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Modeling and Control of VSC-HVDCLinks Connected to Weak AC Systems

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Modeling and Control of VSC-HVDC
Links Connected to Weak AC Systems
Lidong Zhang

ROYAL I NSTITUTE OF T ECHNOLOGY
S CHOOL OF E LECTRICAL E NGINEERING
E LECTRICAL M ACHINES AND P OWER E LECTRONICS
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

A
This document was prepared using LTEX.

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 linecommutated 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 VSCHVDC system for weak-ac-system connections, a novel control method, i.e., powersynchronization 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 unified 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 righthalf 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 powersynchronization 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, Banverket. The financial funding is greatly acknowledged.
My acknowledgements also go to the members of the steering group: Gunnar Asplund (ABB Power Systems), Pablo Rey (ABB Power Systems), Hongbo Jiang (Banverket), Torbj¨ rn Thiringer (Chalmers University of Technology). During the last two and
o
half years, I had many inspiring discussions with the steering group members. Their fruitful 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 retirement 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 H¨ fner, Magnus Ohrstr¨ m,
a
o
Cuiqing Du, and Rolf Ottersten for interesting discussions as well as many helps with
thesis writing. Ying-Jiang H¨ fner carefully reviewed the manuscript of the thesis and
a
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 Sadarangani for reviewing the manuscript of the thesis. I am also grateful to Hailian Xie for her
help with thesis writing, to Peter L¨ nn for his computer support, to Eva Pettersson and
o
Brigitt H¨ gberg for their help with the administrative work.
o
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

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Acknowledgements

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Contents

ix

1 Introduction
1.1 Background . . . . . . . . . . . . . . . .
1.2 Project objectives and outline of the thesis
1.3 Scientific contributions of the thesis . . .
1.4 List of publications . . . . . . . . . . . .

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2 High-Voltage Direct-Current Transmission
2.1 DC versus AC transmission . . . . . . . . . . . . . . . . . . . . . . . .
2.2 HVDC transmission using line-commutated current-source converters . .
2.3 HVDC transmission using forced-commutated voltage-source converters .

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3 Control Methods for VSC-HVDC Systems
3.1 Power-angle control . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Vector current control . . . . . . . . . . . . . . . . . . . . . . .
3.3 Power-synchronization control . . . . . . . . . . . . . . . . . .
3.3.1 Power-synchronization mechanism in ac systems . . . .
3.3.2 Power-synchronization control of grid-connected VSCs .
3.3.3 Bumpless-transfer and anti-windup schemes . . . . . . .
3.3.4 Negative-sequence current control . . . . . . . . . . . .
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Dynamic Modeling of AC/DC Systems
4.1 Jacobian transfer matrix for ac-system modeling . . . . . .
4.1.1 Power-system stability and dynamic modeling . .
4.1.2 Feedback-control view of power systems . . . . .
4.2 Grid-connected VSCs using power-synchronization control

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ix

Contents

4.3
4.4
4.5
4.6

4.2.1 Impedance-source neglecting the ac capacitor at the filter bus .
4.2.2 Impedance-source including the ac capacitor at the filter bus .
4.2.3 AC-source feeding from a series-compensated ac line . . . . .
Grid-connected VSCs using vector current control . . . . . . . . . . .
Jacobian transfer matrix for dc-system modeling . . . . . . . . . . . .
Summary of the properties of the Jacobian transfer matrix . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Control of VSC-HVDC Links Connected to High-Impedance AC Systems
5.1 General aspects of high-impedance ac systems . . . . . . . . . . . . . .
5.2 Comparison of power-synchronization control and vector current control
5.3 Multivariable feedback designs . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Internal model control . . . . . . . . . . . . . . . . . . . . . .
5.3.2 H∞ control . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Performance and robustness comparison . . . . . . . . . . . . .
5.4 Direct-voltage control . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Controller design . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 DC-capacitance requirement . . . . . . . . . . . . . . . . . . .
5.5 Interconnection of two very weak ac systems . . . . . . . . . . . . . .
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
6.1 General aspects of low-inertia ac systems . . . . . . . . . . . . . .
6.2 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Frequency droop control . . . . . . . . . . . . . . . . . . .
6.2.2 Alternating-voltage droop control . . . . . . . . . . . . . .
6.3 Dynamic modeling and linear analysis of a typical island system . .
6.3.1 Jacobian transfer matrix . . . . . . . . . . . . . . . . . . .
6.3.2 Integrated linear model . . . . . . . . . . . . . . . . . . . .
6.3.3 Linear analysis . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Jacobian transfer matrix for other input devices . . . . . . . . . . .
6.5.1 Synchronous generator . . . . . . . . . . . . . . . . . . . .
6.5.2 Induction motor . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Subsynchronous characteristics . . . . . . . . . . . . . . . . . . . .
6.6.1 Frequency-scanning method . . . . . . . . . . . . . . . . .
6.6.2 Large ac-system connection . . . . . . . . . . . . . . . . .
6.6.3 Island operation . . . . . . . . . . . . . . . . . . . . . . . .
6.6.4 Summary of the subsynchronous characteristics . . . . . . .
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x

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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
A.1 Fundamentals of the phasor theory . . . . . . . . . . .
A.2 Fundamentals of the space-vector theory . . . . . . . .
A.3 Implementation of αβ and dq transformations . . . . .
A.3.1 abc-αβ transformation . . . . . . . . . . . . .
A.3.2 αβ-dq transformation . . . . . . . . . . . . . .

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B Jacobian Transfer Matrix
B.1 Derivation of the transfer functions in Table 4.1 . . . . .
B.1.1 Transfer function JPθ (s) . . . . . . . . . . . . .
B.1.2 Transfer function JQθ (s) . . . . . . . . . . . . .
B.1.3 Transfer function JUf θ (s) . . . . . . . . . . . .
B.1.4 Transfer function JPV (s) . . . . . . . . . . . . .
B.1.5 Transfer function JQV (s) . . . . . . . . . . . . .
B.1.6 Transfer function JUf V (s) . . . . . . . . . . . .
B.2 Proof of the instability of the dc Jacobian transfer matrix

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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 gradually 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 alternating 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 contribution 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, synchronous 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 current 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 conditions.
The application of high power-electronic devices, such as HVDC systems and
FACTS devices also imposes new challenges for power-system stability analysis and dynamic 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 systems where the electromagnetic transients have negligible effects on the stability issue of
concern. However, for high power-electronic devices, such a simplification is not acceptable. 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 powerelectronic 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 spacevector theory is mainly applied for analyzing electrical machines and control of powerelectronic 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 threephase linear current controller is analyzed in the frequency domain based on the spacevector approach. In [20], the space-vector theory is applied for modeling of three-phase
dynamic systems using the transfer matrix concept. For subsynchronous torsional interaction (SSTI) analysis, the ac network is normally required to be modeled by the spacevector 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 between high power-electronic devices. In [12], the dynamic interaction between an LCCHVDC 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 highfrequency 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 approach 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 systems.
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., powersynchronization 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 powersynchronization 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 multivariable feedback-control designs, i.e., internal model control (IMC) and H∞ control 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 control, is proposed. The VSC using power-synchronization control basically resembles the dynamic behavior of a synchronous machine. A group of additional control functions, such as high-pass current control, current limitation function, antiwindup schemes, etc. are proposed to deal with various practical issues during operation.
• A novel modeling concept, i.e., the Jacobian transfer matrix, is proposed as a unified dynamic modeling technique for ac/dc systems. With the proposed concept, a
synchronous power system is viewed upon as a multivariable feedback control system. 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 transfer 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 modeling
of 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 control 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 conventional 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 feedback 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 are
compared for VSC-HVDC links connected to weak ac systems, where it is concluded that power-synchronization control is more suitable for weak-ac-system connections.
• Two multivariable feedback-control design methods, i.e., IMC and H∞ control are
investigated 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 proposed 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 ac
systems is derived.
• A control structure for VSC-HVDC links interconnecting two very weak ac systems 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 proposed for VSC-HVDC links connected to low-inertia systems.
• A complete linear model is developed for a typical island system which includes
a 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-synchronization control for various operation modes related to island operation.
• The subsynchronous characteristics of a VSC-HVDC converter using power-synchronization 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 gridconnected 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 unbalanced dips from recorded voltage waveshapes”, in International Telecommunication 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, Technical Report no. 322L, ISBN 91-7197-855-0, Chalmers University of Technology,
G¨ teborg, Sweden, 1999.
o
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 converters 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. D¨ fn¨ s, “A novel method to mitigate commutation failures in
o a
HVDC systems”, in International Conference on Power System Technology, Kunming, China, 2002.
XIV L. Zhang, L. Harnefors and P. Rey, “Power system reliability and transfer capability 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 technologies, 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 advantage of ac transmission is that power can be transformed to different voltage levels. By
using transformers, long-distance power transmission becomes possible. In addition, circuit 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 mainland 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 much
higher 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 expensive 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 ac
won the “battle of currents” due to its possibility to transmit power over long distance [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 transmission. 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 the
two interconnected ac systems have the same nominal frequency and operate synchronously, but dc transmission does not have such requirements. Many back-toback 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 technology 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 advantages, 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 interconnected 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 America 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
idc

V1
-

N

-

Ua
Ub
Uc

+

L
L

Ib

+

L

V5

Ia

+

V3

L

udc

Ic

V4

V6

V2

Fig. 2.1 Graetz bridge for LCC-HVDC system.

2.2 HVDC transmission using line-commutated currentsource 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 circuit 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 installations of LCC-HVDC links are expected to grow at least in the near future. However, the
LCC technology suffers from several inherent weaknesses.
11

udc (p.u.)

Chapter 2. High-Voltage Direct-Current Transmission
1

Uab Uac

Ubc

Uba

Uca

Ucb

Uab

Uac

Ubc

Uba

Uca

Ucb

0

-1

Iv (p.u.)

0
2

0.015

V1

V3
V2

V6

V4

1

0.01

V5

0

-2
0

Uv (p.u.)

0.005

0.005

0.01

0.015

Uab Uac

0.02

0.025

0.03

0.035

V1
V4

0.02
Uab

V3
V2

V6

0.025

0.04

0.03

0.035

0.04

0.03

0.035

0.04

Uac

0

-1
0

0.005

0.01

0.015

0.02
0.025
time (sec)

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 rectifier mode or in inverter mode. Depending on the firing angles, the reactive power consumption of an LCC-HVDC converter station is approximately 50 − 60% of the active power. The reactive-power consumption requires compensation by connecting large
ac filters/capacitors at the converter stations. For a common LCC-HVDC link, the filters/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 commutation 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 regain
current, 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 current, 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 system. 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 wellknown 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 overvoltage 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 saturation of the converter transformer or transformers close to the dc station.
Another problem with weak-ac-system connections is the low-order harmonic resonance. 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 frequency, 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
idc

-

N

-

Ua
Ub
Uc

+

L

Ia

+

L

Ib

+

L

Ic

udc

Fig. 2.3 Two-level voltage-source converter.

where ac capacitors are inserted in series between the valves and converter transformers [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 insulation 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 voltagesource 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 commissioned 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 applications, they have been so far limited to three major types: two-level converter, threelevel 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
idc

-

N

-

Ua
Ub
Uc

+

L

Ia

+

L

Ib

+

L

Ic

udc

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. Consequently, 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 switching 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 withstand twice as high voltage than in the three-level case. Accordingly, the total number of
15

Chapter 2. High-Voltage Direct-Current Transmission
A
V1

A

B

B

V2
(a)

Positive arm

A
A

A

...

B

B

B

-

Ua

+

udc
A
B

A
B

...

A

Negative arm

B

(b)

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

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 insulation 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 cascadeconnected 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 from
A 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 does
not contribute with voltage to the phase voltage.
• Both V1 and V2 are turned off, the module is blocked.
16

UL1 (kV)

2.3. HVDC transmission using forced-commutated voltage-source converters
100
0
-100

UL1 (kV)

0

0.02

0.03

0.04

0.05

0.06

0.01

0.02

0.03

0.04

0.05

0.06

0.01

0.02

0.03
time (sec)

0.04

0.05

0.06

100
0
-100
0

UL1 (kV)

0.01

100
0
-100
0

Fig. 2.6 Pulse-width modulation for different converter topologies. Upper plot: two-level converter. Middle plot: three-level converter. Lower plot: M2C with five modules .

The M2C concept is especially attractive for high-voltage applications, since the converter can be easily scaled up by inserting additional modules in each arm. If considerable 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, preliminary 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 converter. 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
Reactive power
Over-voltage
limitation

1 .0

Converter current
limitation
0 .5

0.5

1.0

Active power

Under-voltage
limitation

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 overvoltage 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 capability, 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 control 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 dctransmission 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 Section 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 powersynchronization 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 straightforward controller for grid-connected VSCs [46–48]. The principle of power-angle control
is based on the following well-known equations
U1 U2 sin θ
X
2
U1 − U1 U2 cos θ
Q=
X
P =

(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 converter 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
VSC

E

Ln

Lc
P, Q

Cf

+

ic +
v

uf

-

-

PLL

Uf
U ref +
Q Qref +

ωt

RPC/AVC

P
Pref + APC

∆V

Voltage
reference
control

ref
va
ref
vb
vcref

θv

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: magnitude, 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 alternatingvoltage 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 by
the magnitude of the VSC voltage. A proportional-integral (PI) controller can be
applied, e.g.,
KiQ
Q
∆V = Kp +
[Qref − Q] .
(3.2)
s
where the output ∆V gives the change in magnitude of the VSC reference voltage.
• Alternating-voltage controller. Alternatively, the VSC-HVDC converter controls
the 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 =

U
Kp +

KiU
s

[Uref − Uf ].

(3.3)

• Active-power controller. The active power to/from the VSC is controlled by the
phase angle of the VSC voltage. A proportional-integral (PI) controller can be ap22

3.1. Power-angle control
plied, i.e.,
θv =

P
Kp +

KiP
s

[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 ac
system. 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(ω1 t − ωt)

= sin ω1 t cos ωt − cos ω1 t sin ωt

(3.6)

where sin ω1 t and cos ω1 t 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
uf a = Uf 0 cos(ω1 t)
uf b = Uf 0 cos(ω1 t − 120◦ )

uf c = Uf 0 cos(ω1 t − 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
uf α = Uf 0 cos ω1 t, uf β = Uf 0 sin ω1 t.

(3.8)

Substituting (3.8) into (3.6), yields
e≈

uf β
uf α
cos ωt −
sin ωt.
Uf 0
Uf 0

(3.9)

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

PLL
Kp +

KiPLL
s

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 ωref t.
23

Chapter 3. Control Methods for VSC-HVDC Systems
ωref t +

ωt

+
u fβ

u fa
u fb
u fc

1
Uf0
COS

αβ
abc

SIN

u fα

+
-

PLL
Kp +

K iPLL
s

θ PLL

1
Uf0
Fig. 3.2 PLL for power-angle control.

• Voltage-reference control. By having ∆V , θv and ωt, the three-phase reference
voltages of the VSC can be formulated as:
ref
va = (V0 + ∆V ) cos(ωt + θv )
ref
vb = (V0 + ∆V ) cos(ωt + θv − 120◦ )

ref
vc = (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 powerangle 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

Pref, P (p.u.)

0.15
0.1
Pref

0.05

P
0
-0.05
0

0.5

1
time (sec)

1.5

2

Qref, Q (p.u.)

