synchronization control is flexible for various operation modes

v

related to island operation and handles the mode shifts seamlessly.

Keywords: Control, modeling, multivariable feedback control, HVDC, power systems,

stability, subsynchronous torsional interaction, voltage-source converter, weak ac

systems.

vi

Acknowledgements

First of all, my deepest gratitude goes to my supervisors, Prof. Hans-Peter Nee and

Prof. Lennart Harnefors. It is an honor and a pleasure for me to have Prof. Hans-Peter

Nee as my supervisor. His patience and support helped me to go through the hardest

moments of the research work. It is also a privilege for me to be a student of Prof. Lennart

Harnefors. I am grateful for his generosity to share with me his deep understanding on

scientific work. Without his guidance, this project cannot reach the same level as it is

today.

This work has been carried out within Elektra Project 30630 and has been funded by

Energimyndigheten, ELFORSK, ABB Power Systems, ABB Corporate Research, Ban-

verket. The financial funding is greatly acknowledged.

My acknowledgements also go to the members of the steering group: Gunnar As-

plund (ABB Power Systems), Pablo Rey (ABB Power Systems), Hongbo Jiang (Banver-

ket), Torbjörn Thiringer (Chalmers University of Technology). During the last two and

half years, I had many inspiring discussions with the steering group members. Their fruit-

ful comments and inputs have greatly improved the quality of the research. Especially, I

would like to thank Gunnar Asplund, who was the chairman of the group before his retire-

ment from ABB Power Systems. Gunnar Asplund initiated the project and gave valuable

suggestions at the beginning of the project.

I would like also to thank my supervisor, Prof. Math Bollen, during my Licentiate

study at Chalmers. Prof. Math Bollen brought me into the scientific world. I received

endless support from him during my study at Chalmers and after graduation.

To my colleagues at ABB, I am grateful for all the supports I have received during

this period. In particular, I would like to thank Ying-Jiang Häfner, Magnus Öhrström,

Cuiqing Du, and Rolf Ottersten for interesting discussions as well as many helps with

thesis writing. Ying-Jiang Häfner carefully reviewed the manuscript of the thesis and

gave important suggestions. I would like to give a special thank to Pablo Rey, my group

manager at ABB, for allowing me to be absent from the group for the Ph.D study.

At KTH, I would like to thank all the colleagues in the Electrical Machines and

Power Electronics department. In particular, I would like to thank Prof. Chandur Sadaran-

gani for reviewing the manuscript of the thesis. I am also grateful to Hailian Xie for her

help with thesis writing, to Peter Lönn for his computer support, to Eva Pettersson and

Brigitt Högberg for their help with the administrative work.

vii

Many thanks to my parents and my parents-in-law for their love and support. My

mother-in-law, Prof. Renmu He, is a renowned professor in power systems in China. I

received many helps from her in my professional life as well as my family life for the past

years. Her valuable suggestions during her stay in Sweden shed light on my research and

influenced the content of this thesis. I would like also to thank my sister and nephew for

their love and encouragement for all the time.

Last but not least, I would like to thank my beloved wife and daughters. Yibin,

thank you so much for your endless love, support and understanding. Thank you, Karin

and Vivianne, for the joys you have brought to my life.

Lidong Zhang

Stockholm, Sweden

April 2010

viii

Contents

Abstract v

Acknowledgements vii

Contents ix

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Project objectives and outline of the thesis . . . . . . . . . . . . . . . . . 3

1.3 Scientific contributions of the thesis . . . . . . . . . . . . . . . . . . . . 4

1.4 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 High-Voltage Direct-Current Transmission 9

2.1 DC versus AC transmission . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 HVDC transmission using line-commutated current-source converters . . 11

2.3 HVDC transmission using forced-commutated voltage-source converters . 14

3 Control Methods for VSC-HVDC Systems 21

3.1 Power-angle control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Vector current control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Power-synchronization control . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Power-synchronization mechanism in ac systems . . . . . . . . . 31

3.3.2 Power-synchronization control of grid-connected VSCs . . . . . . 32

3.3.3 Bumpless-transfer and anti-windup schemes . . . . . . . . . . . . 35

3.3.4 Negative-sequence current control . . . . . . . . . . . . . . . . . 41

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Dynamic Modeling of AC/DC Systems 47

4.1 Jacobian transfer matrix for ac-system modeling . . . . . . . . . . . . . . 47

4.1.1 Power-system stability and dynamic modeling . . . . . . . . . . 47

4.1.2 Feedback-control view of power systems . . . . . . . . . . . . . 50

4.2 Grid-connected VSCs using power-synchronization control . . . . . . . . 52

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Contents

4.2.1 Impedance-source neglecting the ac capacitor at the filter bus . . . 52

4.2.2 Impedance-source including the ac capacitor at the filter bus . . . 63

4.2.3 AC-source feeding from a series-compensated ac line . . . . . . . 70

4.3 Grid-connected VSCs using vector current control . . . . . . . . . . . . . 74

4.4 Jacobian transfer matrix for dc-system modeling . . . . . . . . . . . . . . 81

4.5 Summary of the properties of the Jacobian transfer matrix . . . . . . . . . 87

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Control of VSC-HVDC Links Connected to High-Impedance AC Systems 89

5.1 General aspects of high-impedance ac systems . . . . . . . . . . . . . . . 89

5.2 Comparison of power-synchronization control and vector current control . 91

5.3 Multivariable feedback designs . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.1 Internal model control . . . . . . . . . . . . . . . . . . . . . . . 102

5.3.2 H∞ control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.3.3 Performance and robustness comparison . . . . . . . . . . . . . . 111

5.4 Direct-voltage control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.4.1 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.4.2 DC-capacitance requirement . . . . . . . . . . . . . . . . . . . . 119

5.5 Interconnection of two very weak ac systems . . . . . . . . . . . . . . . 122

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Control of VSC-HVDC Links Connected to Low-Inertia AC Systems 131

6.1 General aspects of low-inertia ac systems . . . . . . . . . . . . . . . . . 131

6.2 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2.1 Frequency droop control . . . . . . . . . . . . . . . . . . . . . . 133

6.2.2 Alternating-voltage droop control . . . . . . . . . . . . . . . . . 134

6.3 Dynamic modeling and linear analysis of a typical island system . . . . . 135

6.3.1 Jacobian transfer matrix . . . . . . . . . . . . . . . . . . . . . . 136

6.3.2 Integrated linear model . . . . . . . . . . . . . . . . . . . . . . . 147

6.3.3 Linear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.4 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.5 Jacobian transfer matrix for other input devices . . . . . . . . . . . . . . 155

6.5.1 Synchronous generator . . . . . . . . . . . . . . . . . . . . . . . 155

6.5.2 Induction motor . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.6 Subsynchronous characteristics . . . . . . . . . . . . . . . . . . . . . . . 162

6.6.1 Frequency-scanning method . . . . . . . . . . . . . . . . . . . . 164

6.6.2 Large ac-system connection . . . . . . . . . . . . . . . . . . . . 165

6.6.3 Island operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.6.4 Summary of the subsynchronous characteristics . . . . . . . . . . 177

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

x

Contents

7 Conclusions and Future Work 181

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

References 185

A Fundamentals of the Phasor and the Space-Vector Theory 197

A.1 Fundamentals of the phasor theory . . . . . . . . . . . . . . . . . . . . . 197

A.2 Fundamentals of the space-vector theory . . . . . . . . . . . . . . . . . . 198

A.3 Implementation of αβ and dq transformations . . . . . . . . . . . . . . . 200

A.3.1 abc-αβ transformation . . . . . . . . . . . . . . . . . . . . . . . 200

A.3.2 αβ-dq transformation . . . . . . . . . . . . . . . . . . . . . . . . 201

B Jacobian Transfer Matrix 203

B.1 Derivation of the transfer functions in Table 4.1 . . . . . . . . . . . . . . 203

B.1.1 Transfer function JPθ (s) . . . . . . . . . . . . . . . . . . . . . . 203

B.1.2 Transfer function JQθ (s) . . . . . . . . . . . . . . . . . . . . . . 205

B.1.3 Transfer function JUf θ (s) . . . . . . . . . . . . . . . . . . . . . 206

B.1.4 Transfer function JPV (s) . . . . . . . . . . . . . . . . . . . . . . 207

B.1.5 Transfer function JQV (s) . . . . . . . . . . . . . . . . . . . . . . 207

B.1.6 Transfer function JUf V (s) . . . . . . . . . . . . . . . . . . . . . 208

B.2 Proof of the instability of the dc Jacobian transfer matrix . . . . . . . . . 208

C Technical Data of the Test System 211

C.1 The VSC-HVDC link . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

C.2 The synchronous generator . . . . . . . . . . . . . . . . . . . . . . . . . 212

C.3 The induction motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

D List of Symbols and Abbreviations 215

xi

Contents

xii

Chapter 1

Introduction

This chapter describes the background of the thesis. The aim and the outline, as well as

the major scientific contributions of the thesis are presented. Finally, a list of publications

is given.