0.15
0.1
Q

0.05

ref

Q
0
-0.05
0

0.5

1
time (sec)

1.5

2

Fig. 3.3 Step response of active power (upper plot) and reactive power (lower plot) with powerangle 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 dcmotor control. Vector current control is generally considered as a substantial step in acmotor control [52].
The application of vector current control on grid-connected VSCs is often considered 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
c
vref = αc Lc (iref − ic ) + jω1 Lc ic + HLP (s)uc
c
c
f

(3.12)

where αc is the desired closed-loop bandwidth of the inner-current controller, iref is the
c
converter current reference, and vref is the voltage reference of the VSC. The superscript
c denotes the converter dq frame. The term jω1 Lc ic is used to remove the so-called crossc
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
VSC-q
v

VSC-d

uf
E

Grid-d

Fig. 3.4 Converter dq frame of vector current control.

capability of the current controller. HLP (s) has the following expression
αf
HLP (s) =
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
dis
= vs − us
(3.14)
f
dt
where the superscript s denotes the stationary reference frame. If it is assumed that
Lc

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
us = uc ejω1 t , is = ic ejω1 t , vs = vc ejω1 t .
f
f
c
c

(3.16)

Substituting (3.16) into (3.14) yields the dynamic equation in the converter dq frame
dic
c
= vc − uc − jω1 Lc ic .
(3.17)
f
c
dt
c
If the switching-time delay is neglected and it is assumed that |vref | does not exceed the
c
maximum voltage modulus, then vc = vref . Substituting (3.12) into (3.17) yields
Lc

Lc
26

dic
s
c
= αc Lc (iref − ic ) −
uc .
c
dt
s + αf f

(3.18)

3.2. Vector current control
By writing (3.18) in component form and applying Laplace transform (s = d/dt), yields
s
αc ref
id −
uc
s + αc
Lc (s + αc )(s + αf ) f d
s
αc ref
ic =
iq −
uc .
cq
s + αc
Lc (s + αc )(s + αf ) f q

ic =
cd

(3.19)

The above design approach for the inner-current controller is often referred to as internalmodel 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 implementations. 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 uc (ic )∗ , Q = Im uc (ic )∗ .
f c
f c

(3.20)

Linearizing (3.20) yields the following expressions
∆P =

∆Q =

ic
cd0

T

∆uc d
f

ic
cq0
ic
d0
−ic
q0

∆uc q
f
T

∆uc q
f
∆uc d
f

+

+

uc d0
f

T

uc q0
f
uc d0
f
uc q0
f

∆ic
cd
∆ic
cq

T

−∆ic
cq
∆ic
cd

(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.,
uc d0 = Uf 0 , uc q0 = 0.
f
f

(3.22)

If the dynamics of the filter-bus voltage are neglected, it follows that
∆uc d = ∆uc q = 0.
f
f

(3.23)

By substituting (3.22) and (3.23) into (3.21), the expressions of ∆P and ∆Q can be
simplified as
∆P = Uf 0 ∆ic , ∆Q = Uf 0 ∆ic .
(3.24)
cd
cq
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

=

αc
Uf 0 s+αc

0

∆iref
d

0

αc
Uf 0 s+αc

∆iref
q

.

(3.25)

J(s)

The transfer matrix J(s) is called Jacobian transfer matrix in this thesis, which is a general 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 ∆iref while ∆Q is only related to ∆iref , and no cross-coupling between
q
d
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 be
neglected. 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-acsystem connections will be given in Chapter 5.
Fig. 3.5 shows the main-circuit and control block diagram of a VSC-HVDC converter 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 iref and iref as outputs. However, the PLL can be deq
d
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
K PLL
PLL
θPLL = Kp + i
Im{uc }.
(3.26)
f
s
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
ref
ref
reference of the VSC is formulated by vd , vq and ωt. This is essentially the same as
power-angle control where the reference of the VSC voltage is formulated by the magnitude, 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 − abc
blocks 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
VSC
E

Ln

Lc
P, Q

+

Cf

ic +
v

uf

-

-

Uf
U ref + -

Q
Qref + -

RPC/AVC

P
Pref + -

APC
PLL

ref
iq

ref
id

uf

ic

ref
ref

vd
Innerαβ v ref
ref
vq dq
β
current
controller

abc
αβ

ref
va
ref
vb
vcref

ωt

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

c
f

c
f

Im{u }

K

PLL
p

K PLL θ PLL +
+ i
s

+

ωt

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 VSCHVDC 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 [|iref | =
c

(iref )2 + (iref )2 ] reaches the current limit Imax immediately after the fault occurrence.
q
d
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 damping 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

P (p.u.)

1
0.5

0.2

1

0.4

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

i ref
d

0.5

ref

ref

id , iq (p.u.)

0
0

i ref

0

q

0

0.2

0.4

0.2

0.4

| ic| (p.u.)

2
1
0
0

time (sec)

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 connections, 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 combination 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 synchronize 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 mechanism 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
J1

dωm1
dω m 2
= Tm1 − Te1
J2
= Te 2 − Tm 2
dt
dt
SM 2
SM1
X

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 consisting 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 phasors E1 and E2 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
E1 E2 sin θ
P =
(3.27)
X
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 mechanical angle of the rotor of SM1 advances, as predicted by the generator-mode swing
equation
dωm1
= Tm1 − Te1
(3.28)
J1
dt
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 SM1
inevitably causes an advance of the phase of the emf of SM1 , as indicated by the phasor

E1 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 E1 , 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
J2

dωm2
= Te2 − Tm2
dt

(3.29)
31

Chapter 3. Control Methods for VSC-HVDC Systems
E1

E2

E1 '

E2

θ
(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 specialists, i.e., the synchronization process is achieved by means of a transient power transfer. 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 acnetwork 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 transfer. 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. Consequently, 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 initial proposal of the control law only considers the damping issue, while the fault-currentlimitation 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

kp
(Pref − P ).
(3.30)
s
where θv supplies the synchronization input to the VSC, i.e., ωt = ωref t + θv . The
power-synchronization loop is essentially an emulation of the swing equation, however, 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 proposed power-synchronization law in (3.30) employs only a single integration.
θv =

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

ku
(Uref − Uf ).
(3.31)
s
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 machine. 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.
∆V =

• Reactive-power control (RPC). When operating against a weak ac system, the
VSC-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 is
proposed as
c
vref = (V0 + ∆V ) − HHP (s) ic
(3.32)
c
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) =

kv s
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 automatically 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
(V0 + ∆V ) − HHP (s) ic − HLP (s)uc − jω1 Lc ic + ic .
c
f
c
c
αc Lc

(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
dic
Lc c = vc − uc − jω1 Lc ic .
(3.35)
f
c
dt
c
Assuming vc = vref , substituting (3.12) into (3.35) yields,

dic
s
Lc
= αc Lc (iref − ic ) −
uc .
c
dt
s + αf f

(3.36)

By setting the time derivative and the Laplace operator s to zero, it is found that
iref = ic .
c

(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 control. The “Current reference control” block corresponds to the control law described by (3.34), while the “Current
controller” block corresponds to the control law described by (3.12).
34

3.3. Power-synchronization control
U ref +
Qref +

RPC

Q-

∆U ref +
Uf

-

k u ∆V
s
AVC
VSC

E

Ln

Lc
P, Q
Cf

+

+

ic

uf

v

-

i ref
Current- d
ref
reference iq
control

P
Pref + -

kp
s
PSL

θv

ref
vd

Current
v ref
controller q

+

ref


αβ
dq

ref


ref
va

abc

αβ

ref
vb
vcref

ωt

+

ωref t

Fig. 3.10 Main-circuit and control block diagram of a VSC-HVDC converter using powersynchronization control.

3.3.3 Bumpless-transfer and anti-windup schemes
In some situations, power synchronization cannot be applied, and a back-up PLL has to
be used instead. Those situations are:
1. Before the VSC is de-blocked, the back-up PLL provides the synchronization input
to the VSC. After the converter is de-blocked, the PLL is replaced by the PSL.
This procedure is similar to the auto-synchronization process used for synchronous
machines before they are connected to the grid.
2. During severe ac-system faults, the control system has to limit the converter current. Thus, the PSL cannot be applied.
According to the control law given by (3.32), the converter dq frame created by the PSL
has the d-axis aligned with the voltage vector v of the VSC. However, the dq frame created
by the PLL has the d-axis aligned with the voltage vector uf of the filter bus. As shown in
Fig. 3.11, the two dq frames differ by an angle θc . Consequently, the dq frame created by
the PLL should be corrected to minimize the “bump” of transfer when the synchronization
input of the VSC is switched from the PSL to the PLL. The angle θc can be derived from
the power-angle equation in (3.27) with the p.u. expresseion
θc = arcsin(

P Xc
) ≈ arcsin(P Xc )
Uf V

(3.38)
35

Chapter 3. Control Methods for VSC-HVDC Systems
where Xc is the reactance of the phase reactor. Uf and V are the magnitudes of the filterbus voltage and the VSC voltage, which are close to 1.0 p.u. in normal operation. Fig. 3.12
shows the block diagram of the backup PLL, which is essentially the same as the PLL for
vector current control in Fig. 3.6 but with angle correction.
However, if the synchronization input of the VSC is switched to the PLL, the PSL
still controls the angle output θv . At the time when the synchronization input is switched
back from the PLL to the PSL, θv might not necessarily be the same as θPLL . This most
likely happens during severe ac-system faults, where the PSL tends to increase θv rapidly.
In Fig. 3.12, a bumpless-transfer scheme is proposed to handle the above situation. Once
ˆ
there is an error between θv and θv , the scheme integrates the error and feeds back a term
to cancel the integral action of the PSL. From the scheme, the following relationship is
established
γp ˆ
kp
θv =
(Pref − P ) +
θv .
(3.39)
s + γp
s + γp
ˆ
If γp is sufficiently large, (3.39) yields θv ≈ θv . In other words, once the synchronization

input of the VSC is switched from the PSL to the PLL, the bumpless-transfer scheme will
make sure that the output of the PSL tracks the output of the PLL to avoid the transfer
“bump” when the synchronization input is switched back to the PSL. 1
In the following, the anti-windup issue of the alternating-voltage controller is discussed. With power-synchronization control, the alternating-voltage controller controls
ref
the d component of the voltage reference vd [cf. (3.32)], which is essentially the magniref
tude of the VSC voltage. However, vd might be limited by either of the two fundamental
limitations of the VSC:
1. Converter-current limitation. This mainly happens during ac-system faults as it has
been described above.
ref
2. Modulation-index limitation. Occasionally, the magnitude of vd is above the maximum value of voltage that the VSC can produce. Thus, the control has to limit the
ref
magnitude of vd to prevent over-modulating the valve. Modulation-index limitation tends to be reached more often with weak-ac-system connections, where a
relatively higher VSC voltage magnitude is necessary to keep the filter-bus voltage
to be nominal. Usually, the tap-changer of the converter transformer is used to prevent modulation-index limitation. However, it can still occur with transient voltage
swings, or if the tap-changer of the converter transformer has reached the limit of
the tapping range.

The principle of the anti-windup scheme of the alternating-voltage control is basically the
same as the bumpless-transfer scheme of the PSL, as shown in Fig. 3.13. Whatever the
1

Interestingly, this is essentially the same problem as transient stability phenomena in ac systems where
synchronous machines accelerate the speed of their rotors during ac-system faults. Thus, the well-known
equal-area criterion might be viewed as an integrator-windup problem. Unfortunately, no bumpless-transfer
or anti-windup scheme can ever be designed for a real machine.

36

3.3. Power-synchronization control

Grid-q
PLL-q
PSL-q
PSL-d
Angle
correction

v

PLL-d

uf

θc
E

Grid-d

Fig. 3.11 Converter dq frames created by the PSL and the PLL. The two dq frames differ by the
angle θc .

Angle
correction

θc
u

c
f

Im{u cf }

PLL
Kp +

K

+

θ PLL +

PLL
i

'
θ PLL

s

ωref t

Backup-PLL
Pref +

P

kp
s
PSL

ˆ
θv +

θv

+

+

ωt

+
-

+

γp
s

Fig. 3.12 Bumpless-transfer scheme for switching the synchronization input of the VSC.

37

Chapter 3. Control Methods for VSC-HVDC Systems
Currrent
limitation
U ref +

Uf

ku ∆V +
s
+
AVC

Modulation
index
limitation

ref
vd

+

ˆ ref
vd

+
-

c
V0 − H HP ( s )icd

+

γE
s

Fig. 3.13 Anti-windup scheme for the alternating-voltage controller.

ref
reason of the voltage limitation, the anti-windup scheme integrates the error of vd and
vd , and feeds back a term to cancel the integrator of the alternating-voltage controller. By
ˆref
ref
choosing γE large enough, the output vd tracks the limited d component reference vd .
ˆref
Once the voltage limitation lifts, the alternating-voltage control gets a smooth re-start.
In the following, some simulation results from PSCAD/EMTDC are shown to demonstrate the performance of a VSC-HVDC link using power-synchronization control. The
VSC-HVDC link is connected to a very weak ac system with SCR = 1.0. The converter
station at the other side of the VSC-HVDC link controls the direct voltage, and the converter is assumed to be connected to a strong system. It is also assumed that the bandwidth
of the direct-voltage controller at the other station is high enough such that the variation
of the dc-link voltage has negligible effects on the dynamics of the converter connected
to the weak ac system. The issue of interconnection of two very weak ac systems will be
discussed in Chapter 5, where the direct-voltage control plays a central role.
Fig. 3.14 shows the converter deblocking and power ramping-up process. In the
lower plot, it is shown that the VSC uses the PLL as its synchronization input before the
converter is deblocked. The output of the PSL, θv , is forced to be equal to the output of

the PLL, θPLL . After the converter is deblocked at 0.1 s, the PSL takes over the synchronization input of the VSC, and ramps up the active power to 0.86 p.u. The PLL tracks
the PSL as the power is ramping up, and is exactly identical to the PSL when the system
reaches the steady state.
In Fig. 3.15, a three-phase ac-system fault with duration 0.2 s is applied close to
the filter bus. By detecting the ac-system fault (by current limitation or magnitude drop of
the filter-bus voltage), the control system switches the synchronization input of the VSC
to the backup-PLL. In the middle plot, it is shown that the PSL initially quickly increases
the angle output θv , but after the back-up PLL takes over the synchronization input of the
VSC, the bumpless transfer scheme re-directs the output of the PSL to follow the output
of the PLL. Once the ac-system fault is cleared, the PSL takes over the synchronization
input and brings the power back to the pre-fault level. The PLL again tracks the PSL and

38

1
0

f

U (p.u.)

3.3. Power-synchronization control

-1
0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.8

1

P (p.u.)

1
0.5
0

θv, θ´ PLL (rad)

0
1.5
1

PSL

0.5
PLL

0
0

0.2

0.4

0.6
time (sec)

Fig. 3.14 Converter deblocking and power ramping up of the VSC-HVDC link using powersynchronization control. Upper plot: three-phase filter-bus voltages. Middle plot: active
power of the VSC-HVDC link. Lower plot: outputs of the PSL and the PLL.

equals the PSL in the steady state.
The lower plot of Fig. 3.15 demonstrates the anti-windup scheme of the alternatingvoltage controller. During the fault period, the d component of the VSC voltage reference
ref
vd is limited by the current controller. Similar to the bumpless-transfer scheme of the
PSL, the anti-windup scheme of the alternating-voltage controller also re-directs the voltage output of the AVC to follow the limited voltage reference. After the fault is cleared,
the AVC takes over the voltage control.
Fig. 3.16 shows the actions of the current controller during the fault. After the threephase ac-system fault is applied at 0.1 s, the modulus of the current reference [|iref | =
(iref )2 + (iref )2 ] reaches the current limit Imax . The control system seamlessly switches
q
d

to the control law (3.12), where the limited iref and iref become the inputs to the current
q
d
controller. Only a short current spike is observed on the valve current |ic | at the fault
occurrence stage, which usually does no harm to the converter valve. After detecting the
faults, the current controller reduces the valve current to half of the maximum load current
Imax or any other desired values to minimize the short-circuit current contribution to the
ac system. After the fault is cleared at 0.3 s, the current limitation lifts, and the voltagevector control law is back to (3.32).
As shown by the time simulation in this section, power-synchronization control,
in fact, uses vector current control and the PLL during severe ac-system faults, since
the power-synchronization law is not applicable. A question is naturally raised: if vector
39

Chapter 3. Control Methods for VSC-HVDC Systems

U (p.u.)