1.1 Background

In 1954, the first commercial high-voltage direct-current (HVDC) link between mainland

Sweden to Gotland island was commissioned. Since then, the accumulated installed power

of HVDC transmission worldwide has increased steadily, and recently a dramatic increase

in volume has been initiated. So far, most of the HVDC systems installed worldwide are

line-commutated converter (LCC) systems using thyristor valves. However, with grad-

ually reduced losses and costs, the recently developed voltage-source converter (VSC)

technology has shown to be more advantageous in many aspects [1–4].

The conventional line-commutated HVDC technology has an inherent weakness,

i.e., the commutation of the converter valves is dependent on the stiffness of the alter-

nating voltage. The converter cannot work properly if the connected ac system is weak.

Substantial research has been performed in this field [5–9]. The most outstanding con-

tribution on this subject is [5], which recommends to use short-circuit ratio (SCR) as a

description of the strength of the ac system relative to the power rating of the HVDC

link. Both [8] and [9] conclude that, for ac systems with an SCR lower than 1.5, syn-

chronous condensers have to be installed to increase the short-circuit capacity of the ac

system. However, synchronous condensers can substantially increase the investment and

maintenance costs of an HVDC project.

In contrast to the conventional LCC-HVDC system, the VSC-HVDC system is

based on self-commutated pulse-width modulation (PWM) technology, i.e., a VSC can

produce its own voltage waveform independent of the ac system. Thus, a VSC-HVDC

system has the potential to be connected to very weak ac systems. However, with the

traditional vector current control the potential of the VSC is not fully utilized [10–12],

1

Chapter 1. Introduction

e.g., Ref. [11] shows that the maximum power that a VSC-HVDC link using vector cur-

rent control can transmit to the ac system with SCR = 1.0 is 0.4 p.u. Ref. [12] shows

that the inner-current controller of vector current control may interact with low-frequency

resonances that are typically present in weak ac systems. In addition, the phase-locked

loop (PLL) dynamics of vector current control might also have a negative impact on the

performance of VSC-HVDC links for weak ac-system connections [10, 11, 13]. The poor

performance of vector current control for weak-ac-system connections has become an

obstacle for VSC-HVDC transmission to be applied in more challenging ac-system con-

ditions.

The application of high power-electronic devices, such as HVDC systems and

FACTS devices also imposes new challenges for power-system stability analysis and dy-

namic modeling. For classical power-system stability analysis, the phasor theory is the

major mathematical tool. With the phasor approach, the electromagnetic transients of

the ac network are neglected. This is a practical solution for conventional power sys-

tems where the electromagnetic transients have negligible effects on the stability issue of

concern. However, for high power-electronic devices, such a simplification is not accept-

able. The dynamic frequency range of high power-electronic devices is much higher than

that of the conventional power-system components. In this frequency, the phasor theory

cannot properly reflect the dynamic interaction between the ac system and the power-

electronic devices on the one hand, and between different power-electronic devices on

the other hand. For example, it has been shown by [14] that the conclusions drawn by

the phasor-based quasi-static analysis might not always agree with the results obtained by

time simulations with electromagnetic-transient programs.

The space-vector theory is based on instantaneous values, and therefore it is able to

represent the electromagnetic transients of the ac network [15]. Traditionally, the space-

vector theory is mainly applied for analyzing electrical machines and control of power-

electronic devices [15–17]. Several methodologies for dynamic modeling of three-phase

systems based on the space-vector theory have been been proposed. In [18], the complex

transfer functions are applied for analyzing three-phase ac machines. In [19], a three-

phase linear current controller is analyzed in the frequency domain based on the space-

vector approach. In [20], the space-vector theory is applied for modeling of three-phase

dynamic systems using the transfer matrix concept. For subsynchronous torsional inter-

action (SSTI) analysis, the ac network is normally required to be modeled by the space-

vector approach to take into account the electromagnetic transients [21, 22]. In recent

years, the space-vector theory has also been applied to study the dynamic interactions be-

tween high power-electronic devices. In [12], the dynamic interaction between an LCC-

HVDC link and a VSC-STATCOM in the frequency domain is analyzed based on the

space-vector theory.

While the space-vector theory has been applied successfully for analyzing high-

frequency stability phenomena in power systems, the theoretical work to connect the

2

1.2. Project objectives and outline of the thesis

high-frequency stability to the classical power-system stability defined by the phasor ap-

proach is missing in the literature. Such a connection is, however, necessary for HVDC

systems and FACTS applications. On the one hand, the dynamic frequency range of such

devices is high. On the other hand, the ratings of those devices are often high enough

to have a significant impact on most of the classical power-system stability phenomena,

such as angle stability and voltage stability. In the foreseeable future, the number of such

devices in power systems is expected to increase considerably. Thus, there is a need for a

unified modeling approach to address both the high-frequency and low-frequency stability

phenomena.

1.2 Project objectives and outline of the thesis

The objectives of the project are:

1. Develop a new control method for VSC-HVDC links connected to weak ac sys-

tems.

2. Develop a unified approach for dynamic modeling of ac/dc systems.

3. Investigate various modeling and control issues for VSC-HVDC links connected

to high-impedance ac systems.

4. Investigate various modeling and control issues for VSC-HVDC links connected

to low-inertia ac systems.

The project is conducted by both theoretical analysis and time simulations. The outline of

the thesis is:

Chapter 2 A short introduction of various technologies for HVDC transmission is

given.

Chapter 3 Two existing control methods for VSC-HVDC systems, i.e., power-angle

control and vector current control are described. A novel control method, i.e., power-

synchronization control, is proposed to solve the problem for VSC-HVDC links

connected to weak ac systems.

Chapter 4 A unified dynamic modeling approach, i.e., the Jacobian transfer matrix,

is proposed for modeling of ac/dc systems. Grid-connected VSCs using power-

synchronization control and vector current control are modeled by the proposed

concept.

Chapter 5 The control issues for VSC-HVDC links connected to high-impedance ac

systems are investigated. The dynamic performance of a VSC-HVDC link using

3

Chapter 1. Introduction

power-synchronization control and vector current control are compared. Two mul-

tivariable feedback-control designs, i.e., internal model control (IMC) and H∞ con-trol are investigated. A direct-voltage controller is proposed. A control structure for

interconnection of two very weak ac systems is proposed.

Chapter 6 Power-synchronization control is applied to VSC-HVDC links connected to

low-inertia ac systems. A frequency droop controller and a voltage droop controller

are proposed. A linear model of a typical island system is developed for tuning the

control parameters of the VSC-HVDC link. The subsynchronous characteristics of

a VSC-HVDC converter are analyzed for both the large ac-system connection and

island operation.

Chapter 7 Summarizes the thesis and provides suggestions for future work.

1.3 Scientific contributions of the thesis

The main contributions of the thesis are:

• A novel control method for grid-connected VSCs, i.e., power-synchronization con-trol, is proposed. The VSC using power-synchronization control basically resem-

bles the dynamic behavior of a synchronous machine. A group of additional con-

trol functions, such as high-pass current control, current limitation function, anti-

windup schemes, etc. are proposed to deal with various practical issues during op-

eration.

• A novel modeling concept, i.e., the Jacobian transfer matrix, is proposed as a uni-fied dynamic modeling technique for ac/dc systems. With the proposed concept, a

synchronous power system is viewed upon as a multivariable feedback control sys-

tem. The proposed concept is intended to be a unified framework for analyzing both

the low-frequency and high-frequency stability phenomena in power systems.

• The theoretical connections between the stability defined by the Jacobian trans-fer matrix concept and the classical power-system stability defined by the phasor

approach are analyzed. It is discovered that the transmission zeros of the Jacobian

transfer matrix have a close relationship with angle and voltage stability in power

systems.

• A similar modeling concept, i.e., the dc Jacobian matrix, is proposed for modelingof dc systems. By using a π-link dc model, it is mathematically proven that the

dc system (constructed by VSCs) is an inherently unstable process, where the dc

resistance gives a destabilizing effect.

4

1.3. Scientific contributions of the thesis

• Grid-connected VSCs using power-synchronization control and vector current con-trol are modeled by the Jacobian transfer matrix concept. The transfer functions are

validated with frequency-scanning results from PSCAD/EMTDC.

• The Jacobian transfer matrix concept is also applied for modeling of two conven-tional power components, i.e., the synchronous generator and the induction motor.