1

f

0

θv, θ´ PLL (rad)

-1
0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

3
2

PSL

PLL

1
0
0

0.2

0.4

ref

vd (p.u.)

1.5
1

Controller vref
d

ref

0.5
0
0

Limited vd
0.2

0.4
time (sec)

Fig. 3.15 Bumpless-transfer and anti-windup schemes of power-synchronization control during
a three-phase ac-system fault. Upper plot: three-phase filter-bus voltages. Middle plot:
outputs of the PSL and the PLL. Lower plot: d component of the VSC voltage reference.

P (p.u.)

1
0.5

ref ref
id , iq (p.u.)

0
0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

1

ref
id

0
-1
0

ref
iq

| ic| (p.u.)

2
1
0
0

time (sec)

Fig. 3.16 Current limitation of power-synchronization control during a three-phase ac-system
fault. Upper plot: active power of the VSC-HVDC link. Middle plot: dq components
of the current reference. Lower plot: modulus of the converter current.

40

3.3. Power-synchronization control
current control does not work well with weak-ac-system connections, could instability
occur during ac-system faults?
As will be shown in Chapter 5, the stability of the current controller deteriorates
with weak-ac-system connections but to an acceptable level. What is not acceptable is
using current references to control the active power and the alternating voltage. The justification of using power-synchronization control is that it is meaningless to just control the
current during normal operation. However, it is enough during severe ac-system faults.
As shown in Fig. 3.16, vector current control works fairly stable during ac-system faults
given the fact that the ac system only has SCR = 1.0.
VSCs using power-synchronization control basically emulate the operation of a
synchronous machine. Therefore, it contributes with short-circuit capacity to the ac system at the PCC. In order to obtain a simple estimate of the impact of the VSC on the
short-circuit capacity of ac system, the effects of the alternating-voltage control and the
ac filter can be disregarded. Doing so, the short-circuit capacity of the ac system including
the VSC at the PCC can be expressed as
2
Uf 0 (Ln + Lc )
.
Sac =
ω1 Ln Lc

(3.40)

However, VSCs do not necessarily increase short-circuit currents to the ac system during
ac-system faults thanks to the current limitation function.

3.3.4 Negative-sequence current control
In reality, there is often phase unbalance in the three-phase voltages of the ac system. The
phase unbalance may appear in the steady state due to unevenly connected loads in the
ac system, but more often caused by unbalanced voltage dips [34]. The zero-sequence
voltage in the ac system cannot pass through the converter transformer where Y − ∆ type
connection is commonly used, but the negative-sequence voltage can create unbalanced
converter currents, which may overload the converter and produce double-grid-frequency
ripples on the direct voltage [58]. In this section, a negative-sequence current controller is
proposed to mitigate the phase-unbalance problem.
Fig. 3.17 shows the control block diagram of the proposed negative-sequence current controller, which is principally the same as the positive-sequence current controller in
Fig. 3.5 and Fig. 3.10 but with negative-sequence variables. The outputs of the negativeref
ref
ref
sequence current controller va− , vb− and vc− are directly added to the outputs of the
ref
ref
ref
positive-sequence current controller va , vb and vc . Sometimes, the two current controllers together are referred to as dual vector current controller [59, 60]. The minus sign
in the subscripts of the variables in Fig. 3.17 represents the negative-sequence quantities.
However, the dq frame of the negative-sequence current controller rotates in the opposite
direction to the converter dq frame. Therefore, −ωt is used as the synchronization input.
The proposed control law of the negative-sequence current controller is expressed as
41

Chapter 3. Control Methods for VSC-HVDC Systems
u f − ic −
ref
id −
ref
iq −

ref
vα −

ref
vd −

Innerref
vq −
current
controller

αβ
dq

ref
vβ −

abc

αβ

ref
va −
ref
vb −
vcref


− ωt

Fig. 3.17 Negative-sequence current controller.
ref
v− = αc− Lc iref − ic + jω1 Lc ic + uc −

c−
c−
f

(3.41)

where αc− is the desired bandwidth of the negative-sequence current controller. Eq. (3.41)
is essentially the same expression as (3.12) but with negative-sequence variables. The
negative-sequence current reference iref is usually set to zero if there is no other special

purpose.
Thus, the principle of the negative-sequence current controller is to create phase
unbalance on the VSC voltages so as to minimize the negative-sequence current flowing
through the converter. There are different ways to obtain negative-sequence variables. The
following is a common approach [61].
In a stationary reference frame, if the zero-sequence component is disregarded, a
complex vector ys (t) can be expressed as
ys (t) = Y+ ejωt + Y− e−jωt
s
y+ (t)

(3.42)

s
y− (t)

s
where Y+ is the magnitude of the positive-sequence vector y+ (t), while Y− is the mags
nitude of the negative-sequence vector y− (t). The quantity ω is the angular speed of the
s
rotating vector y+ (t). It follows from (3.42) that a voltage vector delayed with a quarter
period (Tp /4) and multiplied by j is

jys (t − Tp /4) = j[Y+ ejω(t−Tp /4) + Y− e−jω(t−Tp /4) ]

(3.43)

jys (t − Tp /4) = Y+ ejωt − Y− e−jωt .

(3.44)

or
s
Accordingly, the negative sequence vector y− (t) can be obtained from (3.42) and (3.44)
as
1
s
y− (t) = Y− ejωt = [ys (t) − jys (t − Tp /4)].
(3.45)
2

Fig. 3.18 demonstrates the effect of the negative-sequence current controller by time
simulations in PSCAD/EMTDC. At 0.1 s, a negative-sequence voltage source with 10%
magnitude of the positive-sequence voltage is added on the ac source E, which makes the
42

3.4. Summary

Uf (p.u.)

1
0
-1
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.25

0.3

0.35

0.4

0.2
0.25
time (sec)

0.3

0.35

0.4

v ref, v ref (p.u.)

0.1

ref

vq-

d-

q-

0
-0.1
ref

-0.2
0

vd0.05

0.1

0.15

0.05

0.1

0.15

0.2

0.6

c

| i | (p.u.)

0.8

0.4
0

Fig. 3.18 Effects of negative-current control in the steady state. Upper plot: three-phase filter-bus
voltages. Middle plot: dq components of the negative-sequence VSC voltage reference.
Lower plot: modulus of the converter current.

converter current unsymmetrical. The lower plot in Fig. 3.18 shows the modulus of the
converter-current vector in the converter frame that oscillates with double-grid frequency
caused by the negative-sequence component. If the operating point of the VSC is close to
the maximum valve-current limit, this phase unbalance might overload the converter. At
0.3 s, the negative-sequence current controller is activated, which successfully removes
the negative-sequence current component. Fig. 3.19 and Fig. 3.20 show the effect of the
negative-sequence current controller during a single-line-to-ground fault in the ac system.
With the negative-current controller activated, as shown in Fig. 3.20, the unbalance of
valve currents is much reduced.
It should be noted that, since unbalanced VSC voltage need to be produced in order
to reduce the negative-sequence current, the magnitudes of some phases of the VSC voltage might go above the modulation-index limitation. This is the reason why there is still
remaining negative-sequence component on the converter current during the unbalanced
fault even with the negative-sequence current controller applied in Fig. 3.20.

3.4 Summary
In this chapter, two existing control methods, i.e., power-angle control and vector current
control, for VSC-HVDC systems are reviewed, and a novel control method, i.e., powersynchronization control, is proposed. Power-angle control is simple and straightforward
to implement. However, power-angle control has no general means to damp resonances in
43

Chapter 3. Control Methods for VSC-HVDC Systems

Uf (p.u.)

1
0

ref ref
id , iq (p.u.)

-1
0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

1
ref

id

ref

iq

0
-1
0

0.2

0.4

0.2

0.4

| ic| (p.u.)

2
1
0
0

time (sec)

Fig. 3.19 Single-line-to-ground fault without the negative-sequence current controller applied.
Upper plot: three-phase filter-bus voltages. Middle plot: dq components of the current
reference. Lower plot: modulus of the converter current.

Uf (p.u.)

1
0

ref ref
id , iq (p.u.)

-1
0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

1

-1
0

ref
id

ref
iq

0

0.2

0.4

0.2

0.4

| ic| (p.u.)

2
1
0
0

time (sec)

Fig. 3.20 Single-line-to-ground fault with the negative-sequence current controller applied.

44

3.4. Summary
the ac system and the converter may get over current during ac-system faults. Vector current control is the most popular control method used for grid-connected VSCs. However,
for HVDC applications, vector current control has poor performance for weak-ac-system
connections. VSCs using power-synchronization control basically resemble 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.

45

Chapter 3. Control Methods for VSC-HVDC Systems

46

Chapter 4
Dynamic Modeling of AC/DC Systems
This chapter discusses the dynamic modeling issue for ac/dc systems. In Section 4.1, a
new concept, i.e., the so-called Jacobian transfer matrix, is introduced for modeling of ac
systems. By the proposed modeling concept, an synchronous ac system is viewed upon as
one multivariable feedback control system where the Jacobian transfer matrix is defined
as the controlled process. In Section 4.2 and Section 4.3, the Jacobian transfer matrix concept is applied to model grid-connected VSCs using power-synchronization control and
vector current control. In Section 4.4, the Jacobian transfer matrix concept is extended to
modeling of dc systems. The properties of the Jacobian transfer matrix are summarized in
Section 4.5. The major results of this chapter are summarized in Section 4.6. Some results
of this chapter are included in [62, 63].

4.1 Jacobian transfer matrix for ac-system modeling
In this section, a new concept for dynamic modeling of ac systems is described. In Section 4.1.1, the existing concepts for power-system stability and dynamic modeling are reviewed. In Section 4.1.2, the Jacobian transfer matrix is introduced as a unified dynamicmodeling concept for ac systems.

4.1.1 Power-system stability and dynamic modeling
Power-system stability has been a major subject for power engineering for many years.
The definitions of power-system stability and dynamic-modeling techniques have continuously been extended with the evolvement of power systems [32, 64–66].
In the early days, the stability problem was one of maintaining synchronous operation. Thus, transient stability was the most classical stability problem of concern. Transient stability is defined as the ability of a power system to maintain synchronism when
subjected to severe transient disturbance [67]. Transient stability is basically an issue of
energy balance of the rotors of synchronous machines, where the equal-area criterion is
47

Chapter 4. Dynamic Modeling of AC/DC Systems
often used to describe the mechanism.
Small-signal stability is another stability phenomenon related to synchronous machines. In today’s power systems, small-signal stability is largely a problem of insufficient
damping of oscillations [32]. The fast excitation control has been the traditional culprit
for reduced or even negative damping torque [68]. Insufficient damping may also be a
consequence of large power systems interconnected by weak ties [69]. The power-system
stabilizer (PSS) are generally considered as an easy and effective solution for small-signal
stability issues, at least for the local-mode oscillations [70]. HVDC systems and flexible
ac transmission systems (FACTS) have also been shown to be effective in damping interarea oscillations [4, 28, 71]. For small-signal stability, modal analysis based on linearized
ac-system models is the most popular tool for analyzing the phenomenon and evaluating
various damping solutions [32].
The above two forms of stability are mainly related to the dynamic behavior of
synchronous machines. Consequently, they are often referred to as “rotor-angle stability”.
Voltage stability was identified as a problem in the 1970s. Voltage stability is defined as the ability of a power system to maintain acceptable voltage at all buses in the
system under normal operating conditions and after being subjected to a disturbance [67].
Voltage stability is often caused by lack of reactive-power support in heavy loaded areas [66]. In contrast to rotor-angle stability, voltage stability is more related to the dynamic behavior of loads and their interactions with the ac system. Dynamic load modeling
is generally believed to be of central importance for voltage-stability study [72, 73].
For voltage-stability analysis, an interesting methodology was proposed, i.e., the
Jacobian matrix. The Jacobian matrix was originally applied for solving power equations
using the Newton-Raphson algorithm in power-flow programs. It has been found that the
singularities of the Jacobian matrix have close relationships with voltage-stability problems. For instance, it can easily be shown that the critical points on the well-known P −V
curves exactly correspond to the operation conditions when the Jacobian matrix becomes
singular. Based on the Jacobian matrix, a modal analysis technique that is similar to the
one used for small-signal analysis was developed [74]. However, the applications of the
Jacobian matrix and other related methods in voltage-stability analysis have been questioned for their mathematical foundations [75]. Voltage instability, like any other kind
of instability, is a dynamic phenomenon. A rigorous analysis, therefore, would require a
dynamic system formulation and the application of appropriate dynamic criteria, but the
Jacobian matrix is merely a static matrix that describes the power-flow deviations related
to voltage and angle variations in the system at a certain operating point.
One argument to support the Jacobian matrix and the related methods is that the Jacobian matrix might be viewed as a dynamic description of the ac system if the frequency
range of the stability of concern is “quasi-static” [76]. As it will be shown later by the
Jacobian transfer matrix modeling technique introduced by this thesis, such an explanation does make sense. However, the question is, if the Jacobian matrix can be viewed as
48

4.1. Jacobian transfer matrix for ac-system modeling
a dynamic description of the ac system that is valid in the “quasi-static” frequency range,
then what is the dynamic description of the ac system that is valid in the whole frequency
range, and how are they related mathematically ?
The subsynchronous resonance (SSR) problem appeared also in the 1970s. The first
SSR problem was experienced in 1970 resulting in the failure of a turbine-generator shaft
at the Mohave plant in southern California, USA [32]. In the beginning, it was believed
that SSR occurs only in series-compensated transmission systems, where the long shaft of
the rotor of thermal units may interact with the series capacitor at a frequency lower than
the fundamental frequency. However, it was later discovered that similar interactions can
also occur with an HVDC converter [77]. Therefore, subsynchronous torsional interaction
(SSTI) is suggested to be a more general definition [78].
For rotor-angle and voltage stability, the phasor theory is the major mathematical
tool for dynamic modeling and analysis. With the phasor approach, electromagnetic transients of the ac network are neglected. Such a simplification is acceptable for rotor-angle
and voltage stability but not for SSTI, since the dynamic frequency range of SSTI is
much higher than rotor-angle and voltage stability. For analyzing SSTI, the electromagnetic transients of the ac network have to be properly represented. Thus, the space-vector
theory is commonly applied to formulate the dynamic model of the ac network.
There are predominantly two methods for SSTI analysis: eigenvalue analysis and
frequency scanning [22, 79, 80]. Eigenvalue analysis is a rigorous tool for power-system
modeling, since the whole power system is modeled as one single state-space representation. The computation of eigenvalues and eigenvectors is an excellent method of providing
crucial information about the nature of the power system [21]. The major drawback of the
eigenvalue analysis is its complexity once the order of the system is high. The frequencyscanning method is based on the complex-torque concept [81]. The risk of SSTI can thus
be evaluated by plotting the damping-torque curve of the generator in the frequency domain. Although frequency scanning is a frequency-domain concept, the damping-torque
curve is also possibly obtained from time-simulation programs by perturbing the rotor
speed with a small sweeping-frequency sinusoidal signal. As the aforementioned reason,
the time-simulation program used for frequency scanning should be of electromagnetictransient type, such as EMTP or PSCAD/EMTDC.
In the foreseeable future, the main challenges for power-system stability analysis
and dynamic modeling are imposed by high power-electronic devices, such as HVDC
systems and FACTS if the number of such devices is considerably increased in power
systems. On the one hand, the ratings of those devices are often high enough to have a
significant impact on all the above-mentioned stability phenomena. On the other hand,
their dynamic frequency range are much higher than the traditional power-system components. They might interact with power systems and with each other in a more complex
way.
From the above brief review of power-system stability and the related dynamic49

Chapter 4. Dynamic Modeling of AC/DC Systems
Generator

K1

K5
K2

FACTS

HVDC
system

Jacobian transfer
matrix J (s)
K3

Constant
power load K
4

Induction
motor

Fig. 4.1 An ac network connected to various input devices.

modeling techniques, it can be concluded:
1. Due to historical and practical reasons, power-system stability has traditionally
been defined mainly based on physical phenomena, but the mathematical relationships between different forms of stability are not rigourously clarified. The dynamic
modeling techniques are also versatile for various forms of stability.
2. For any dynamic system, the stability of the system is usually closely related to
feedback control. However, power-system stability has rarely been interpreted from
the feedback-control point of view.
3. High power-electronic devices might interact with the ac system and each other
in a more complex way than conventional power components. New modeling and
analysis techniques are required to meet the challenges in the future.
In the next sub-section, the Jacobian transfer matrix is introduced as a unified ac-system
dynamic modeling technique to address these issues.