It is discovered that the transmission zeros of the Jacobian transfer matrix are useful

for interpreting some classical concepts, such as the synchronizing torque for the

synchronous generator and the pull-out slip for the induction motor, from a feed-

back control point of view. The transfer functions of the Jacobian transfer matrices

are also validated with frequency-scanning results from PSCAD/EMTDC.

• The performance of power-synchronization control and vector current control arecompared for VSC-HVDC links connected to weak ac systems, where it is con-

cluded that power-synchronization control is more suitable for weak-ac-system con-

nections.

• Two multivariable feedback-control design methods, i.e., IMC and H∞ control areinvestigated for VSC-HVDC links connected to high-impedance ac systems. The

performance and robustness of various control designs are compared and discussed.

• A two-degree-of-freedom direct-voltage controller for VSC-HVDC system is pro-posed where a prefilter is applied to remove the overshoot of the direct voltage. A

notch filter is proposed to reduce the dc-resonance peak.

• The requirement of dc capacitance for VSC-HVDC links connected to weak acsystems is derived.

• A control structure for VSC-HVDC links interconnecting two very weak ac sys-tems is proposed. The linear model is validated with time simulations from PSCAD/-

EMTDC for each major design step.

• A frequency droop controller and an alternating-voltage droop controller are pro-posed for VSC-HVDC links connected to low-inertia systems.

• A complete linear model is developed for a typical island system which includesa synchronous generator, an induction motor, a VSC-HVDC link and some RLC

loads. The root-locus technique is applied to tune the control parameters of the

VSC-HVDC link.

• Simulation studies are performed to demonstrate the flexibility of power-synchroni-zation control for various operation modes related to island operation.

• The subsynchronous characteristics of a VSC-HVDC converter using power-synch-ronization control are analyzed using the frequency-scanning method.

5

Chapter 1. Introduction

1.4 List of publications

- The doctoral thesis has resulted in the following publications:

I L. Zhang and H.-P. Nee, “Multivariable feedback design of VSC-HVDC connected

to weak ac systems”, in PowerTech 2009, Bucharest, Romania, 2009.

II L. Zhang, L. Harnefors and H.-P. Nee, “Power-synchronization control of grid-

connected voltage-source converters”, IEEE Trans. Power Systems, vol. 25, no. 2,

pp. 809-820, May 2010.

III L. Zhang, L. Harnefors and H.-P. Nee, “Modeling and control of VSC-HVDC

links connected to island systems” accepted for publication at IEEE Power and

Energy Society General Meeting, 2010, Minneapolis, USA.

IV L. Zhang, L. Harnefors and H.-P. Nee, “Interconnection of two very weak ac

systems by VSC-HVDC links using power-synchronization control”, accepted for

publication in IEEE Trans. Power Systems.

V L. Zhang, H.-P. Nee and L. Harnefors, “Analysis of stability limitations of a

VSC-HVDC link using power-synchronization control”, submitted to IEEE Trans.

Power Systems.

- The author has co-authored the following publication during the course of the Ph.D

study:

VI L. Harnefors, L. Zhang and M. Bongiorno, “Frequency-domain passivity-based

current controller design”, IET Power Electron., vol. 1, no. 4, pp. 455-465, 2008.

- During the course of the licentiate study, the author has authored and co-authored the

following publications:

VII L. Zhang and M. H. J. Bollen, “A method for characterizing unbalanced voltage

dips with symmetrical components”, IEEE Power Engineering Letter, pp. 50-52,

July 1998.

VIII L. Zhang and M. H. J. Bollen, “A method for characterization of three-phase un-

balanced dips from recorded voltage waveshapes”, in International Telecommuni-

cation Energy Conference, Copenhagen, Danmark, 1999.

IX L. Zhang and M. H. J. Bollen, “Characteristics of voltage dips in power systems”,

IEEE Trans. Power Delivery, vol. 15, no. 2, pp. 827-832, April 2000.

X L. Zhang, “Three-phase unbalance of voltage dips”, Licentiate thesis, Techni-

cal Report no. 322L, ISBN 91-7197-855-0, Chalmers University of Technology,

Göteborg, Sweden, 1999.

6

1.4. List of publications

XI M. H. J. Bollen and L. Zhang, “Analysis of voltage tolerance of ac adjustable-speed

drives for three-phase balanced and unbalanced sags”, IEEE Trans. Ind. Applicat.,

vol. 36, no. 3, pp. 904-910, May/June 2000.

XII M. H. J. Bollen, J. Svensson, and L. Zhang, “Testing of grid-connected power con-

verters for the effects of short circuits in the grid”, in European Power Electronics

Conference, Lausanne, Switzerland, 1999.

- In addition, during the working time in ABB, the author has authored and co-authored

the following publications that are relevant to the subjects of the doctoral and licentiate

studies:

XIII L. Zhang and L. Döfnäs, “A novel method to mitigate commutation failures in

HVDC systems”, in International Conference on Power System Technology, Kun-

ming, China, 2002.

XIV L. Zhang, L. Harnefors and P. Rey, “Power system reliability and transfer capa-

bility improvement by VSC-HVDC (HVDC Light)”, in Cigre Regional meeting,

Tallin, Estonia, 2007.

XV M. H. J. Bollen and L. Zhang, “Different methods for classification of three-phase

unbalanced voltage dips due to faults”, Electric Power Systems Research, vol. 66,

no. 1, pp. 59-69, July 2003.

7

Chapter 1. Introduction

8

Chapter 2

High-Voltage Direct-Current

Transmission

This chapter presents general aspects of HVDC transmission. Two major HVDC tech-

nologies, i.e., HVDC transmission using line-commutated current-source converters and

HVDC transmission using forced-commutated voltage-source converters are described.

2.1 DC versus AC transmission

The history of electric power systems began with direct-current (dc) transmission. In

1882, Thomas Edison built the first power system with dc transmission with a low voltage

level. However, dc transmission was quickly replaced by three-phase alternating current

(ac) transmission because of several advantages of the latter. The most prominent advan-

tage of ac transmission is that power can be transformed to different voltage levels. By

using transformers, long-distance power transmission becomes possible. In addition, cir-

cuit breakers for alternating current can take advantage of the natural current zeros that

occur twice per cycle, and ac motors are cheaper and more robust than dc motors.

In spite of the principal use of ac transmission in power systems, the interests on dc

transmission still remain [23]. In 1954, the first commercial HVDC link between main-

land Sweden to Gotland island was commissioned. Since then, the accumulated installed

power of HVDC transmission systems worldwide has increased steadily, and recently a

dramatic increase in volume has been initiated. Given the extra costs and losses related

to the converter stations, HVDC transmission is justified by some particular conditions

where the dc technology is the most feasible or may be the only solution:

• Power transmission via cables. Due to their physical structures, cables have muchhigher capacitance than overhead lines. The capacitive current in cables created by

the alternating voltage makes ac power transmission over long distance impossible.

Even for a moderate length (50km), the losses created by the capacitive current can

9

Chapter 2. High-Voltage Direct-Current Transmission

be so high that reactive compensating equipment has to be installed in the middle

of the cable [24]. However, installation of reactive compensating equipment is ex-

pensive and not always practical, e.g., with submarine cable transmission under sea.

On the other hand, if the power is transmitted by direct currents, there will be no

losses related to capacitive currents. Therefore, for long-distance submarine cable

transmission, HVDC transmission is the only feasible technical solution.

• Bulk-power transmission over long-distance. Interestingly, given the fact that acwon the “battle of currents” due to its possibility to transmit power over long dis-

tance [25], HVDC transmission wins the battle back after a century. To transmit

the same amount of power, dc transmission needs fewer power lines than ac trans-

mission. Accordingly, the costs and losses of the converter stations get balanced by

savings on the overhead lines where the break-even distance is around 400 km to

700 km depending on the land conditions and project specifications [26]. Besides,

dc transmission does not have the stability limitation related to ac transmission over

long distance.

• Unsynchronized ac-system connection. AC transmission is only possible if thetwo interconnected ac systems have the same nominal frequency and operate syn-

chronously, but dc transmission does not have such requirements. Many back-to-

back HVDC links have been built for such purposes.

Besides the above essential arguments, there are additional benefits by having embedded

HVDC links in ac systems:

• Power-system stability improvement. One of the major features of the HVDC tech-nology is its capability to manipulate large amount of power in a very short time,

which can often be utilized to improve the stability of the ac system. One example is

the improvement of transient stability by running up or running back the dc power

for emergency power supports [4, 27]. Another example is that HVDC system can

be used to damp low-frequency oscillations in ac systems by having an auxiliary

damping controller [28].

• Firewall function. Large interconnected ac systems have many well-known advan-tages, e.g., the possibility to use larger and more economical power plants, reduction

of reserve capacity in the systems, utilization of the most efficient energy resources,

as well as achieving an increase in system reliability [25,29]. However, larger inter-

connected ac systems also increase the system complexity from the operation point

of view. One of the consequence of such complexity is the large blackouts in Amer-

ica and Europe [30]. In this aspect, HVDC links have the “firewall” function in

preventing cascaded ac-system outages spreading from one system to another [31].

10

2.2. HVDC transmission using line-commutated current-source converters

-

-

-

+

+

+

aU

N

bU

cU

L

L

L

aI

bI

cI

Ldci

dcu

V1

V6

V5

V2V4

V3

Fig. 2.1 Graetz bridge for LCC-HVDC system.

2.2 HVDC transmission using line-commutated current-

source converters

The converter technology used for HVDC transmission in the early days was based on

mercury valves. The major problem with mercury-arc technology was ark-back fault

which destroyed the rectifying property of the converter valve and consequently triggered

other problems [23]. In the late 1960s, the thyristor valve technology was developed that

overcame the problems of mercury-arc technology. Converters based on either mercury

valves or thyristor valves are called line-commutated converters (LCCs), or current-source

converters (CSCs). The basic module of an LCC is the three-phase full-wave bridge cir-

cuit shown in Fig. 2.1. This topology is known as the Graetz bridge. Although there are

several alternative configurations possible, the Graetz bridge has been universally used

for LCC-HVDC converters as it provides better utilization of the converter transformer

and a lower voltage across the valve when not conducting [32].

The Graetz bridge can be used for transmitting power in two directions, i.e., the

rectifier mode and the inverter mode. This is achieved by applying different firing angles

on the valves. If the firing angle is lower than 90◦, the direct current is flowing from the

positive terminal of the dc circuit, thus the power is following from the ac side to the dc

side; If the firing angle is higher than 90◦, the direct voltage changes polarity, thus the

direct current is flowing from the negative terminal of the dc circuit. The power is then

flowing from the dc side to the ac side. An HVDC link is essentially constructed by two

Graetz bridges, which are interconnected on the dc sides. The interconnection could be

an overhead line, a cable, or a back-to-back connection.

The application of LCC-HVDC technology has been very successful and the instal-

lations of LCC-HVDC links are expected to grow at least in the near future. However, the

LCC technology suffers from several inherent weaknesses.

11

Chapter 2. High-Voltage Direct-Current Transmission

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

-1

0

1

ud

c (

p.u

.)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-2

0

2

I v (

p.u

.)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

-1

0

1

Uv (

p.u

.)

time (sec)

V5 V1 V3 V1 V3

V4 V6 V2 V4 V6 V2

Uab Ubc Uca Ucb Uac Ubc Uca Ucb Uba Uac Uab Uba

Uab Uac Uac Uab Uab

Fig. 2.2 Commutation failures of an LCC-HVDC inverter. Upper plot: direct voltage. Middle plot:

valve currents. Lower plot: valve voltage.

One problem is that the LCC always consumes reactive power, either in recti-

fier mode or in inverter mode. Depending on the firing angles, the reactive power con-

sumption of an LCC-HVDC converter station is approximately 50 − 60% of the ac-tive power. The reactive-power consumption requires compensation by connecting large

ac filters/capacitors at the converter stations. For a common LCC-HVDC link, the fil-

ters/capacitors not only increase the costs, but also occupy large amounts of space of

the converter stations. Besides, large filters/capacitors also contribute to the temporary

overvoltage (TOV) and low-order harmonic resonance problems of the HVDC link when

connects to a weak ac system [5].

Another well-known problem of the LCC-HVDC system is the occurrence of com-

mutation failures at the inverter station typically caused by disturbances in the ac system.

Either depressed voltage magnitude or phase-angle shift of the alternating voltage may

reduce the extinction volt-time area of the inverter valve [33, 34]. If the extinction angle

of the inverter valve is smaller than 5 − 6◦, the previously conducted valve will regaincurrent, which will end up with a commutation failure. Fig. 2.2 shows plots of a typical

commutation failure. A disturbance on Uab appears during the commutation between V1

and V3. Because V1 does not get the reverse voltage that is needed to switch off the cur-

rent, V1 continues to be conducted and the valve current of V3 goes down to zero again.

When the next commutation occurs between V2 and V4, V1 and V4 conduct at the same

time. From Fig. 2.2 it can be observed that the commutation failure, in fact, creates a short

circuit on the dc side, which essentially temporarily stops the power transmission.

Commutation failures are common phenomena of LCC-HVDC systems. A single

12

2.2. HVDC transmission using line-commutated current-source converters

commutation failure generally does no harm to either the converter valves or the ac sys-

tem. However, a number of repeated commutation failures may force the HVDC link to

trip [35].

While the above two problems can be mitigated by relatively easy measures, the

third problem is more fundamental, which can become a limiting factor for LCC-HVDC

applications. For LCCs, the successful commutation of the alternating current from one

valve to the next relies on the stiffness of the alternating voltage, i.e., the network strength

of the ac system. If the ac system has low short-circuit capacity relative to the power

rating of the HVDC link, i.e., low SCR, more problematic interactions between the ac

and the dc systems are expected. Besides, the SCR of the ac system also imposes an

upper limitation on the HVDC power transmission, which is often described by the well-

known maximum power curve (MPC) [5]. As mentioned before, an LCC-HVDC link

normally requires reactive-power compensation by connecting larger ac filters/capacitors

at the converter stations. These ac filters/capacitors create additional problems in weak ac

systems as described below.

One such problem is the aforementioned TOV issue. In case of a sudden change in

the active power, or the blocking of converter, the large filters/capacitors at the converter

station together with the high inductance of the ac system cause a temporary overvolt-

age before the protection system disconnects the filters/capacitors. The magnitude of the

overvoltage is directly related to the strength of the ac system. Ref. [5] gives the following

estimation of the fundamental components of TOV (TOVfc) regarding the SCR of the ac

system:

• SCR > 3: TOVfc lower than 1.25 p.u.

• 2 < SCR < 3: TOVfc higher than 1.25 p.u. but lower than 1.4 p.u.

• SCR < 2: TOVfc higher than 1.4 p.u.

TOVs can influence the design and costs of the dc stations. TOVs can also lead to satura-

tion of the converter transformer or transformers close to the dc station.

Another problem with weak-ac-system connections is the low-order harmonic res-

onance. The high inductance of the ac system and the large filters/capacitors of the HVDC

link create a resonance with low frequency fres, which can be estimated approximately by

fres ≈ f1√

2 · SCR (2.1)

where f1 is the nominal frequency of the ac system. That is to say, the resonance frequency

tends to be lower for weak ac systems. Generally speaking, the lower the resonance fre-

quency, the greater the risk for harmful interaction with the converter control system.

An improved topology of the LCC-HVDC system to overcome part of the above

mentioned problems is the capacitor-commutated converter (CCC)-HVDC technology,

13

Chapter 2. High-Voltage Direct-Current Transmission

dci

dcu

-

-

-

+

+

+

aU

N

bU

cU

L

L

L

aI

bI

cI

Fig. 2.3 Two-level voltage-source converter.

where ac capacitors are inserted in series between the valves and converter transform-

ers [36]. The series-connected capacitors not only supply the reactive power consumed

by the valves, it also improve the dynamic performance of the HVDC system. However,

the major drawback of the CCC concept is that the series capacitors increase the insula-

tion costs of the valves. Thus, the CCC-HVDC technology has been so far only applied

to back-to-back HVDC links, where the voltage level of the valves is much lower.

2.3 HVDC transmission using forced-commutated voltage-

source converters

Voltage-source converters are a new converter technology for HVDC transmission [2].

The first commercial VSC-HVDC (HVDC Light) link with a rating of 50 MW was com-

missioned in 1999 in Gotland island of Sweden, close to the world’s first LCC-HVDC

link.

Voltage-source converters (VSCs) utilize self-commutating switches, e.g., gate turn

off thyristors (GTOs) or insulated-gate bipolar transistors (IGBTs), which can be turned

on or off freely. This is in contrast to the LCC where the thyristor valve can only be

turned off by reversed line voltages. Therefore, a VSC can produce its own sinusoidal

voltage waveform using pulse-width modulation (PWM) technology independent of the

ac system.

Many different topologies have been proposed for VSCs. However, for HVDC ap-

plications, they have been so far limited to three major types: two-level converter, three-

level converter, and modular multilevel converter (M2C) [37–39].