4.1.2 Feedback-control view of power systems
The fundamental idea of the Jacobian transfer matrix modeling is that a synchronous
power system is modeled as one multivariable feedback-control system, where the feedback controllers and the controlled process of the power system are explicitly defined.
As shown in Fig. 4.1, for a normal power system, there are various power components
50

4.1. Jacobian transfer matrix for ac-system modeling
r
+

K (s )

J (s )

y

-

Fig. 4.2 Feedback-control view of a power system.

connected to it, such as generators, induction motors, HVDC systems, FACTS devices,
and loads, etc. The controllers, as will be defined later, are pulled out from those power
components and form a controller transfer matrix as


K1 (s)




K2 (s)




K(s) = 

..


.


Kn (s)

and the rest of the power system, i.e., the ac network and the electrical parts of those
power components form another transfer matrix J(s). A synchronous power system thus
is modeled by the feedback-control system shown in Fig. 4.2. The transfer matrix J(s) is
called Jacobian transfer matrix in this thesis, i.e., the Jacobian transfer matrix is defined
as the controlled process of a power system.1 With the proposed modeling concept, the
stability of a power system is uniquely defined as the stability of the closed-loop system
formed by the controller transfer matrix K(s) and the Jacobian transfer matrix J(s). Some
terminologies related to the proposed modeling concept are defined in below:
Input device: Any power component connected to a power system which has a feedbackcontrol property. The aforementioned synchronous generators, induction motors,
HVDC systems, and FACTS are all treated as input devices. An input device consists of two parts: the electrical part and the controller.
AC network: The passive power components in a power system, such as transmission
lines, transformers, line inductors, shunt capacitors, resistive-inductive-capacitive
(RLC) loads, etc.
Controller transfer matrix: A transfer matrix formed by all the controllers from each individual input device. The controller transfer matrix is usually a diagonal or blockdiagonal transfer matrix unless there is cross controls between the input devices.
The controller as defined in this thesis is a generalized terminology. For example,
the rotors of synchronous generators and induction motors are treated as controllers
even though physically they are not intentionally implemented controllers as such.
1

The name “Jacobian ” will be justified in the following sections.

51

Chapter 4. Dynamic Modeling of AC/DC Systems

Pref +

-

kp

∆θ v

s
PSL
U ref +

-

ku
s
AVC

∆V
V0

∆P
Jacobian
transfer
matrix
J (s )

∆U f

Fig. 4.3 Closed-loop system for grid-connected VSCs using power-synchronization control

Jacobian transfer matrix: A transfer matrix represents the ac network and the electrical parts of the input devices. One exception is vector control of grid-connected
VSCs, which is also formed into the Jacobian transfer matrix for mathematical convenience even though they are not electrical.
In the next few sections, the Jacobian transfer matrix concept will be applied for modeling of grid-connected VSCs connected to some simplified ac-network configurations,
where the VSC is the only input device of the ac system. In Chapter 6, some other input
devices, such as synchronous generators and induction motors, are also modeled to create
a complete linear model for an island system with several input devices.

4.2 Grid-connected VSCs using power-synchronization control
In this section, the Jacobian transfer matrix is developed for grid-connected VSCs using
power-synchronization control. Fig. 4.3 shows the closed-loop system for grid-connected
VSCs using power-synchronization control, i.e., the Jacobian transfer matrix is the controlled process which represents the ac circuit of the VSC and the ac network.

4.2.1 Impedance-source neglecting the ac capacitor at the filter bus
Fig. 4.4 shows the main circuit of a VSC connected to an ac network represented by a
simple impedance source. Lc and Rc are the inductance and resistance of the phase reactor
of the VSC, and Ln and Rn are the inductance and resistance 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. E0 , Uf 0 , and V0
are their corresponding voltage magnitudes. The ac source, which is a constant-frequency
stiff voltage source, is used as the voltage reference, and the phase angles of uf and v are
52

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

E0∠00

Ln

Rn

P, Q
in

+

Lc

U f 0∠θ u 0 Rc

V0∠θv 0
ic

+

E

uf

-

+
v

-

Cf

-

Fig. 4.4 Main circuit of a VSC connected to an impedance ac source.
P, Q
E = E∠ 0

R

X = ω1L

I

V = V∠ θ

Fig. 4.5 Equivalent circuit by neglecting the ac capacitor for angle-stability analysis.

θu0 and θv0 respectively. P and Q are the active and reactive powers from the VSC to the
ac system. The quantity ic is the current vector of the phase reactor, and in is the current
vector to the ac source. At the dc side, the VSC is assumed to connect to a stiff directvoltage source. The ac capacitor at the filter bus of the VSC is neglected in the analysis in
this subsection. Such a simplification is useful for explaining some basic concepts of the
proposed modeling technique in an analytical way. Especially, it is easier to demonstrate
how angle and voltage stability defined in the classical power-system theory can be cast
into the Jacobian transfer matrix modeling frame work.
Stability analysis by the phasor theory
In the following, the stability of the small power system is analyzed by the classical phasor
theory. Fig. 4.5 shows the equivalent circuit of Fig. 4.4 by neglecting the ac capacitor,
¯
¯
where E and V represent the phasor of the ac source and the VSC. R and X are the
equivalent resistance and reactance between the ac source and the VSC. Neglecting the
resistance, the angle stability of the equivalent system is given by the well-known powerangle equation
EV
P =
sin θ.
(4.1)
X
From (4.1) it follows that the power that can be transmitted to/from the VSC is limited
by the fact that −1 ≤ sin θ ≤ 1. Fig. 4.6 shows the power-angle curve with different line
reactances.
For voltage-stability analysis, suppose that the VSC controls the reactive power to
the ac system. Neglecting the resistance, the reactive power Q can be expressed by the
53

Chapter 4. Dynamic Modeling of AC/DC Systems
4
3.5

X = 0.3 p.u.

3

P (p.u.)

2.5
2
X = 0.6 p.u.
1.5
X = 1 p.u.

1
0.5
0
0

50

100
θ (deg.)

150

200

Fig. 4.6 Power-angle curves with different line reactance.

following expression:
V 2 − EV cos θ
.
X
By eliminating the angle θ in (4.1) and (4.2), the following equation is derived:
Q=

V2

2

− 2QX + E 2 V 2 + X 2 P 2 + Q2 = 0.

(4.2)

(4.3)

To get real solutions of V 2 in (4.3), the required condition is
2QX + E 2

2

− 4X 2 P 2 + Q2 ≥ 0.

(4.4)

If (4.1) and (4.2) are substituted back into (4.4), as
V ≥

E
.
2 cos θ

(4.5)

If the condition (4.5) is fulfilled, two solutions of V in (4.3) are obtained
V =

E2
+ QX ±
2

E4
− X 2 P 2 + XE 2 Q.
4

(4.6)

The relation in (4.6) is plotted in Fig. 4.7 where
tan φ = −

Q
.
|P |

(4.7)

These plots are commonly called P − V curves, or “nose curves”. Besides the P − V
54

4.2. Grid-connected VSCs using power-synchronization control

tanφ= -0.5
1
tanφ= -0.2
tanφ= 0

0.8

V/E

tanφ= 0.4
0.6

tanφ= 1.5

0.4

0.2

0
0

0.2

0.4

0.6

0.8

1

|P|X/E2

Fig. 4.7 P − V curves with different tan φ of the load.

curve, another way to determine the voltage-stability limit is by the singularity of the
power-flow Jacobian matrix [74]. Jacobian matrix J is expressed as



∆P
∆Q



=

JPθ JPV
JQθ JQV
J




∆θv
∆V
V




(4.8)

where ∆P , ∆Q, ∆θv , and ∆V are the incremental changes in the active power, the reactive power, the phase angle, and the magnitude of the bus respectively. JPθ , JPV , JQθ , JQV
are the partial derivatives of the power-flow equations with respect to the voltage angle
and magnitude of the bus.
For the simple ac system in Fig. 4.5, the Jacobian matrix J at the VSC bus can be
derived analytically from the active-power and reactive-power expressions (4.1) and (4.2).
Accordingly,
∂P
EV cos θ
=
∂θ
X
∂P
EV sin θ
=
V =
∂V
X
∂Q
EV sin θ
=
=
∂θ
X
∂Q
2V 2 − EV cos θ
=
V =
.
∂V
X

JPθ =
JPV
JQθ
JQV

(4.9)
55

Chapter 4. Dynamic Modeling of AC/DC Systems
The singularity of the Jacobian matrix J is satisfied if its determinant equals zero, i.e.,
det [J] = JPθ JQV − JPV JQθ = 0

(4.10)

giving
V =

E
.
2 cos θ

(4.11)

Comparing (4.11) to (4.5), it is easily found that the operating conditions where the Jacobian matrix becomes singular exactly correspond to the “critical voltage” points on the
P − V curves in Fig. 4.7.
Stability analysis by the Jacobian transfer matrix approach
In the next, the ac system and the VSC in Fig. 4.4 is modeled by the proposed Jacobian
transfer matrix concept. As mentioned before, VSCs have two feasible operation modes,
i.e., alternating-voltage control and reactive-power control. For the two operation modes,
the process models can be written in the following input-output form:






∆P
∆Uf
∆P
∆Q



=


=

JPθ (s)

JPV (s)

JUf θ (s) JUf V (s)
JPU (s)

JPθ (s) JPV (s)
JQθ (s) JQV (s)
JPQ (s)







∆θv
∆V
V0

∆θv
∆V
V0







(4.12)

where the definitions of ∆P , ∆Q, ∆θv , and ∆V , in fact, are the same as previously
defined for the power-flow Jacobian matrix (4.8), and ∆Uf is the incremental change of
the voltage magnitude of the filter bus. The transfer matrices JPU (s) and JPQ (s) are the
Jacobian transfer matrices for the two operation modes. JPQ (s) has the exact input and
output variables as the Jacobian matrix in (4.8). However, a fundamental difference is that
Jacobian transfer matrix is a dynamic description of the ac system in all frequency range,
while Jacobian matrix is only valid in the “quasi-static” frequency range.
Appendix B.1 gives the detailed procedure for deriving the transfer functions in
(4.12) by applying the space-vector theory. The six transfer functions of JPU (s) and
JPQ (s) generally have the following form
Jxx (s) =
56

a0 s2 + a1 s + a2
(sL + R)2 + (ω1 L)2

(4.13)

4.2. Grid-connected VSCs using power-synchronization control
where the a-coefficients are summarized in Table 4.1. The k parameters are defined as
k1 = V0 Uf 0 cos (θv0 − θu0 )
k2 = V0 Uf 0 sin (θv0 − θu0 )
k3 = E0 V0 cos θv0

k4 = E0 V0 sin θv0 .

(4.14)

Table 4.1 Coefficients of the transfer functions in the Jacobian transfer matrices.

a0
JPθ (s)

a1

L
ω1
R
ω1

(k3 − k1 )
(k3 − k1 ) − Lk2

a2
a0
JQθ (s)

ω1 Lk3 − Rk4
L
ω1
R
ω1

a1

(k4 − k2 )
(k4 − k2 ) − Lk1

a2
a0
JUf θ (s)

ω1 L (k4 − 2k2 ) + R (k3 − 2k1 )
−LLn k2 /Uf 0

a1

−RLn k2 /Uf 0

a2
a0

ω1 Lk4 + Rk3

a0
JQV (s)

a1

L
ω1
R
ω1

a2

JPV (s)

2
−ω1 LLn k2 /Uf 0 − RLn k1 /Uf 0

L
ω1
R
ω1

a1

(k4 − k2 )
(k4 − k2 ) + Lk1
(k1 − k3 )
(k1 − k3 ) − Lk2

a2
a0

LLn k1 /Uf 0

a1

RLn k1 /Uf 0

a2

JUf V (s)

ω1 L (2k1 − k3 ) + R (k4 − 2k2 )

2
ω1 LLn k1 /Uf 0 − RLn k2 /Uf 0

It should be noted that the output variables of the transfer functions derived for
the Jacobian transfer matrices are the active power, the reactive power and the voltage
magnitude at the filter bus. To compare with the results from the phasor analysis, it is
necessary to substitute Uf 0 = V0 and θu0 = θv0 into the coefficients in Table 4.1, i.e.,
the condition for the phasor analysis is considered as a special case that the filter bus is
identical to the VSC bus.
57

Chapter 4. Dynamic Modeling of AC/DC Systems
By substituting Uf 0 = V0 and θu0 = θv0 into the transfer functions of JPQ (s) , and
further assuming R = 0 and s = 0, the four transfer functions JPθ (s) , JPV (s) , JQθ (s),
and JQV (s), are identical to the four elements of the Jacobian matrix J defined in (4.8).
In other words, the power-flow Jacobian matrix is the static form of the Jacobian transfer
matrix JPQ (s) that is valid in the “quasi-static” frequency range.
Eq. (4.13) shows that all the six transfer functions have a pair of resonant (complex)
poles
R
p1,2 = − ± jω1 .
(4.15)
L
and two zeros. The resonant poles will be further discussed in the next subsection. In
this subsection, the transmission zeros of JPU (s) and JPQ (s) are analyzed, since their
locations have close relationships with angle and voltage stability defined by the phasor
approach.
The computation of the transmission zeros for a general multi-input multi-output
(MIMO) process is a complex issue [82]. However, for a square matrix, the transmission
zeros can be obtained simply by its determinant, i.e., the transmission zeros of JPU (s)
are the values of s = z that satisfy
det [JPU (s)] = JPθ (s) JUf V (s) − JPV (s) JUf θ (s) = 0.

(4.16)

If the resistance R is neglected, the solutions to (4.16) are given by
z1,2 = ±ω1

E0 cos θu0
.
Uf 0 − E0 cos θu0

(4.17)

Graphically, the locations of the transmission zeros of JPU (s) can be divided by the following borders:
• The border where JPU (s) gets zeros at the origin. This is equivalent to
E0 cos θu0 = 0

(4.18)

giving θu0 = ±90◦ .
• The border where JPU (s) gets zeros at infinity. This is equivalent to
Uf 0 − E0 cos θu0 = 0

(4.19)

giving
θu0 = ± arccos

Uf 0
E0

.

(4.20)

• The border where JPU (s) gets zeros at ±ω1 . This is equivalent to
E0 cos θu0
=1
Uf 0 − E0 cos θu0
58

(4.21)

4.2. Grid-connected VSCs using power-synchronization control
100
80
Origin border
60

20

±ω1 border

Infinity border

0

θ

u0

(deg.)

40

-20
-40
-60
Origin border
-80
-100
0

0.5

1
U /E
f0

1.5

2

0

Fig. 4.8 Locations of the transmission zeros of JPU (s) .

giving
θv0 = ± arccos

Uf 0
2E0

.