Fig.2.3 shows a two-level grid-connected VSC. The two-level bridge is the simplest

topology that can be used in order to build up a three-phase forced-commutated VSC

bridge. The bridge consists of six valves and each valve consists of a switching device

14

2.3. HVDC transmission using forced-commutated voltage-source converters

dci

dcu

-

-

-

+

+

+

aU

N

bU

cU

L

L

L

aI

bI

cI

Fig. 2.4 Three-level neutral-point-clamped voltage-source converter.

and an anti-parallel diode. For an HVDC link, two VSCs are interconnected on the dc

side. For high-voltage applications, series connection of switching devices is necessary.

The operation principle of the two-level bridge is simple. Each phase of the VSC can be

connected either to the positive dc terminal, or the negative dc terminal. By adjusting the

width of pulses, the reference voltage can be reproduced, as shown in the upper plot of

Fig. 2.6. After filtering by phase reactors and shunt filters, this series of voltage pulses

resembles the voltage waveform of the reference voltage.

The three-level VSC shown in Fig. 2.4 is also called neutral-point-clamped (NPC)

converter. The key components that distinguish this topology from the two-level converter

are the two clamping diodes in each phase. These two diodes clamp the switch voltage to

half of the dc voltage. Thus, each phase of the VSC can switch to three different voltage

levels, i.e., the positive dc terminal, the negative dc terminal and the mid-point. Con-

sequently, voltage pulses produced by a three-level VSC match closer to the reference

voltage. Therefore, the three-level NPC converter has less harmonic content as shown in

the middle plot of Fig. 2.6. Additionally, the three-level NPC converter has lower switch-

ing losses. Compared to two-level VSCs, three-level NPC VSCs require more diodes for

neutral-point clamping. However, the total number of switching components does not

necessarily have to be higher. The reason for this is that, for HVDC applications, a valve

consists of many series-connected switches. In the two-level case a valve has to with-

stand twice as high voltage than in the three-level case. Accordingly, the total number of

15

Chapter 2. High-Voltage Direct-Current Transmission

A

B

A

B

A

BA

B

A

B...

-

+aU

A

B

A

B

A

B

...

dcu

(a)

(b)

Positive arm

Negative arm

V1

V2

Fig. 2.5 Modular multilevel voltage-source converter. (a) One M2C module. (b) One phase topol-

ogy.

switches is approximately equal.

The NPC concept can be extended to higher number of voltage levels, which can

result in further improved harmonic reduction and lower switching losses [40]. However,

for high-voltage converter applications, the neutral-clamped diodes complicate the insu-

lation and cooling design of the converter valve. Therefore, NPC concepts with a number

of voltage levels higher than three has never been considered for HVDC applications [37].

The recently proposed modular multilevel converter (M2C) concept [39, 41–43]

attracts significant interests for high-voltage converter applications. Fig. 2.5 shows the

M2C topology for one phase. Compared to the above two topologies, one major feature

of the M2C is that no common capacitor is connected at the dc side. Instead, the dc

capacitors are distributed into each module, while the converter is built up by cascade-

connected modules.

Fig. 2.5(a) shows an M2C module. Each M2C module consists of two valves which

can be switched in three different ways:

• V2 is turned on and V1 is turned off, the capacitor is inserted into the circuit fromA to B. The module contributes with voltage to the phase voltage. The capacitor is

charged if the current is from A to B, and discharged otherwise.

• V1 is turned on and V2 is turned off, the capacitor is by passed. The module doesnot contribute with voltage to the phase voltage.

• Both V1 and V2 are turned off, the module is blocked.

16

2.3. HVDC transmission using forced-commutated voltage-source converters

0 0.01 0.02 0.03 0.04 0.05 0.06

-100

0

100

UL

1 (

kV

)

0 0.01 0.02 0.03 0.04 0.05 0.06

-100

0

100

UL

1 (

kV

)

0 0.01 0.02 0.03 0.04 0.05 0.06

-100

0

100

time (sec)

UL

1 (

kV

)

Fig. 2.6 Pulse-width modulation for different converter topologies. Upper plot: two-level con-

verter. Middle plot: three-level converter. Lower plot: M2C with five modules .

The M2C concept is especially attractive for high-voltage applications, since the con-

verter can be easily scaled up by inserting additional modules in each arm. If consider-

able amounts of modules are cascaded (approximately 100 modules would be common

for HVDC applications), each module theoretically only needs to switch on and off once

per period, which greatly reduces the switching losses of the valves. However, prelimi-

nary investigation indicates that slightly higher switching frequencies are necessary. The

lower plot of Fig. 2.6 shows the voltage waveform produced by a five-module (five for

each arm) M2C. With only five modules, the waveform already resembles much better

the sinusoidal voltage reference than the other two topologies. With M2C, the harmonic

content of the voltage produced by the VSC is so low that additional filtering equipment

is almost unnecessary.

An additional benefit of the M2C is that the control system has an extra freedom in

dealing with faults at the dc side. The dc capacitors are not necessarily discharged during

faults. Thus, the fault recovery can be faster [39].

Compared to the other two topologies, the major drawback of the M2C topology is

that the required switching components are doubled since only one of the valves of each

module contributes to the phase voltage when the module is inserted in. In addition, the

design and control of the M2C are generally more complex at least than the two-level con-

verter. However, since the switching frequency of the M2C can be kept very low switches

with higher blocking voltages may be used, which in turn limits the increase in number

of switches. On the other hand, the reduction of switching losses and savings on filtering

equipment of the M2C may eventually justify its application for HVDC transmission.

17

Chapter 2. High-Voltage Direct-Current Transmission

5.0

0.1

5.0 0.1

Reactive power

Over-voltage

limitation

Under-voltage

limitation

Converter current

limitation

Active power

Fig. 2.7 PQ diagram for a typical VSC-HVDC converter.

No matter what converter topology is used, the VSC can always be treated as an

ideal voltage source where the control system has the freedom to specify the magnitude,

phase, and frequency of the produced sinusoidal voltage waveform. However, for control

design and stability analysis, it is important to take into account the limitation of the

converter in terms of active and reactive power transfer capability.

One such limit is the converter-current limitation, which is imposed by the current

carrying capability of the VSC valves. Since both the active power and the reactive power

contribute to the current flowing through the valves, this limitation is manifested as a

circle in a PQ diagram. Accordingly, if the converter is intended to support the ac system

with reactive-power supply/consumption, the maximum active power has to be limited to

make sure that the valve current is within the limit.

Another limitation which determines the reactive-power capability of the VSC is

the over/under voltage magnitude of the VSC (modulation index limitation). The over-

voltage limitation is imposed by the direct-voltage level of the VSC. The under-voltage

limit, however, is limited by the main-circuit design and the active-power transfer capa-

bility, which requires a minimum voltage magnitude to transmit the active power. In this

respect, the tap-changer of the converter transformer can play an important role to extend

the reactive-power limitation of the VSC. This could be an argument to have converter

transformers in VSC-HVDC systems. Fig. 2.7 shows the PQ diagram with the above

mentioned limitations for a typical VSC-HVDC converter [44].

VSC-HVDC technology overcomes most of the weaknesses of the LCC-HVDC

technology. In addition, it supports the ac system with reactive-power supply/consumption.

18

2.3. HVDC transmission using forced-commutated voltage-source converters

Similar to an LCC-HVDC system, a VSC-HVDC system can quickly run up or run back

the active power for ac system emergency-power support, but it can also instantly reverse

the active power [4].

Since the direct voltage of a VSC-HVDC system varies in a much smaller range

than that of a LCC-HVDC system, extruded cables can be used for VSC-HVDC systems.

The extruded cable reduces the cable cost and the construction cost. The latter makes

long-distance land-cable transmission possible [2].

Besides the above features, the most essential one is that a VSC-HVDC system

has an unlimited connection capability with ac systems, i.e., with properly designed con-

trol systems, VSC-HVDC system has the potential to be connected to any kind of ac

system with any number of links. This outstanding property will eventually bring the dc-

transmission technology to ever broader application fields!

19

Chapter 2. High-Voltage Direct-Current Transmission

20

Chapter 3

Control Methods for VSC-HVDC

Systems

This chapter describes various control methods used for VSC-HVDC systems. In Sec-

tion 3.1 and Section 3.2 two existing control methods, i.e., power-angle control and

vector current control are described. A novel control method, i.e., the so-called power-

synchronization control, is introduced in Section 3.3. The major results of this chapter are

summarized in Section 3.4. Some results of this chapter are included in [45].