(4.22)

The ±ω1 border gives an idea about how much the zeros limit the achievable bandwidth of the control system, even though it is not a “real” border.
From Fig. 4.8, it is easily observed that the transmission zeros of JPU (s) depend very
much on the operating points. Both Uf 0 /E0 and θu0 affect their locations. However, within
moderate voltage levels, e.g., 0.8 ≤ Uf 0 /E0 ≤ 1.2, they are affected mainly by θu0 . The
tendency is that, with higher θu0 , the zeros get closer to the origin border. In control theory,
a process which has right-half plane (RHP) zeros is called non-minimum-phase system.
From a feedback-control perspective, the RHP zero of the process causes an additional
time delay, which imposes a fundamental limitation on the achievable bandwidth of the
control loop [82], i.e., the closed-loop system cannot achieve a higher bandwidth than
the location of the RHP zero. When the RHP zero reaches the origin, i.e., θu0 = ±90◦ ,
it causes 180◦ phase shift even in the steady state, which means that tight control at low
frequencies is not possible [82]. If θu0 is replaced by θv0 in (4.18), it is easily found
that the operating conditions where the transmission zeros of the Jacobian transfer matrix
JPU (s) reach the origin exactly correspond to the angle-stability limit defined by the
phasor approach.
From the above analysis, it can be clearly seen that the ac system is, in fact, a
non-minimum-phase system, where the angle-stability limitation is a consequence of the
non-minimum-phase effect. How can this phenomenon be interpreted physically? A brief
59

Chapter 4. Dynamic Modeling of AC/DC Systems
Imaginary

V'

jωL I '

U'f

Instant
V
jω L I

Uf
∆θ

I'

Slow

I

E
Real

Fig. 4.9 Non-minimum-phase phenomena hidden by the phasor diagram.

analysis of the physical mechanism is given below.
Fig. 4.9 shows the phasor diagram of the ac system, where the resistance is neglected. Now, suppose that the voltage vector of the VSC rotates with the phase angle
∆θv and its magnitude is kept constant during the angle rotation. The current magnitude
¯
¯
|I| naturally increases to |I ′ | in response to such angular rotation, but the voltage magni¯′
¯
tude along the ac line drops instead, e.g., |Uf | < |Uf |. This can be easily observed from
the phasor diagram but it can also be proven by the transfer function JUf θ (s) in (B.28).
However, the phasor diagram hides one important fact, i.e., the voltage change is abrupt,
but the current does not increase instantly due to the inductance of the ac line.
Similar to JPU (s), the transmission zeros of JPQ (s) are the values of s = z that
satisfy
det [JPQ (s)] = JPθ (s) JQV (s) − JPV (s) JQθ (s) = 0.

(4.23)

If the resistance R is neglected, the solutions to (4.23) are given by
z1,2 = ±ω1

2
2E0 Uf 0 cos θu0 − E0
.
2
2
E0 + Uf 0 − 2E0 Uf 0 cos θu0

(4.24)

The locations of the transmission zeros of JPQ (s) can also be divided by the following
borders:
• The border where JPQ (s) gets transmission zeros at the origin. This is equivalent
to
2
2E0 Uf 0 cos θu0 − E0 = 0

(4.25)

giving
cos θu0 =

60

E0
.
2Uf 0

(4.26)

4.2. Grid-connected VSCs using power-synchronization control
• The border where JPQ (s) gets transmission zeros at ±ω1 . This is equivalent to
2
2E0 Uf 0 cos θu0 − E0
=1
2
2
E0 + Uf 0 − 2E0 Uf 0 cos θu0

(4.27)

giving
θu0 = ± arccos

E0
Uf 0
+
2Uf 0 4E0

.

(4.28)

The ±ω1 border gives an idea about how much the transmission zeros limit the
achievable bandwidth of the VSC controller.
From Fig. 4.10, it is easily observed that the locations of the transmission zeros of
JPQ (s) also depend on the operating points, in the same way as for JPU (s). But besides
the dependency on the phase angle, the locations of the transmission zeros of JPQ (s) also
strongly depend on the voltage magnitude of the filter bus. The tendency is that the zeros
are closer to the origin border with higher θu0 and lower Uf 0 /E0 .
For the special condition where the filter bus is identical to the VSC bus, the transmission zeros of JPQ (s) become
s = ±ω1

2
2E0 V0 cos θv0 − E0
.
2
E0 + V02 − 2E0 V0 cos θv0

(4.29)

Accordingly, the origin border of JPQ (s) becomes
cos θv0 =

E0
.
2V0

(4.30)

Comparing (4.30) with (4.5) and (4.11), it is easily found that the operating conditions where the transmission zeros of JPQ (s) reach the origin exactly correspond to the
voltage-stability limit defined by the phasor approach.
By comparing Fig. 4.8 with Fig. 4.10, it can be found that within moderate voltage
levels, the transmission zeros of JPQ (s) are closer to the origin than the transmission
zeros of JPU (s), which implies that the achievable bandwidth of the VSC is higher in
alternating-voltage control mode than in reactive-power control mode.
To interpret the physical mechanism of voltage stability, the phasor diagram in
Fig. 4.9 can still be used. Different from in alternating-voltage control mode, in reactivepower control mode, the voltage magnitude of the VSC is no longer held constant during
the phase-angle rotation. On the contrary, in order to keep the reactive power constant, the
VSC decreases the voltage magnitude following the phase-angle rotation. This behavior
obviously makes the non-minimum-phase effect worse, and the effect also becomes more
dependent on the VSC voltage magnitude.
Remark 1: The above analysis clearly shows the close relationships of the transmission zeros of the Jacobian transfer matrices with angle and voltage stability in power
61

Chapter 4. Dynamic Modeling of AC/DC Systems

80
60

Origin border

40

θu0 (deg.)

20

±ω1 border

0
-20
-40
-60
-80
0

0.5

1

1.5

2

2.5

3

3.5

Uf0/E0

Fig. 4.10 Locations of the transmission zeros of JPQ (s).

systems. It also explains why the singularity of the power-flow Jacobian matrix can be
used as a criterion to determine voltage instability in power systems. But, strictly speaking, the only correct criterion for power-system instability is the poles of the closed-loop
system. A Jacobian transfer matrix having transmission zeros at the origin only indicates
that the process is impossible to be tightly controlled at low frequencies. Instability, however, can only be caused by the feedback control.
Remark 2: For the particular case in this section, the physical reason to the nonminimum phase phenomena are explained. However, non-minimum phenomena cannot
be generalized to the whole power system as a reason to angle and voltage stability, since
zeros are input dependent. For instance, in Chapter 6, it will be shown that the Jacobian
transfer matrix for induction motors only has a left-half plane (LHP) real zero moving
towards the origin as the slip of the induction motor increases.
Remark 3: For the particular case in this section, it has been shown that the powerflow Jacobian matrix is the static form of the Jacobian transfer matrix. However, this
cannot be generalized either. One reason is that the Jacobian transfer matrix includes
the ac network as well as the electrical parts of the input devices, while the power-flow
Jacobian matrix normally only includes the ac network. Another reason is that, as it will
be shown in later chapters, the outputs of the controllers of different input devices are
versatile, while the power-flow Jacobian matrix only has voltage magnitude and phase
angle as inputs.
Remark 4: Based on the analysis in this section, it can be concluded that reactivepower control basically is not a suitable operation mode for VSC-HVDC links connected
62

4.2. Grid-connected VSCs using power-synchronization control
to weak ac systems. Therefore, in the later sections, only the transfer matrix JPU (s) will
be derived, and with a short name J (s).

4.2.2 Impedance-source including the ac capacitor at the filter bus
In this section, the Jacobian transfer matrix will be derived for the main circuit shown
in Fig. 4.4 by including the ac capacitor at the filter bus. If the high-pass current control
HHP (s) is disregarded, the voltage-vector control law of the VSC in (3.32) becomes
c
vref = V0 + ∆V.

(4.31)

In a synchronous grid dq reference frame with the d axis chosen aligned with the ac source
c
E, if the switching-time delay is neglected and it is assumed that |vref | does not exceed
the maximum voltage modulus, then
c
v = vref ejθv = (V0 + ∆V )ejθv .

(4.32)

With the voltage vector v expressed by (4.32), the dynamic equations of the main circuit
in Fig. 4.4 can be written as
dic
= (V0 + ∆V )ejθv − uf − Rc ic − jω1 Lc ic
dt
duf
Cf
= ic − in − jω1 Cf uf
dt
din
Ln
= uf − E − Rn in − jω1 Ln in
dt
Lc

(4.33)
(4.34)
(4.35)

and in dq-component form
dicd
dt
dicq
Lc
dt
duf d
Cf
dt
duf q
Cf
dt
dind
Ln
dt
dinq
Ln
dt
Lc

= (V0 + ∆V ) cos θv − uf d − Rc icd + ω1 Lc icq
= (V0 + ∆V ) sin θv − uf q − Rc icq − ω1 Lc icd
= icd − ind + ω1 Cf uf q
= icq − inq − ω1 Cf uf d
= uf d − E0 − Rn ind + ω1 Ln inq
= uf q − Rn inq − ω1 Ln ind .

(4.36)

The output variables are the active power P and the voltage magnitude Uf at the filter bus.
In per unit form these two quantities are defined as
P = Re {uf i∗ } , Uf =
n

u2 d + u2 q .
f
f

(4.37)
63

Chapter 4. Dynamic Modeling of AC/DC Systems
The state-space model can be obtained by linearizing (4.36) and (4.37), which yields the
following form
d
x = Ax + Bu
dt
y = Cx + Du

(4.38)

where


c
− Rc
L

ω1

1
− Lc

0

0

0


1
 −ω1 − Rc
0 − Lc
0
0

Lc

1
 1
0
0
ω1 − Cf
0
 Cf

A=
1
1
−ω1
0
0
− Cf
 0
Cf


1
n
 0
0
0 − Rn ω1
Ln
L

1
n
0
0
0
−ω1 − Rn
Ln
L
B=

C=

D=
x=

− V0 sin θv0
Lc
V0 cos θv0
Lc

0 0 ind0
0 0
0 0
0 0

uf d0
Uf 0

V0 cos θv0
Lc
V0 sin θv0
Lc

T

0 0 0 0
















0 0 0 0

inq0 uf d0 uf q0
uf q0
Uf 0

, u=

0
∆θv

0
, y=

∆V
V0

∆P
∆Uf

∆icd ∆icq ∆uf d ∆uf q ∆ind ∆inq

T

.

(4.39)

The state-space representation (4.38) can also be written in input-output transfer matrix
form
y = C (sI − A)−1 B + D u
which yields




∆P
∆Uf



=

JPθ (s)

JPV (s)

JUf θ (s) JUf V (s)
J(s)

(4.40)



∆θ
∆V
V0



.

(4.41)

Due to the order of the system, the analytical expressions of the transfer functions of the
Jacobian transfer matrix J (s) in (4.41) are difficult to obtain. However, the analytical
expressions of the transmission zeros and poles are possible to be derived from the statespace representation.
64

4.2. Grid-connected VSCs using power-synchronization control
The transmission zeros of J (s) can be derived by the QZ method suggested in [83,
84]. A polynomial system matrix P (s) is defined as
P (s) =

A − sI B
C

D

.

(4.42)

The transmission zeros are then the values s = z for which P (s) loses rank, resulting in
zero output for some non-zero input values, i.e., the zeros are found by the values s = z
that satisfy
det[P (s)] = 0.
(4.43)
By neglecting the resistances Rc and Rn , the solutions to (4.43) are found to be exactly
identical to those of (4.17). In other words, the ac capacitor at the filter bus has no influence on the transmission zeros of the Jacobian transfer matrix!
As will be explained below, this somewhat surprising result is reasonable. In Section 4.2.1, it was shown that the transmission zeros of the Jacobian transfer matrix have
a close relationship with angle stability of the ac system. As long as the filter-bus voltage is controlled, the angle stability of the ac system is indeed only determined by the
phase angle between the ac source and the filter bus, and independent of the ac capacitor. Of course, in practice, the ac capacitor will affect how much reactive power the VSC
needs to supply in order to keep the filter-bus voltage constant, but this will not affect the
stability unless the converter is forced into voltage limitation.
The poles of J (s) are the eigenvalues of the A matrix. If the resistances Rc and Rn
are neglected also in this case, the poles can be solved analytically with the expressions
p1,2 = ±jω1
p3,4 = ±j

1
1
+
− ω1
Ln Cf
Lc Cf

p5,6 = ±j

1
1
+
+ ω1
Ln Cf
Lc Cf

.

(4.44)

The two pairs of poles p3,4 and p5,6 are related to the ac capacitor connected at the filter
bus. However, the pole pair p1,2 at the grid frequency is not related to any ac capacitor. In
Fig. 3.3, the resonance at the grid frequency has already been observed during the power
step response of the VSC-HVDC link using power-angle control.
In contrast to the transmission zeros, the poles of the Jacobian transfer matrix are
independent of the operating points, but are usually very poorly damped due to the low
resistance in transmission systems. For grid-connected VSCs, the control system has to
provide damping to these poles to achieve a reasonable bandwidth.
In the following, it is demonstrated how these resonant poles are damped by the
high-pass current control HHP (s) proposed for power-synchronization control in Chapter 3.
65

Chapter 4. Dynamic Modeling of AC/DC Systems
By including HHP (s), the voltage vector v of the VSC is expressed as
c
v = vref ejθv

= [(V0 + ∆V ) − HHP(s)ic ]ejθv
c
= (V0 + ∆V )ejθv − HHP (s)ic .

(4.45)

Thus, the dynamic equation in (4.33) is modified as
Lc

dic
= (V0 + ∆V )ejθv − HHP (s) ic − uf − Rc ic − jω1 Lc ic .
dt

(4.46)

To eliminate the Laplace transform variable s in HHP (s) which in (4.46) should be interpreted as s = d/dt, a new state variable ρc needs to be introduced. With the new state
variable ρc , the dynamic equation of the phase reactor in (4.46) is expressed as
dic
= Lc ρc + (V0 + ∆V )ejθv − (Rc + kv + αv Lc ) ic − uf − jω1 Lc ic
dt
dρc
Lc
= αv (V0 + ∆V )ejθv − Rc ic − uf − jω1 Lc ic .
(4.47)
dt
Lc

Replacing the dynamic equations of the phase-reactor in (4.33) by (4.47) and following
the same procedure, the Jacobian transfer matrix J (s) including HHP (s) can be obtained.
For mathematical convenience, HHP (s) has been treated as a part of the Jacobian
transfer matrix, i.e., the controlled process, even though it physically belongs to the control system. Fig. 4.11 shows the pole-zero map of J (s) with a variation of the gain kv of
the high pass-current control. The system parameters of the Jacobian transfer matrix are
given in Table 4.2. As shown in Fig. 4.11, the effect of HHP (s) is to shift the resonant
poles of J (s) towards the left-half plane (LHP). This damping effect is general for any
resonances in the ac system since HHP (s) basically emulates the dynamic behavior of a
physical resistor.
From Fig. 4.11 it can also be observed that the two transmission zeros are not
affected by HHP (s). This property, in fact, can also be analytically verified by applying
the QZ method for zero calculation.
Figs. 4.12-4.15 show the Bode plots of the four transfer functions JPθ (s), JPV (s),
JUf θ (s), and JUf V (s) overlapped with plots produced by frequency-scanning results from
PSCAD/EMTDC (dashed lines). The transfer functions with and without HHP (s) are
placed side by side in the figures. It clearly shows the damping effects of HHP (s) to all
the resonant poles for all the transfer functions.
The resonant poles p3,4 and p5,6 caused by the ac capacitor at the filter bus appear
in all the four transfer functions. However, it seems that the resonant pole pair p1,2 at the
grid frequency only affects JPθ (s) and JPV (s), but not JUf θ (s) and JUf V (s). The reason is
that, by neglecting the resistances Rn and Rc , JUf θ (s) and JUf V (s) get exactly pole-zero
cancelation at the grid frequency.
66

4.2. Grid-connected VSCs using power-synchronization control

p5

2000
1500

p3

ω (rad/sec)

1000
500

p1

z1

z2

0
-500

p2

-1000
-1500

p4

-2000

p6
-2000 -1500 -1000 -500
0
500
ω (rad/sec)

1000

1500

2000

Fig. 4.11 Damping effects of HHP (s) on the resonant poles of J (s). Variations of the gain kv
from 0.0 p.u. to 0.6 p.u.

Table 4.2 Parameters of the Jacobian transfer matrix for a grid-connected VSC using powersynchronization control connected to an impedance-source. Per unit based on 350 MVA
and 195 kV.

Parameters
Rc
ω 1 Cf

0.17 p.u.
0.01 p.u.

ω1 Ln

1.0 p.u.

E0

1.0 p.u.

V0

1.0 p.u.

θu0 (θv0 )

30◦ (35.8◦ )

kv
High-pass current control

0.2 p.u.

Rn

Initial conditions

0.01 p.u.

ω1 Lc
Main-circuit parameters

Value

0.45 p.u.

αv

40 rad/s

67

Chapter 4. Dynamic Modeling of AC/DC Systems

2

2

10
|JPθ(jω)|

|JPθ(jω)|

10

0

10

-2

-2

1

0
-200
-400
-600
-800

2

10

10

3

10



10

ARG J (jω) (deg.)



ARG J (jω) (deg.)

10

0

10

1

10

2

10
ω (rad/sec)

3

10

1

10

0
-200
-400
-600
-800

1

10

(a) Without HHP (s).

2

10

2

10
ω (rad/sec)

3

10

3

10

(b) With HHP (s).

Fig. 4.12 Bode plots of JPθ (s) (solid: linear models, dashed: frequency-scanning results from
PSCAD/EMTDC).

2

2

10
|JPV(jω)|

|JPV(jω)|

10

0

10

-2

-2

1

2

10

100
0

PV

-100
-200
-300

10

3

10

(jω) (deg.)

10

1

10

2

10
ω (rad/sec)

(a) Without HHP (s).

3

10

ARG J

ARG J

PV

(jω) (deg.)

10

0

10

1

10

2

10

3

10

100
0
-100
-200
-300

1

10

2

10
ω (rad/sec)

3

10

(b) With HHP (s).

Fig. 4.13 Bode plots of JPV (s) (solid: linear models, dashed: frequency-scanning results from
PSCAD/EMTDC).

68

4.2. Grid-connected VSCs using power-synchronization control

2

2

10
|JU θ(jω)|

|JU θ(jω)|

10

0

f

f

0

10

-2

-2

1

10

2

10

-200
-400
-600
1

10

2

10
ω (rad/sec)

10

3

10

3

10

ARG J θ(jω) (deg.)
U
f

ARG J θ(jω) (deg.)
U
f

10

10

1

10

2

10

3

10

-200
-400
-600
1

10

(a) Without HHP (s).

2

10
ω (rad/sec)

3

10

(b) With HHP (s).

Fig. 4.14 Bode plots of JUf θ (s) (solid: linear models, dashed: frequency-scanning results from
PSCAD/EMTDC).

2

2

|JU V(jω)|

10

0

10

-2
2

10

-100
-200
1

10

2

10
ω (rad/sec)

(a) Without HHP (s).

10

3

10

0

-300

10

-2

1

10

3

10

ARG J V(jω) (deg.)
U
f

10
ARG J V(jω) (deg.)
U
f

0

f

f

|JU V(jω)|

10

1

10

2

10

3

10

0
-100
-200
-300

1

10

2

10
ω (rad/sec)

3

10

(b) With HHP (s).

Fig. 4.15 Bode plots of JUf V (s) (solid: linear models, dashed: frequency-scanning results from
PSCAD/EMTDC).

69

Chapter 4. Dynamic Modeling of AC/DC Systems
P, Q
E0∠00

+

Cn
un

Rn

Ln

+

U f 0∠θu 0 Rc

VSC
Lc

V0∠θ v 0

+
in

uf

E

+
ic

-

v

-

-

Fig. 4.16 AC-source feeding from a series-compensated ac line.

This property can also be confirmed by the analytical expressions of JUf θ (s) and
JUf V (s) in Table 4.1 for the network configuration without considering the ac capacitor
at the filter bus. Substituting R = 0 into the transfer functions it is found that
JUf θ (s) = −
JUf V (s) =

Ln
V0 sin (θv0 − θu0 )
L

Ln
V0 cos (θv0 − θu0 )
L

(4.48)

i.e., the poles are canceled by the zeros.

4.2.3 AC-source feeding from a series-compensated ac line
Fig. 4.16 shows a network configuration with an ac source feeding from a series-compensated transmission line, where Cn is the capacitance of the series ac capacitor, while un
is the voltage vector across the series ac capacitor. The ac capacitor at the filter bus of
the VSC is neglected in this configuration in order to obtain analytical expressions of the
transmission zeros and poles of the Jacobian transfer matrix.
In a synchronous dq reference frame where the d axis is chosen aligned with the ac
source, the dynamic equations of the main circuit in Fig. 4.16 can be expressed as
di
= v − un − E − Ri − jω1 Li
dt
dun
Cn
= i − jω1 Cn un
dt
L

(4.49)
where L = Lc + Ln , R = Rc + Rn , and i = ic = in . In component form, (4.49) can be
70

4.2. Grid-connected VSCs using power-synchronization control
written as
did
dt
diq
L
dt
dund
Cn
dt
dunq
Cn
dt
L

= vd − und − E0 − Rid + ω1 Liq
= vq − unq − Riq − ω1 Lid
= id + ω1 Cn unq
= iq − ω1 Cn und .
(4.50)

Since the ac capacitor at the filter bus is neglected, the filter-bus voltage vector uf is
neither a state variable nor an input. Such a variable has to be represented by other states
and/or inputs. For this network configuration, uf can be solved from the following two
equations
di
= uf − un − Rn i − jω1 Ln i
(4.51)
dt
di
Lc = v − uf − Rc i − jω1 Lc i.
(4.52)
dt
By dividing (4.51) with Ln and (4.52) with Lc , and further subtracting (4.52) from (4.51),
it is possible to express uf as
Ln

uf =

Ln
Lc
v + Rx i + un .
L
L

(4.53)

where Rx = (Rn Lc − Rc Ln )/L.
Following the same procedure as the previous section, the state-space model in
the form of (4.38) can be obtained by linearizing (4.50) and (4.37) with ∆uf d and ∆uf q
derived from the linearized component form of (4.53). The corresponding matrices and
vectors in (4.38) are expressed as
 R

1
− L ω1 − L
0


T
1
 −ω1 − R
0 −L 
− V0 sin θv0 V0 cos θv0 0 0
L
Lc
Lc


A=
, B =
V0 cos θv0
V0 sin θv0
 1
0
0
ω1 
0 0
Lc
Lc
 Cn



C=
D=
x=

0

1
Cn

−ω1

0

id0 Lc
L

iq0 Lc
L

uf d0 Lc
Uf 0 L

uf q0 Lc
Uf 0 L

id0 Rx + uf d0 iq0 Rx + uf q0
uf d0 Rx
Uf 0
Ln V0
(iq0
L
Ln V0
(uf q0
LUf 0

uf q0 Rx
Uf 0

cos θv0 − id0 sin θv0 )
cos θv0 − uf d0 sin θv0 )

∆id ∆iq ∆und ∆unq

T

.

Ln V0
(id0
L
Ln V0
(uf d0
LUf 0




cos θv0 + iq0 sin θv0 )
cos θv0 + uf q0 sin θv0 )
(4.54)
71

Chapter 4. Dynamic Modeling of AC/DC Systems
The definitions of u and y are the same as in (4.39). Similarly, the Jacobian transfer matrix
J(s) can be obtained from the state-space representation by applying (4.40).
The poles of J(s) are the eigenvalues of the A matrix. These are given by
p1,2 = −

R
1
±
2L 2

R2
4
2
− 4ω1 −
+4
2
L
LCn

2
2
4ω1
R 2 ω1

LCn
L2

p3,4 = −

1
R
±
2L 2

R2
4
2
− 4ω1 −
−4
2
L
LCn

2
2
4ω1
R 2 ω1

.
LCn
L2

(4.55)

If the resistance R is neglected, (4.55) can be simplified as
1
LCn
1
= ±j ω1 + √
LCn

p1,2 = ±j ω1 − √
p3,4

.

(4.56)

For series compensation, the reactance of the series capacitor is always smaller than the

reactance of the ac line. Consequently, 1/ LCn is always smaller than ω1 . Thus, (4.56)
shows that J(s) has two pairs of poles, one pair (p1,2 ) in the subsynchronous frequency
range, and the other pair (p3,4 ) in the supersynchronous frequency range. If Cn = ∞, these
two pairs of poles correspond to the pole pair at the grid frequency for the impedancesource configuration in the previous subsection. The smaller the ac capacitance, i.e., the
higher degree that the ac line is compensated, the further the pole pairs are separated.
The pole pair p1,2 in the subsynchronous frequency range is usually troublesome.
Besides the SSR problem with the rotor shaft of the thermal plant as mentioned before,
they can also create problems for power-electronic devices connected in the vicinity since
the subsynchronous frequency range is also where the control systems of most powerelectronic devices are active.
If the resistances Rc and Rn are neglected, the transmission zeros of J(s) can be
derived analytically by applying the QZ method, which gives
z1,2 = ±
z3,4 = ±


2
Uf 0 (ω1 Ln Cn − 1) + b
1
2
− ω1 +
2Ln Cn (Uf 0 − E0 cos θu0 )
Ln Cn

2
Uf 0 (ω1 Ln Cn − 1) − b
1
2
− ω1 +
2Ln Cn (Uf 0 − E0 cos θu0 )
Ln Cn

(4.57)

where the term b is expressed as
2
2
2
b = 8E0 cos θu0 ω1 Ln Cn (2E0 cos θu0 − 3Uf 0 ) − Uf 0 + Uf 0 (ω1 Ln Cn + 3)2 .

(4.58)

Fig. 4.17 shows the loci of the two pairs of zeros as the load angle θu0 is increased. For
the Jacobian transfer matrix, the main-circuit parameters are chosen as given by Table 4.2
72

4.2. Grid-connected VSCs using power-synchronization control

600

p

3

ω (rad/sec)

400
z3

200
0
-200

p1

z1
θu0=32

°

θu0=41

°

θu0=54

°

p

z2
°

2

θu0=54 θu0=41

°

θu0=32

°

z4
-400
p
-600
-600

-400

-200

4

0
ω (rad/sec)

200

400

600

Fig. 4.17 Loci of the transmission zeros with increased load angles.

but with a 60% degree of series compensation, i.e., the capacitance of the series capacitor
is chosen as ω1 Cn = 1.667 p.u. The zero pair z1,2 is on the real axis, while z3,4 is on the
imaginary axis. With increased load angles, the resonant zero pair z3,4 moves from p1,2
towards p3,4 . The real zero pair z1,2 moves towards the origin. If θu0 = 90◦ is substituted
into the analytical expression of z1,2 in (4.57), it precisely gives z1,2 = 0. It can also be
analytically verified that, if Cn = ∞ is substituted into (4.57), then the mathematical
expression of z1,2 is identical to (4.17), while z3,4 = ∞.
In contrast to the resonant poles, for which HHP (s) provides significant damping,
HHP (s) has no effect on the resonant zero pair z3,4 , which might negatively affect the
phase margin of the control system around the subsynchronous frequency range. Care
must be taken if the bandwidth of the control system of the VSC is intended to be higher
than the frequency of z3,4 . Fig. 4.18 shows the Bode plot of the open-loop transfer function
of the power-synchronization loop, which shows the effect of HHP (s). The open-loop
transfer function of the power-synchronization loop is expressed as
HPSL (s) = JPθ (s)

kp
.
s

(4.59)

During the process of describing the modeling of VSCs using power-synchronization
control with the concept of Jacobian transfer matrix, simplified ac-network configurations
have been used. This has been done in order to develop a basic understanding of the Jacobian transfer matrix modeling concept, as well as to understand the physical meanings
73

Chapter 4. Dynamic Modeling of AC/DC Systems
2

10

0

10

-2

10

|HPSL(jω)|

10

-4

Without HHP(s)
With HHP(s)
10

1

10

2

10

3

10

1

10
ω (rad/sec)

2

10

3

arg HPSL(jω) (deg.)

0

-100

-200

-300

Fig. 4.18 Bode plot of the open-loop transfer function HPSL (s), kp = 130 rad/s. Initial conditions: θu0 = 24.9◦ (θv0 = 37◦ ).

of the transmission zeros and poles of the Jacobian transfer matrix in an analytical way.
Within the investigated ac-network configurations so far, a constant-frequency ac
voltage source with a voltage vector E is assumed. The d axis of the grid dq frame has
been chosen aligned with the voltage vector for analysis. Such voltage sources, of course,
do not exist in the real system. However, if the VSC is connected to a large ac system,
assuming such an equivalent network configuration is acceptable. In case that the VSC is
connected to an island system, a common ac-network R − I frame needs to be defined
which is not related to any voltage sources. Such issues will be discussed in Chapter 6.

4.3 Grid-connected VSCs using vector current control
In Chapter 3, the Jacobian transfer matrix was developed for a grid-connected VSC using
vector current control. However, the model was based on the assumption that the filter-bus
voltage is stiff, whereas such assumption is not valid if the VSC is connected to a weak ac
system. In this section, the Jacobian transfer matrix is developed taking into account the
dynamics of the filter bus, as well as the dynamics of the PLL. The methodology is independent of ac-network configurations, but for simplicity the impedance-source network
configuration shown in Fig. 4.4 is used for model development.
Fig. 4.19 shows the small-signal block diagram of vector current control, where the
Jacobian transfer matrix is defined as the transfer matrix which has the current reference
∆iref and ∆iref as inputs, and the active power ∆P and the the filter-bus voltage ∆Uf as
q
d
74

4.3. Grid-connected VSCs using vector current control
Jacobian transfer matrix J (s)

c
cd

∆i

ref
∆id

ref
∆iq

∆v

c
∆icq

∆u c
fd
∆u

c
fq

G cc ( s)

∆P
∆U f
∆icd
∆icq
∆u fd

ref
d

c
dref

∆v
Mv

c
∆vqref

J n ( s)

ref
∆vq

∆u fq

∆θ PLL
∆u fd
∆θ PLL

∆u c
fd

G PLL ( s) ∆u
fq

∆u fd
Mu

∆u c
fq

∆u fq

∆θ PLL
c
∆icd

c
∆icq

∆icd
Mi

∆icq

∆θ PLL

Fig. 4.19 Definition of the Jacobian transfer matrix for a grid-connected VSC using vector current
control.

outputs.2
By defining the Jacobian transfer matrix, Fig. 4.20 shows the closed-loop system of a grid-connected VSC using vector current control, where the minus sign in the
alternating-voltage control is due to the direction of the phase-reactor current reference.
In the following, the mathematical expressions of the three transfer matrices Jn (s),
GPLL (s), Gcc (s), and the three real matrices Mv , Mu , and Mi shown in Fig. 4.19 are
described, where the superscript c on the variables represents the converter dq frame.
1. Network Jacobian transfer matrix Jn (s). This is the ac-network Jacobian transfer matrix which has the dq components of the VSC voltage ∆vd and ∆vq as the
inputs, and the active power ∆P , the filter-bus voltage magnitude ∆Uf , dq components of the phase-reactor current ∆icd and ∆icq , and the dq components of the
filter-bus voltage ∆uf d and ∆uf q as the outputs. To distinguish it from the final
Jacobian transfer matrix for vector current control, the subscript n is used. Jn (s)
2

The procedure of model development is the same if reactive power ∆Q is used as output instead of
∆Uf , but the alternating-voltage control mode is focused on as it is the preferred operation mode for weakac-system connections.