3.1 Power-angle control

Power-angle control is also called voltage-angle control. It is perhaps the most straightfor-

ward controller for grid-connected VSCs [46–48]. The principle of power-angle control

is based on the following well-known equations

P =U1U2 sin θ

X

Q =U21 − U1U2 cos θ

X(3.1)

where P and Q are the active and reactive powers between two electrical nodes in ac

systems with voltage magnitudes U1 and U2. The quantities θ and X are the phase-angle

difference and line reactance between the two nodes. From (3.1) it follows that the active

power is mainly related to the phase angle θ, while the reactive power is more related to

the voltage-magnitude difference. These mathematical relationships are the foundation of

power-angle control, i.e., the active power is controlled by the phase angle of the VSC

voltage, while the reactive power or filter-bus voltage is controlled by the magnitude of

the VSC voltage.

Fig. 3.1 shows the main-circuit and control block diagram of a VSC-HVDC con-

verter using power-angle control. Lc is the inductance of the phase reactor, and Ln is the

21

Chapter 3. Control Methods for VSC-HVDC Systems

vθ

nL cL

-+refPP

QP,

E

VSC

ref

av

ref

bvref

cv

+

-fu

fC

APC

PLL

RPC/AVC-+refQQ V∆

tω

ci

v

+

-

Voltage

reference

control

-+

fU

refU

Fig. 3.1 Main-circuit and control block diagram of a VSC-HVDC converter using power-angle

control.

inductance of the ac system. Cf is the ac capacitor connected at the filter bus. The bold

letter symbols E, uf , and v represent the voltage vectors of the ac source, the filter bus,

and the VSC respectively. P and Q are the active power and reactive power from the VSC

to the ac system. The quantity ic is the current vector of the phase reactor.

To produce three-phase alternating voltages, the VSC needs three variables: magni-

tude, phase angle and frequency. With power-angle control, these three variables are given

by three different controllers, i.e., the reactive-power controller (RPC) or the alternating-

voltage controller (AVC), the active-power controller (APC), and the phase-locked loop

(PLL). The above controllers are briefly described in below:

• Reactive-power controller. The reactive power to/from the VSC is controlled bythe magnitude of the VSC voltage. A proportional-integral (PI) controller can be

applied, e.g.,

∆V =

(KQp +

KQis

)[Qref −Q] . (3.2)

where the output ∆V gives the change in magnitude of the VSC reference voltage.

• Alternating-voltage controller. Alternatively, the VSC-HVDC converter controlsthe filter-bus voltage instead of the reactive power. The output of the AVC is the

same as that of the RPC. A PI controller can be applied, e.g.,

∆V =

(KUp +

KUis

)[Uref − Uf ]. (3.3)

• Active-power controller. The active power to/from the VSC is controlled by thephase angle of the VSC voltage. A proportional-integral (PI) controller can be ap-

22

3.1. Power-angle control

plied, i.e.,

θv =

(KPp +

KPis

)[Pref − P ]. (3.4)

where the output θv gives the change in phase angle of the VSC reference voltage.

• Phase-locked loop. The function of the PLL is to synchronize the VSC to the acsystem. Below a description of a PLL design suitable for power-angle control is

given.

If ω1 is the angular frequency of the ac system, and ω is the angular frequency of the

VSC, a PLL controller has the objective to follow the phase angle of the filter-bus

voltage by minimizing

e = (ω1 − ω)t. (3.5)

If the error e in (3.5) is sufficiently small, (3.5) can be approximated by

e ≈ sin(ω1t− ωt)= sinω1t cosωt− cosω1t sinωt (3.6)

where sinω1t and cosω1t can be obtained by αβ transformation of the filter-bus

voltage as shown below.

The phase quantities of the filter-bus voltage can be defined as

ufa = Uf0 cos(ω1t)

ufb = Uf0 cos(ω1t− 120◦)ufc = Uf0 cos(ω1t− 240◦). (3.7)

The corresponding real and imaginary parts of the vector uf in the stationary αβ

reference frame (see Appendix A) can be written as

ufα = Uf0 cosω1t, ufβ = Uf0 sinω1t. (3.8)

Substituting (3.8) into (3.6), yields

e ≈ ufβUf0

cosωt− ufαUf0

sinωt. (3.9)

A PI controller can be used to minimize the error e, i.e.,

θPLL =

(KPLLp +

KPLLis

)e. (3.10)

Fig. 3.2 shows the control block diagram of the PLL. The angle change θPLL is

added to a reference frequency signal ωreft.

23

Chapter 3. Control Methods for VSC-HVDC Systems

abc

αβ

trefω

fau

fbu

fcu

COS

SIN

0

1

fU

0

1

fU

s

KK ip

PLLPLL +

tω

αfu

βfu

+

- PLLθ

+

+

Fig. 3.2 PLL for power-angle control.

• Voltage-reference control. By having ∆V , θv and ωt, the three-phase referencevoltages of the VSC can be formulated as:

vrefa = (V0 + ∆V ) cos(ωt+ θv)

vrefb = (V0 + ∆V ) cos(ωt+ θv − 120◦)vrefc = (V0 + ∆V ) cos(ωt+ θv − 240◦) (3.11)

where V0 is a nominal voltage reference, e.g., V0 = 1.0 p.u.

As shown in this section, the design and implementation of power-angle control

is simple and straightforward. However, power-angle control practically has never been

applied to any real VSC-HVDC system, since it suffers from two fundamental problems:

1. The control system has no general means to damp the various resonances in the

ac system. Therefore, the bandwidth of the controller is very much limited by the

resonances in the ac system, especially the one at the grid frequency [48].

Fig. 3.3 shows a plot of active-power and reactive-power step responses with power-

angle control. The resonance at the grid frequency can be easily observed. Although

the resonance can be damped out by applying notch filters in the active-power and

reactive-power controllers, or canceled by some model-based control designs [49],

the effects of such measures are doubtful since the ac system is a highly uncertain

process where not all of the resonance frequencies are known.

2. The control system does not have the capability to limit the valve current of the

converter. This is a serious problem, as the converters of a VSC-HVDC link usually

do not have over-current capability. It is very important for the control system to

limit the valve current to prevent the converters from being blocked (tripped) at

disturbances.

24

3.2. Vector current control

0 0.5 1 1.5 2-0.05

0

0.05

0.1

0.15

time (sec)

Pre

f, P

(p

.u.)

Pref

P

0 0.5 1 1.5 2-0.05

0

0.05

0.1

0.15

time (sec)

Qre

f, Q

(p

.u.)

Qref

Q

Fig. 3.3 Step response of active power (upper plot) and reactive power (lower plot) with power-

angle control. Observe the resonance at the grid frequency.

3.2 Vector current control

Vector current control of VSCs has initially been applied to variable-speed drives, where

the VSC is connected to an ac motor [50,51]. By utilizing the dq decoupling technique, the

control system is able to control the torque and flux independently. Once field orientation

is obtained the ac motor can be controlled using principles quite similar to those of dc-

motor control. Vector current control is generally considered as a substantial step in ac-

motor control [52].

The application of vector current control on grid-connected VSCs is often consid-

ered as a dual problem of drive control [17,53]. The basic principle is to control the active

power and the reactive power independently through an inner-current control loop [54,55].

As shown in Fig. 3.4, the essence of vector current control is that the control system

creates a converter dq frame, where a PLL is applied to make sure that the d-axis of the

converter dq frame is always aligned with the filter-bus voltage in order to synchronize

the VSC to the ac system. A simple implementation of a current controller is achieved by

the proportional-type control law

vcref = αcLc (iref − icc) + jω1Lcicc +HLP(s)ucf (3.12)

where αc is the desired closed-loop bandwidth of the inner-current controller, iref is the

converter current reference, and vcref is the voltage reference of the VSC. The superscript

c denotes the converter dq frame. The term jω1Lcicc is used to remove the so-called cross-

coupling. The function HLP(s) is a low-pass filter to improve the disturbance rejection

25

Chapter 3. Control Methods for VSC-HVDC Systems

Grid-q

Grid-d

fu

v

E

VSC-d

VSC-q

Fig. 3.4 Converter dq frame of vector current control.

capability of the current controller. HLP(s) has the following expression

HLP(s) =αf

s + αf(3.13)

where αf is typically chosen with a bandwidth (40 − 100 rad/s) [13].By applying the control law in (3.12), vector current control is claimed to be able to

control the active power and the reactive power independently by the d and q components

of the current reference iref . However, this is a conditional correct conclusion, which is

only valid if the filter-bus voltage is sufficiently stiff such that its dynamics are negligible.

A brief analysis is given below.