75

Chapter 4. Dynamic Modeling of AC/DC Systems

Pref +

-

KP
K + i
s
APC
P
p

U ref +

K iU
)
s
AVC

− (KU +
p
-

ref
∆id

∆P
Jacobian
transfer
matrix

ref
∆iq

J ( s)

∆U f

Fig. 4.20 Closed-loop system for VSCs using vector current control.

is similar to the Jacobian transfer matrix developed for power-synchronization control in Section 4.2.2, but differs by the inputs and the outputs. Those matrices in the
state-space representation that are different from (4.39) are


0 0 ind0 inq0 uf d0 uf q0


 0 0 uf d0 uf q0
0
0 


Uf 0
Uf 0


T


1 0 0 0 0 0
 1 0 0
0
0
0 

B=
, C=


0 1 0 0 0 0
 0 1 0
0
0
0 




 0 0 1
0
0
0 


0 0 0
1
0
0
u=

∆vd ∆vq

T

, y=

∆P ∆Uf ∆icd ∆icq ∆uf d ∆uf q

T

.
(4.60)

The network Jacobian transfer matrix Jn (s) thus can be obtained from (4.40), i.e.,
y = Jn (s)u.

(4.61)

2. Phase-locked loop (PLL) transfer matrix GPLL (s). As described in Chapter 3, the
control law of the PLL is given by
θPLL =

PLL
Kp +

KiPLL
s

Im{uc } = FPLL (s)uc q .
f
f

(4.62)

FPLL (s)

The filter-bus voltage uc in the converter dq frame is related to uf in the grid dq
f
frame by
uc = uf e−jθPLL
(4.63)
f
76

4.3. Grid-connected VSCs using vector current control
or in component form
uc d + juc q = (uf d + juf q )(cos θPLL − j sin θPLL )
f
f

(4.64)

uc q = uf q cos θPLL − uf d sin θPLL .
f

(4.65)

which yields

Substituting (4.65) into (4.62), yields
θPLL = FPLL (s)(uf q cos θPLL − uf d sin θPLL )

(4.66)

which is further linearized as
∆θPLL =

−FPLL (s) sin θu0
∆uf d
1 + FPLL (s)(uf d0 cos θu0 + uf q0 sin θu0 )
APLL (s)

+

FPLL (s) cos θu0
∆uf q
1 + FPLL (s)(uf d0 cos θu0 + uf q0 sin θu0 )

(4.67)

BPLL (s)

where θu0 is the angle output of the PLL in the steady state, which corresponds
to the phase angle of the filter-bus voltage in the grid dq frame. Eq. (4.67) can be
further expressed in transfer matrix form as

∆θPLL =

APLL (s) BPLL (s)
GPLL (s)

∆uf d
∆uf q

.

(4.68)

3. Frame-transformation matrices Mv , Mu , and Mi . The VSC voltage reference vref ,
the filter-bus voltage uf , and the phase-reactor current ic , are related in the converter
dq frame and the grid dq frame as
c
vref = vref ejθPLL , uc = uf e−jθPLL , ic = ic e−jθPLL .
f
c

(4.69)

By writing (4.69) in component form and applying linearization, (4.69) can be ex77

Chapter 4. Dynamic Modeling of AC/DC Systems
pressed in input-output matrix form as

ref
∆vd
ref
∆vq

∆uc d
f
∆uc q
f

∆ic
cd
∆ic
cq

=

=

=

c
c
cos θu0 − sin θu0 (−vd0 sin θu0 − vq0 cos θu0 )

sin θu0

cos θu0

cos θu0

sin θu0

− sin θu0 cos θu0
cos θu0

sin θu0

− sin θu0 cos θu0



c
∆vdref





 ∆v c 
qref 

c
c
(vd0 cos θu0 − vq0 sin θu0 )
∆θPLL
Mv


∆uf d
(−uf d0 sin θu0 + uf q0 cos θu0 ) 

 ∆uf q 


(−uf d0 cos θu0 − uf q0 sin θu0 )
∆θPLL
Mu


∆icd
(−icd0 sin θu0 + icq0 cos θu0 ) 

 ∆icq  .


(−icd0 cos θu0 − icq0 sin θu0 )
∆θPLL
Mi

(4.70)

4. Current-control transfer matrix Gcc (s). The current control law is given by (3.12).
After writing (3.12) in component form and applying linearization, the current controller can be expressed in input-output transfer matrix form as

c
∆vdref
c
∆vqref

=

αc Lc

0

−αc Lc −ω1 Lc HLP (s)

0

αc Lc

ω1 Lc

−αc Lc

0

0
HLP (s)

Gcc (s)














∆iref
d




∆iref 
q


∆ic 
cd 
.
∆ic 
cq 

∆uc d 
f 
∆uc q
f
(4.71)

If the switching-time delay of the converter is neglected and it is assumed that |vref | does
not exceed the maximum voltage modulus, then v = vref . The above derived transfer/real
matrices can be interconnected by the block diagram shown in Fig. 4.19. Accordingly, the
Jacobian transfer matrix for vector current control is obtained as


78

∆P
∆Uf



=

JPId (s)

JPIq (s)

JUf Id (s) JUf Iq (s)
J(s)




∆iref
d
∆iref
q



.

(4.72)

4.3. Grid-connected VSCs using vector current control

p

2000
1500

5

p3

ω (rad/sec)

1000
500

p1

z1

z2

0
p2

-500
-1000
p4

-1500
-2000

p6
-2000 -1500 -1000 -500
0
500
ω (rad/sec)

1000

1500

2000

Fig. 4.21 Damping effects of the current controller on the resonant poles of J (s) for vector current
control. Variations of αc from 1200 rad/s to 2500 rad/s.

Table 4.3 Control parameters of the Jacobian transfer matrix for a grid-connected VSC using
vector current control connected to an impedance source.

Parameters

2500 rad/s
80 rad/s

PLL
Kp

20

KiPLL

PLL

αc
αf

Inner-current control

Value

20

Similar to the HHP(s) function of power-synchronization control, the current controller of vector current control also provides damping to the resonant poles of the ac
network, as shown in Fig. 4.21. The main-circuit parameters and initial conditions of the
Jacobian transfer matrix for vector current control are chosen the same as those given
in Table 4.2 for power-synchronization control, and the control parameters of the inner
current control and the PLL are given in Table 4.3.
The higher the bandwidth of the current controller, the more damping it adds. For a
modern transistor PWM converter with a switching frequency of 1−2 kHz, the bandwidth
of the current controller can be chosen as αc = 1000 − 2500 rad/s [51], which is enough
for damping purpose.
When analyzing the transmission zeros of the Jacobian transfer matrix, an inter79

Chapter 4. Dynamic Modeling of AC/DC Systems
0

10

0

q

|JPI (jω)|

d

|JPI (jω)|

10

-1

-2

2

d

0

ARG J

1

10

2

10
ω (rad/sec)

3

10

1

10

-600
1

10

|JU I (jω)|

0

10

-2

2

-200

1

10

2

10
ω (rad/sec)

(c) JUf Id (s).

3

10

1

10

2

10

3

10

-200

UI

-400
-600

10

3

10

(jω) (deg.)

10

fq

1

10

ARG J

(jω) (deg.)

3

10

fq

fd

|JU I (jω)|

0

-1

fd

2

10
ω (rad/sec)

(b) JPIq (s).

10

10

UI

3

10

-400

(a) JPId (s).

ARG J

2

10

-200

PI

PI

-100
-200

10

3

10

(jω) (deg.)

10

q

1

10

ARG J

(jω) (deg.)

10

-400
-600

1

10

2

10
ω (rad/sec)

3

10

(d) JUf Iq (s).

Fig. 4.22 Bode plots of the transfer functions of the Jacobian transfer matrix for vector current control (solid: linear models, dashed: frequency-scanning results from
PSCAD/EMTDC). Parameters and initial conditions are given in Table 4.2 and Table 4.3.

esting observation can be made. With the same main-circuit parameters and initial conditions, by comparing Fig. 4.21 with Fig. 4.11, it can be easily found that the locations
of the transmission zeros of the Jacobian transfer matrix for vector current control are
exactly identical to those for power-synchronization control! Thus, from the fundamental limitation point of view, i.e., the bandwidth limitation by the RHP zero, there is no
difference between vector current control and power-synchronization control.
Fig. 4.22 shows the Bode plots of the transfer functions of the Jacobian transfer
matrix with overlapped frequency-scanning plots from PSCAD/EMTDC. The two plots
generally show good agreement for all the transfer functions.
80

4.4. Jacobian transfer matrix for dc-system modeling
Pac2

VSC2

Pdc2
Pdc3

VSC1

VSC4

Pac3

Pdc1

Pac1

Pac4

VSC3

DC-Jacobian
transfer matrix
G dc ( s )

Pdc4

Pdc5

VSC5

Pac5

Fig. 4.23 A dc network connected to multiple VSCs.
u

r
+

ref
dc

+

K dc ( s )
-

+

K ac ( s )
-

− ∆Pac = ∆Pdc
J (s )

G dc ( s )

∆u dc

y

AC-Jacobian
transfer matrix

DC-Jacobian
transfer matrix

Fig. 4.24 Feedback-control view of the ac/dc combined system.

4.4 Jacobian transfer matrix for dc-system modeling
The concept of Jacobian transfer matrix for dc-system modeling is similar to ac-system
modeling. As shown in Fig. 4.23, a dc system is also treated as a multivariable feedback
control system where the Jacobian transfer matrix Gdc (s) is defined as the controlled
process. In this thesis, only VSCs are considered as input devices to the dc system. The
Jacobian transfer matrix of a dc system thus represents the passive components in the dc
system. The input variables to Gdc (s) are the dc powers from the VSCs, while the output
variables from Gdc (s) are the direct voltages at each VSC terminal.
Fig. 4.24 shows the feedback-control view of the ac/dc combined system, where
Kac (s) is the controller transfer matrix of the ac system while Kdc (s) is the controller
transfer matrix of the dc system, i.e., the direct-voltage controllers. In other words, the
dc Jacobian transfer matrix Gdc (s) is treated as an additional controlled process of the
81

Chapter 4. Dynamic Modeling of AC/DC Systems
Pac1

VSC1

Pdc1

Pdc2

VSC2

Pac2

Pac1

VSC1

Pdc1

+

Rdc

idc

Cdc

udc1

-

Pdc2

VSC2

Pac2

+

+

udc

Ldc

udc2

Cdc1

-

Cdc2

-

(b)

(a)

Fig. 4.25 DC-link representation for a two terminal VSC-HVDC link. (a) Single dc capacitor. (b)
π-link.

closed-loop system of the combined ac/dc system. The ac Jacobian transfer matrix J(s)
and the dc Jacobian transfer matrix Gdc (s) are connected by the equivalence of the acpower vector Pac and the dc-power vector Pdc , i.e.,
Pac = −Pdc

(4.73)

where the minus sign is due to the definition of the power directions. By using the equivalence in (4.73), it should be pointed out that the losses of the converter valves can introduce an error, which will be discussed later.
In the following, a dc link connected to two VSCs, i.e., a two-terminal VSC-HVDC
link, is modeled by the dc Jacobian transfer matrix modeling concept. Fig. 4.25(a) shows
an dc-link circuit that is represented by a single dc capacitor. The capacitor bank at the
dc link is an energy storage. The time derivative of the stored energy must equal the sum
of the instantaneous power infeed from the two converters (neglecting the losses). The
direct-voltage dynamics can thus be written as
d (u2 )
1
dc
Cdc
= Pdc1 + Pdc2
2
dt

(4.74)

where Cdc is the dc-link capacitance and udc is the direct voltage. Pdc1 and Pdc2 are the
instantaneous powers from VSC1 and VSC2 . If the direct-voltage controller were to operate directly on the error uref − udc , the closed-loop dynamics would be dependent on the
dc
operating point udc0 . This inconvenience is avoided by selecting the direct-voltage con2
troller operating instead on the error uref − u2 as suggested in [13,85]. Consequently,
dc
dc
the dc-link dynamics can be written in the linearized form
∆u2 =
dc

2
(∆Pdc1 + ∆Pdc2 ) .
sCdc

(4.75)

Gdc (s)

If the dc transmission line is a long overhead line, then the resistance and the inductance
of the dc line have to be taken into account. Fig. 4.25(b) shows a dc link represented by
82

4.4. Jacobian transfer matrix for dc-system modeling
a π-link model. Based on Kirchhoff’s voltage and current laws, the dynamic equations of
the dc circuit can be written as
dudc1
Pdc1
Cdc1
=
− idc
dt
udc1
didc
= udc1 − udc2 − Rdc idc
Ldc
dt
dudc2
Pdc2
Cdc2
= idc +
(4.76)
dt
udc2
where Cdc1 and Cdc2 are lumped capacitances at the two VSC stations. The quantities
udc1 and udc2 are the direct voltages. Pdc1 and Pdc2 are the instantaneous powers from
VSC1 and VSC2 . Rdc and Ldc are the resistance and the inductance of the dc line. The
quantity idc is the direct current of the dc line. If the linearized deviation variables ∆Pdc1
and ∆Pdc2 of Pdc1 and Pdc2 are chosen as the inputs, and ∆u2 and ∆u2 of udc1 and
dc1
dc2
udc2 are chosen as the outputs, a state-space model can be obtained by linearization of
(4.76)
d
x = Ax + Bu
dt
y = Cx + Du
(4.77)
where





A=

C=
u=

1
− u2Pdc10
− Cdc1
Cdc1



0

dc10

1
Ldc

dc
− Rdc
L

− L1
dc

0

1

− u2Pdc20
dc20 Cdc2

Cdc2

2udc10 0
0

0

, D=

0 2udc20

∆Pdc1 ∆Pdc2

T

, y=



, B =


0 0
0 0

1
udc10 Cdc1

0

0

0

0

1
udc20 Cdc2

, x=

2
∆u2
dc1 ∆udc2

T

T

∆udc1 ∆idc ∆udc2
.

T

(4.78)

The state-space representation in (4.77) can also be written in input-output transfer matrix
form by (4.40) which yields




∆u2
∆Pdc1
Gdc11 (s) Gdc12 (s)
dc1

=
.

(4.79)
∆u2
∆Pdc2
Gdc21 (s) Gdc22 (s)
dc2
Gdc (s)

The poles of Gdc (s) can be solved analytically for the operating point where Pdc10 =
Pdc20 = 0 with the expression
p1 = 0
p2,3 = −

(4.80)
Rdc
±j
2Ldc

2
Rdc
Cdc1 + Cdc2

.
2
4Ldc Ldc Cdc1 Cdc2

83

Chapter 4. Dynamic Modeling of AC/DC Systems
Table 4.4 Parameters of the dc Jacobian transfer matrix for the π-link model. Per unit based on
350 MW and 150 kV.

Parameters
Rdc
Ldc

0.0025 p.u.
0.0077 p.u.

Cdc2

0.0077 p.u.

Udc10
Initial conditions

0.04 p.u.

Cdc1

1.0 p.u.

Pdc10

Main-circuit parameters

Value

0.0 p.u.