In a stationary frame, the dynamic equation of the phase reactor of the VSC can be

described by Kirchhoff’s voltage law as

Lcdis

dt= vs − usf (3.14)

where the superscript s denotes the stationary reference frame. If it is assumed that

ω = ω1 (3.15)

at all times, i.e., the angular frequency ω of the converter dq frame equals the angular

frequency ω1 of the grid, the following relations are established

usf = ucfe

jω1t, isc = icce

jω1t, vs = vcejω1t. (3.16)

Substituting (3.16) into (3.14) yields the dynamic equation in the converter dq frame

Lcdiccdt

= vc − ucf − jω1Lcicc. (3.17)

If the switching-time delay is neglected and it is assumed that |vcref | does not exceed themaximum voltage modulus, then vc = vcref . Substituting (3.12) into (3.17) yields

Lcdiccdt

= αcLc(iref − icc) −s

s + αfucf . (3.18)

26

3.2. Vector current control

By writing (3.18) in component form and applying Laplace transform (s = d/dt), yields

iccd =αc

s+ αcirefd −

s

Lc(s+ αc)(s+ αf )ucfd

iccq =αc

s+ αcirefq −

s

Lc(s+ αc)(s+ αf )ucfq. (3.19)

The above design approach for the inner-current controller is often referred to as internal-

model control (IMC) design [51], since the bandwidth of the inner-current control is

explicitly specified in the control parameters. Another common design approach is the

deadbeat-current control design [56, 57], which can only be realized by digital imple-

mentations. Generally speaking, if the bandwidth αc of IMC is chosen sufficiently high,

IMC and deadbeat-current control give similar results. For either of the control design,

the control bandwidth is basically limited by the switching frequency of the PWM and

the sampling period of the computer. Moreover, both methods rely on a good knowledge

of the value of Lc.

The following analysis will establish the relations between the active/reactive power

and the current references of the inner-current control. Assuming per unit quantities, the

instantaneous active power and reactive power from the VSC to the filter bus are given by

P = Re{ucf(i

cc)

∗}, Q = Im

{ucf (i

cc)

∗}. (3.20)

Linearizing (3.20) yields the following expressions

∆P =

[iccd0

iccq0

]T [∆ucfd

∆ucfq

]+

[ucfd0

ucfq0

]T [∆iccd

∆iccq

]

∆Q =

[icd0

−icq0

]T [∆ucfq

∆ucfd

]+

[ucfd0

ucfq0

]T [ −∆iccq∆iccd

](3.21)

where the subscript 0 denotes the operating-point value. In the converter dq frame, in the

steady state, the q component of the filter-bus voltage equals zero and the d component

equals the voltage magnitude, i.e.,

ucfd0 = Uf0, ucfq0 = 0. (3.22)

If the dynamics of the filter-bus voltage are neglected, it follows that

∆ucfd = ∆ucfq = 0. (3.23)

By substituting (3.22) and (3.23) into (3.21), the expressions of ∆P and ∆Q can be

simplified as

∆P = Uf0∆iccd, ∆Q = Uf0∆i

ccq. (3.24)

27

Chapter 3. Control Methods for VSC-HVDC Systems

Linearizing (3.19) and further substituting it into (3.24) [with the condition in (3.23)]

yields the following relationship in transfer matrix form

[∆P

∆Q

]=

[Uf0

αcs+αc

0

0 Uf0αc

s+αc

]

︸ ︷︷ ︸J(s)

[∆irefd

∆irefq

]. (3.25)

The transfer matrix J(s) is called Jacobian transfer matrix in this thesis, which is a gen-

eral concept for dynamic modeling of ac/dc systems that is to be introduced in Chapter 4.

Eq. (3.25) shows that the Jacobian transfer matrix J(s) is diagonal, i.e., ∆P is

only related to ∆irefd while ∆Q is only related to ∆irefq , and no cross-coupling between

the two loops exists. However, (3.25) is derived based on the assumptions of (3.15) and

(3.23). Both of the assumptions are related to the stiffness of the filter-bus voltage. If the

ac system is strong enough, i.e., Ln ≪ Lc, the dynamics of the filter-bus voltage can beneglected. However, if the ac system is weak, the assumptions in (3.15) and (3.23) no

longer hold. Therefore, the weaker the ac system, the higher the off-diagonal elements in

J(s), i.e., the more interactions between the active-power and the reactive-power control.

Consequently, to analyze the stability of vector current control for VSC-HVDC

links connected to weak ac systems, the dynamics of the filter-bus voltage have to be

considered. That is, the Jacobian transfer matrix J(s) in (3.25) should take into account

the grid inductance Ln and the dynamics of the PLL. Such a model will be developed in

Chapter 4. An in-depth analysis of the difficulty with vector current control for weak-ac-

system connections will be given in Chapter 5.

Fig. 3.5 shows the main-circuit and control block diagram of a VSC-HVDC con-

verter using vector current control. The active-power controller and the reactive-power or

the alternating-voltage controller of vector current control can be designed in a similar

way as power-angle control but with irefd and irefq as outputs. However, the PLL can be de-

signed in a more concise way by utilizing the concept of the converter dq frame , i.e., a PI

controller is applied to minimize the q component of the filter-bus voltage in the converter

dq frame

θPLL =

(KPLLp +

KPLLis

)Im{ucf}. (3.26)

In this way, the VSC is synchronized to the ac system. Fig. 3.6 shows the control block

diagram of the PLL for vector current control. With vector current control, the voltage

reference of the VSC is formulated by vrefd , vrefq and ωt. This is essentially the same as

power-angle control where the reference of the VSC voltage is formulated by the magni-

tude, the phase angle and the frequency. In the former case the rectangular form is used

while the latter uses polar form. The mathematical expressions of the dq-αβ and αβ−abcblocks in Fig. 3.5 are given in Appendix A.3.

For vector current control, given sufficiently high bandwidth, the dq components of

the converter current always follow the corresponding current references. Consequently,

28

3.2. Vector current control

nL cL

-+refPP

QP,

E

+

-fu

fC

APC

PLL

RPC/AVC-+refQQ

tω

ci

v

+

-

Inner-

current

controller

dqαβ abc

αβ

fu ci-

+refU

fU

ref

αv

ref

βv

ref

dv

ref

qvref

di

ref

qi

VSC

ref

av

ref

bvref

cv

Fig. 3.5 Main-circuit and control block diagram of a VSC-HVDC converter using vector current

control.

++

s

KK ip

PLLPLL +

c

fu}{Im cfu

PLLθ tω

trefω

Fig. 3.6 PLL for vector current control.

by limiting the modulus of the current references, the valve current of the converter is

limited. The simulation results in Fig. 3.7 show the fault ride-through capability of a VSC-

HVDC link using vector current control. A three-phase ac-system fault with 0.2 s duration

is applied at 0.1 s close to the filter bus. The modulus of the current reference [|irefc | =√(irefd )

2 + (irefq )2] reaches the current limit Imax immediately after the fault occurrence.

The control system automatically limits the converter current. After the fault is detected,

the control system reduces the fault current to half of the maximum load current (or any

other desired values to minimize the short-circuit current contribution to the ac system)

except a very short current spike at the fault inception. In VSC applications, regardless

of the control principle, the converter always tries to protect itself from excessive over

currents. This fast protection is often implemented as a low-level hardware system, and

its objective is to protect the converter in cases where the higher levels of control fail.

Since the current spike in Fig. 3.7 is so short ( < 1.6 pu in magnitude and < 5 ms in

duration ), it neither does any harm to the converter valve, nor contributes much to the

short-circuit current to the ac system.

Besides the fault-current limitation capability, the current control also has a damp-

ing effect on resonances in the ac system. Therefore, vector current control overcomes

the two fundamental problems of power-angle control. In practice, vector current control

29

Chapter 3. Control Methods for VSC-HVDC Systems

0 0.2 0.4 0.6 0.8 10

0.5

1

P (

p.u

.)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

i re

f

d, i r

ef

q (

p.u

.)

0 0.2 0.4 0.6 0.8 10

1

2

time (sec)

| i c

| (p

.u.)

i ref

d

i ref

q

Fig. 3.7 Fault ride-through capability of vector current control. Upper plot: active power from the

VSC. Middle plot: dq components of the current reference. Lower plot: modulus of the

converter current.

has been successfully applied to a number of commercial VSC-HVDC links. However, a

major drawback of vector current control is its poor performance for VSC-HVDC links

connected to weak ac systems, which becomes an obstacle for VSC-HVDC transmission

to be applied in more challenging ac-system conditions.

To overcome the problem of vector current control with weak-ac-system connec-

tions, a novel control method, i.e., power-synchronization control, is proposed in the next

section. In some sense, power-synchronization control might be viewed upon as a combi-

nation of power-angle control and vector current control.

3.3 Power-synchronization control

With the two control methods described in the previous sections, a PLL is used to synchro-

nize the VSC with the ac system. This has since long been believed to be a pre-condition

for any grid-connected VSC. However, there is, in fact, an internal synchronization mech-

anism in ac systems that the VSC can utilize to synchronize with the ac system. In this

section, a control method based on such type of synchronization is proposed. The major

goal of the proposed control method is to overcome the problem of vector current control

with weak-ac-system connections.

30

3.3. Power-synchronization control

1SM 2SMX

222

2 mem TT

dt

dJ −=

ω11

11 em

m TTdt

dJ −=

ω

Fig. 3.8 Synchronization mechanism between SMs in an ac system.

3.3.1 Power-synchronization mechanism in ac systems

In this sub-section, the power-synchronization mechanism between synchronous machines

(SMs) in ac systems is described. The mechanism is illustrated by a simple system con-

sisting of two interconnected SMs as shown in Fig. 3.8. SM1 operates as a generator and

SM2 operates as a motor. The reactance X is the sum of the reactances of the SMs and

that of the line interconnecting the two SMs. All resistances and other damping effects

are disregarded.

Initially, it is assumed that the two SMs operate at steady state, as described by the

phasor diagram in Fig. 3.9(a). The phasorsE1 andE2 represent the line-to-line equivalents

of the inner emfs of the two SMs respectively. These emfs are assumed to be constant at

all times (even during transients). The electric power transmitted from SM1 to SM2 is

given by

P =E1E2 sin θ

X(3.27)

where θ is the electrical angle separating the two emfs E1 and E2. The mechanical torque

Tm1 of SM1 is now increased by a certain amount for a short duration and then brought

back to its initial value. As a consequence of the temporary increase of Tm1, the me-

chanical angle of the rotor of SM1 advances, as predicted by the generator-mode swing

equation

J1dωm1dt

= Tm1 − Te1 (3.28)

where J1 is the total inertia of the shaft-system of SM1 , ωm1 is the rotor speed, and Te1 is

the electromagnetic torque of of SM1. Since the emf of a synchronous machine is tightly

connected to the rotor position, the advance of the mechanical angle of the rotor of SM1inevitably causes an advance of the phase of the emf of SM1, as indicated by the phasor

E′

1 in Fig. 3.9(b). The initial position of E1 is shown as a dashed line in Fig. 3.9(b). Due

to the phase advancement indicated by E′

1, the phase difference between the emfs of the

two SMs is increased. According to (3.28), this translates into an increase of the electric

power transmitted from SM1 to SM2 . This increase in power is equivalent to an increase

in the electromagnetic torque Te2 of SM2 . Assuming that SM2 has a constant load torque

Tm2, the rotor of SM2 starts to accelerate as dictated by

J2dωm2dt

= Te2 − Tm2 (3.29)

31

Chapter 3. Control Methods for VSC-HVDC Systems

'1E1E 2E

θ2E

(a) (b)

Fig. 3.9 Phasor diagrams describing power synchronization.

where J2 is the total inertia of the shaft-system of SM2 , and ωm2 is the mechanical angular

velocity of SM2 . As the rotor of SM2 starts to accelerate, the same thing occurs with the

phase of E2, as indicated by the arc-shaped arrow in Fig. 3.9(b). The acceleration of the

phasor E2 results in a reduction of the phase difference between the emfs of the two

SMs. After a transient, which in reality involves a certain amount of damping, the phase

difference between the emfs of the two SMs is brought back to its initial value (as the

transmitted electric power), and the system is again at steady state.

The synchronization mechanism described above is known to all power system spe-

cialists, i.e., the synchronization process is achieved by means of a transient power trans-

fer. The same kind of synchronization also appears in large systems of interconnected

synchronous machines.

Due to the fact that the synchronous machines can maintain operation in various ac-

network conditions while the vector-current-controlled VSCs are prone to fail, it makes

sense to suggest a control method based on a synchronization process where the electric

power is the communicating medium. In the next sub-section, a controller based on power

synchronization is proposed.

3.3.2 Power-synchronization control of grid-connected VSCs

From the discussion in the preceding sub-section it is known that the SMs in an ac system

maintain synchronism by means of power synchronization, i.e., a transient power trans-

fer. This power transfer involves a current which is determined by the interconnecting

network. Generally, this current is unknown. If power synchronization should be used to

control a VSC, therefore, it cannot be combined with a vector current controller, which

requires a known current reference. As will be shown below, the active power output from

the VSC is instead controlled directly by the power-synchronization loop and the reactive

power (or alternating voltage) is controlled by the magnitude of the VSC voltage. Conse-

quently, an inner current loop is not necessary from a power and voltage control point of

view.

However, as it was mentioned in Section 3.2, besides power and voltage control,

the current controller is also important in

1. Providing damping effects to poorly-damped resonances in ac systems.

2. Limiting the valve current of the converter during severe ac-system faults.

32

3.3. Power-synchronization control

To make the essence of power-synchronization control easier to be captured, the ini-

tial proposal of the control law only considers the damping issue, while the fault-current-

limitation issue is proposed as a modification of the initial design.

Accordingly, an initial control design based on the power-synchronization law is

proposed as

• Power-synchronization loop (PSL). The control law is given by

θv =kps

(Pref − P ). (3.30)

where θv supplies the synchronization input to the VSC, i.e., ωt = ωreft + θv. The

power-synchronization loop is essentially an emulation of the swing equation, how-

ever, not an exact copy. Since the mechanical angular velocity ωm is the derivative

of the angular position, (3.28) represents a double integration when going from

torque (or electric power) to angular position. This double integration, inherently,

yields a poor phase margin even with considerable damping. Therefore, the pro-

posed power-synchronization law in (3.30) employs only a single integration.

• Alternating-voltage control (AVC). The control law is given by

∆V =kus

(Uref − Uf ). (3.31)

where ∆V gives the change in magnitude of the VSC reference voltage. The AVC

can also be viewed as an emulation of the exciter control of a synchronous ma-

chine. A normal exciter control of a synchronous machine is of proportional type.

However, it is found to be more beneficial to have integral process for the VSC to

suppress high-frequency disturbances. If there are other voltage-controlling devices

connected close to the filter bus, a load compensation should be applied to avoid

voltage hunting. This issue will be discussed in Chapter 6.

• Reactive-power control (RPC). When operating against a weak ac system, theVSC-HVDC converter should preferably be operated in AVC mode to give the ac

system the best possible voltage support. In case reactive-power control is required,

the output of this controller should be added to the alternating-voltage reference,

and the added amount should be limited. The PI-type controller proposed in (3.2)

can be used for the RPC but with voltage reference change ∆Uref as output.

• Voltage-vector control law. The control law of the voltage vector of the VSC isproposed as

vcref = (V0 + ∆V ) −HHP (s) icc (3.32)where V0 is the nominal value, e.g., V0 = 1.0 p.u., and ∆V is given by the AVC.

HHP (s) is a high-pass filter for damping purpose, which is expressed by

HHP (s) =kvs

s+ αv(3.33)

33

Chapter 3. Control Methods for VSC-HVDC Systems

where αv should be chosen low enough to cover all the possible resonances in the

ac system. Typically αv should be chosen between 30 rad/s and 50 rad/s to also

cover the subsynchronous resonance in ac systems. The gain kv determines the level

of the damping effect with typical values between 0.2 p.u. and 0.6 p.u. The effects

of HHP (s) for resonance damping will be further analyzed in Chapter 4.

In the following, a current limitation scheme is proposed as a modification of the

initial design. The principle is to have the control system in current-limitation mode au-

tomatically once the converter current is above the limit.

In case that the converter current is above the maximum limit Imax, the desired

control law of the VSC is the inner-current control law of vector current control in (3.12).

However, instead of giving a constant current reference to (3.12), the value of iref in (3.12)

is given by

iref =1

αcLc

[(V0 + ∆V ) −HHP (s) icc −HLP(s)ucf − jω1Lcicc

]+ icc. (3.34)

The current reference in (3.34) is designed in such a way that the control law in (3.12)

becomes (3.32) in normal operation. This can be easily verified by substituting (3.34)

into (3.12). However, the current reference iref in (3.34) gives an indication of the actual

converter current. During ac-system faults, current limitation is automatically achieved by

limiting the modulus of iref to the maximum current limit Imax. A brief analysis of this is

given below.

The dynamics of the converter current in the converter dq frame can be described

by

Lcdiccdt

= vc − ucf − jω1Lcicc. (3.35)

Assuming vc = vcref , substituting (3.12) into (3.35) yields,

Lcdic

dt= αcLc (iref − icc) −

s

s+ αfucf . (3.36)

By setting the time derivative and the Laplace operator s to zero, it is found that

iref = icc. (3.37)

That is, the current reference is identical to the actual converter current in the steady state.

In other words, by limiting the modulus of the current reference, the converter current is

limited.

Fig. 3.10 shows the overview of power-synchronization cont