As shown in Fig. 4.26, the poles of Gdc (s) are dependent on the operating point.
With the increase of loading, the frequencies of the two complex poles p2,3 are reduced,
and the real pole p1 at the origin moves into the right-half plane. The main-circuit parameters and initial conditions of the dc Jacobian transfer matrix for the π-link are given in
Table 4.4. In control theory, the RHP pole of the process imposes a fundamental lower
limit on the bandwidth of the controller, i.e., the closed-loop system of the direct-voltage
control has to achieve a bandwidth that is higher than the location of the RHP pole of
Gdc (s) to stabilize the process. Recalling also the upper limit of bandwidth imposed by
the RHP transmission zero of the ac Jacobian transfer matrix [cf. (4.17)], it is generally
more complicated to operate grid-connected VSCs at high load angles.
The instability of Gdc (s) is to do with the resistance of the dc link. With dc powers
as the inputs, as the way how VSCs work to a dc link, the dc resistance gives a destabilizing effect. The analytical solutions of the poles of Gdc (s) with other operating point
than Pdc10 = Pdc20 = 0 are difficult to obtain. However, Appendix B.2 gives a rigorous
mathematical proof of the instability of Gdc (s) for other operating points than Pdc10 = 0
or Pdc20 = 0. The mathematical proof also shows the role of the dc resistance to the instability of Gdc (s). Fig. 4.27 shows the root-loci of Gdc (s) by varying the dc-resistance
values. With increased dc resistance, the resonant pole pair p2,3 becomes more damped,
but p0 moves towards the right-half plane. It should be noted that the destabilizing effect
of the dc resistance only becomes apparent if the dc-transmission line is sufficiently long,
e.g., HVDC transmission over long-distance overhead lines.
By applying the QZ method, the transmission zero of Gdc (s) for the π-link model
is derived with a surprisingly simple expression
z=−

Rdc
.
Ldc

(4.81)

In contrast to the zeros of the ac Jacobian transfer matrix, the location of the zero of the
dc Jacobian transfer matrix is independent of the operating points.
84

4.4. Jacobian transfer matrix for dc-system modeling

400
p2

300

ω (rad/sec)

200
100
p

z

1

0
-100
-200
-300
-400
-30

p3
-20

-10

0
ω (rad/sec)

10

20

30

Fig. 4.26 Root-loci of Gdc (s) regarding variations of Pdc10 from 0.0 p.u. to 1.0 p.u.

400
p2

300

ω (rad/sec)

200
100
z

p1

0
-100
-200
-300
-400
-200

p
-150

-100

-50

3

0
ω (rad/sec)

50

100

150

200

Fig. 4.27 Root-loci of Gdc (s) regarding variations of Rdc from 0.0 p.u. to 0.24 p.u. Initial conditions: Pdc10 = 1.0 p.u., udc10 = 1.0 p.u.

85

Chapter 4. Dynamic Modeling of AC/DC Systems
J u d θ ( s)
− ∆PVSC1 = ∆Pdc1

∆θ v
∆V
V0

J (s )

∆U f

∆Pdc2

2
∆udc1

G dc ( s )

2
∆udc2

Fig. 4.28 Definition of the transfer function Jud θ (s).

To verify the proposed modeling concept with frequency-scanning results from time
simulations, a transfer function Jud θ (s) is defined as
Jud θ (s) = JP θ (s)Gdc11 (s)

(4.82)

which is shown graphically in Fig. 4.28. It should be noted that the active power derived
previously for the ac Jacobian transfer matrix is the power from the filter bus, which
is somewhat different from the active power flowing from the VSC due to the energy
stored in the phase reactor. Therefore, the ac power ∆Pac should be obtained from the
linearization of
Pac = PVSC = Re {vi∗ } .
c

(4.83)

In Fig. 4.28, the active power flowing from the VSC is denoted as PVSC to distinguish it
from the active power from the filter bus to the ac system.
Fig. 4.29 shows the Bode plots of Jud θ (s) overlapped with the frequency-scanning
results from PSCAD/EMTDC for the two types of dc-link representations respectively,
where the impedance-source system shown in Fig. 4.4 is used as the ac-network configuration. Besides, the high-pass current control HHP (s) has been applied in the ac Jacobian
transfer matrix.
Some discrepancies can be observed between the plots of the linear models and the
frequency-scanning results, which are mainly due to the lack of valve-loss representation
of the linear models. The most noticeable one is the resonance peak of the π-link model
in Fig. 4.29(b). The frequency-scanning results show that the valve losses have a damping
effect on the resonance peak while the linear model tends to overestimate the impact of
the dc resonance. In addition, the slopes of the magnitudes of the linear models in both
Fig. 4.29(a) and Fig. 4.29(b) are generally steeper than those from the frequency-scanning
results, which can be considered as the resistive effects of the valve losses. To properly
represent the losses of the converter valves, the linear model should take into account the
topology of the converter as well as the applied PWM technique, which is of nonlinear
nature. These issues certainly require further investigations in the future research.
86

4.5. Summary of the properties of the Jacobian transfer matrix
5

5

|JU θ(jω)|

10

0

10

-5
2

10

10

3

10

d

0
-200
-400
-600
-800

10

-5

1

10

ARG JU θ(jω) (deg.)

d

ARG JU θ(jω) (deg.)

10

0

d

d

|JU θ(jω)|

10

1

10

2

10
ω (rad/sec)

(a) Single dc capacitor model.

3

10

0
-200
-400
-600
-800

1

10

1

10

2

10

2

10
ω (rad/sec)

3

10

3

10

(b) π-link model.

Fig. 4.29 Bode plots of Jud θ (s) (solid: linear models, dashed: frequency-scanning results from
PSCAD/EMTDC).

4.5 Summary of the properties of the Jacobian transfer
matrix
In this chapter, the concept of Jacobian transfer matrix is proposed for dynamic modeling of ac/dc systems. By investigating some simplified ac/dc network configurations
connected to VSCs, several properties of the ac/dc Jacobian transfer matrix can be summarized below:
1. The complex poles of the ac/dc Jacobian transfer matrix generally reflect the resonances in the ac/dc network. The poles of the ac Jacobian transfer matrix are independent of the operating point, while the poles of the dc Jacobian transfer matrix
are dependent on the operating point except in the special case when the dc network
is represented by a single dc capacitor.
2. The ac Jacobian transfer matrix is an inherently stable process, although the poles
might be poorly damped and the zeros can move to the origin. Therefore, instability
of an ac system can only be caused by feedback control. On the other hand, the
dc Jacobian transfer matrix is an inherently unstable process or has a pole at the
origin. Therefore, a dc system has to be stabilized by feedback control, i.e., the
direct-voltage control.
3. The zeros (on the real axis) of the ac Jacobian transfer matrix are dependent on the
operating point. By moving to the origin, the zeros impose a fundamental limitation
on power transmission in the ac network. This limitation has manifested as anglestability and voltage-stability phenomena in the classical power-system stability
theory. If the zero of the ac Jacobian transfer matrix appears on the right-half plane,
87

Chapter 4. Dynamic Modeling of AC/DC Systems
it also imposes an upper limitation on the achievable bandwidth of the input device,
i.e., the bandwidth of the power controller of the input device should be lower than
the location of the RHP zero. On the other hand, the dc Jacobian transfer matrix
does not have such operating-point-dependent zero. The above difference of ac and
dc systems explains the fact that power can be transmitted in the dc network up to
the thermal limit but is often limited in the ac network, especially for long-distance
power transmissions.
4. The pole on the right-half plane of the dc Jacobian transfer matrix imposes a fundamental lower limitation on the bandwidth of the direct-voltage controller, i.e., the
bandwidth of the direct-voltage controller should be higher than the location of the
RHP pole to stabilize the process.
It should be pointed out that the above conclusions about the dc system are only valid
for the dc system defined in this thesis, i.e., the VSCs are the only input devices to the
dc network, and dc powers are the inputs while direct voltages are the outputs of the dc
Jacobian transfer matrix. For example, for dc systems connected to LCCs, the dynamic
modeling is certainly different. Such scenarios are, however, out of the scope of this thesis.
In this chapter, the ac Jacobian transfer matrix has only considered the VSC as the
input device. However, the concept of the ac Jacobian transfer matrix is meant for any
type of input device in ac systems. In Chapter 6, the Jacobian transfer matrices for two
other input devices, i.e., synchronous generators and induction motors, are developed.

4.6 Summary
In this chapter, the concept of Jacobian transfer matrix is proposed for dynamic modeling of ac/dc systems. The fundamental idea of the ac Jacobian transfer matrix modeling
is that a synchronous power system is modeled as one multivariable feedback-control
system, where the feedback controllers and the controlled process of the power system
are explicitly defined. The concept has been applied to model grid-connected VSCs using power-synchronization control and vector current control for several simplified acnetwork configurations. The modeling concept is also extended to model dc systems constructed by multiple VSCs. By theoretical analysis, it is found that the poles and zeros
of the Jacobian transfer matrix give useful information about the properties of the ac/dc
system.

88

Chapter 5
Control of VSC-HVDC Links
Connected to High-Impedance AC
Systems
In this chapter, the control of VSC-HVDC links connected to high-impedance weak ac
systems is investigated. In Section 5.1, general aspects of high-impedance ac systems
are described. In Section 5.2, the performance of vector current control and powersynchronization control for weak-ac-system connections are compared based on the Jacobian transfer matrix modeling concept. In Section 5.3, multivariable feedback designs
of power-synchronization control are investigated by two design approaches, i.e., internal model control and H∞ control. In Section 5.4, a direct-voltage controller is proposed
for power-synchronization control. The dc-capacitance requirement for VSC-HVDC links
connected to weak ac systems is discussed. Finally, a control structure for interconnecting
two very weak ac systems is proposed in Section 5.5. The major results of this chapter are
summarized in Section 5.6. Some results of this chapter are included in [63, 86].

5.1 General aspects of high-impedance ac systems
As it was mentioned in Chapter 2, there is an inherent weakness with the conventional
LCC-HVDC system, i.e., the commutations of the thyristor valves are dependent on the
stiffness of the alternating voltage. The converter cannot work properly if the connected
ac system is weak. Substantial research has been performed in this field [5, 6, 8]. The
most outstanding contribution on this subject is [5]. According to [5], an ac system can be
considered as weak from two aspects: 1. ac-system impedance is high. 2. ac-system inertia
is low. Either of the two system conditions may become an obstacle for conventional
LCC-HVDC applications.
If an HVDC link is terminated at the weak point of a large ac system, i.e., the
89

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
equivalent impedance of the ac system is high, the alternating voltage at the filter bus
will become sensitive to power variations of the HVDC link. This difficulty is usually
measured by the short-circuit ratio (SCR), which is a ratio of the ac-system short-circuit
capacity vs. the rated power of the HVDC link. If the ac system is represented as an
impedance source as shown in Fig. 4.4, then the SCR is directly related to the ac-system
inductance Ln . According to [5], SCR is defined as
SCR =

Sac
PdcN

(5.1)

where Sac is the short-circuit capacity of the ac system at the filter bus, while PdcN is the
rated dc power of the HVDC link. The short-circuit capacity of the ac system Sac can be
expressed as
2
2
Uf 0
Uf 0
Sac =

(5.2)
Z
ω1 Ln
where ω1 is the angular frequency of the ac system and Z is the equivalent impedance of
the ac system. To further simplify the expression of SCR in (5.1), the filter-bus voltage
is assumed to be identical to the base value, i.e, Uf 0 = UacN , and the rated power of the
HVDC link PdcN is used as the base power of the ac system, i.e., SacN = PdcN . If ω1 Ln is
expressed in per unit, it follows from (5.1) and (5.2) that SCR can be expressed as
SCR =

1
.
ω1 Ln

(5.3)

The following is a definition of the strength of an ac system based on the classification
of [5]:
• Strong system, if the SCR of the ac system is greater than 3.0.
• Weak system, if the SCR of the ac system is between 2.0 and 3.0.
• Very weak system, if the SCR of the ac system is lower than 2.0.
One of the driving forces to develop the VSC-HVDC technology is to overcome the
weak-ac-system connection problem of the conventional LCC-HVDC system. By applying VSC techniques, the two notorious weak-ac-system related problems for conventional
LCC-HVDC systems, i.e., the transient over voltage (TOV) and the low-order harmonic
resonance, are no longer big issues. For VSC-HVDC systems, large ac capacitors are
not needed for reactive-power compensation. Moreover, as shown in Chapter 4, for both
vector current control and power-synchronization control, the control system can easily
provide damping to resonances in ac systems at any frequency 1 .
However, weak-ac-system connections still represent more challenging operation
conditions for VSC-HVDC systems due to the following reasons:
1

However, as it is shown by [12], the problem of low-harmonics resonance might not be trivial for vector
current control, but it is not a problem for power-synchronization control thanks to the high-pass current
control function.

90

5.2. Comparison of power-synchronization control and vector current control
1. The filter-bus voltage is more sensitive to power variations of the VSC-HVDC link
in weak ac systems. Thus, a weak-ac-system connection requires that the control
system of the VSC-HVDC link must be less dependent on the stiffness of the filterbus voltage.
2. The SCR of the ac system imposes a theoretical limitation on the maximum power
that the VSC-HVDC system is possible to transmit to or from the ac system. This
limitation has traditionally been analyzed by the maximum power curve (MPC) for
conventional LCC-HVDC systems [5, 87]. As it was discussed in Chapter 4, such
power limitations are basic characteristics of the ac system that are related to the
operating-point-dependent zeros of the ac Jacobian transfer matrix. Moreover, for
a VSC-HVDC link, the RHP zero of the ac Jacobian transfer matrix imposes a
fundamental limitation on the achievable bandwidth of the control system.
3. The off-diagonal elements of the Jacobian transfer matrix are larger for weak-acsystem connections, especially with high loadings. This problem is in common for
vector current control and power-synchronization control, but it is more serious for
vector current control. High off-diagonal elements imply more interactions between
the active-power control and the alternating-voltage control.
4. Model uncertainties, either due to variations of ac-network configurations or operating points, are generally higher for weak-ac-system connections. The control
system of the VSC-HVDC link is required to be robust for model variations.
In the following sections, the above issues related to high-impedance ac systems will be
addressed. The low-inertia ac-system connection issue will be discussed in Chapter 6.

5.2 Comparison of power-synchronization control and vector current control
In this section, the dynamic performance of power-synchronization control and vector
current control are compared for VSC-HVDC links connected to high-impedance weak
ac systems. For simplicity, the impedance-source system shown in Fig. 4.4 is assumed
as the ac-network configuration. Except for the quantities that are varied, the parameters
of the network configuration, initial conditions, and control systems, are based on those
defined in Table 4.2 and Table 4.3. The comparisons are performed by analyzing the characteristics of the Jacobian transfer matrices in the frequency domain and by verifications
using time simulations.
Fig. 5.1 shows the Bode plots of the transfer functions of the Jacobian transfer
matrix for vector current control regarding the variations of the SCRs of the ac system.
The VSC-HVDC converter is assumed to operate with zero loading P = 0.0 p.u., i.e.,
91

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

10

10

-2

0

10

-2

q

d

|JPI (jω)|

0

|JPI (jω)|

10

10

10

4

10

10

-2

10
ω (rad/sec)

2

10

4

0

10
ω (rad/sec)

2

10

4

0

10

0

-2

fq

fd

|JU I (jω)|

0

10

2

10
ω (rad/sec)

|JU I (jω)|

10

0

10

0

2

10
ω (rad/sec)

10

4

10

Fig. 5.1 Bode plots of the transfer functions of the Jacobian transfer matrix for vector current
control with P = 0.0 p.u. (solid: SCR = 5.0, dashed: SCR = 2.0, dotted: SCR = 1.0).

10

10

-2

0

10

-2

q

d

|JPI (jω)|

0

|JPI (jω)|

10

10

10

4

10

10
ω (rad/sec)

2

10

4

0

10
ω (rad/sec)

2

10

4

0

10

0

-2

fq

fd

|JU I (jω)|
-2

10

0

10

2

10
ω (rad/sec)

|JU I (jω)|

10

0

10

0

2

10
ω (rad/sec)

10

4

10

Fig. 5.2 Bode plots of the transfer functions of the Jacobian transfer matrix for vector current
control with SCR = 1.0 (solid: P = 0.0 p.u., dashed: P = 0.5 p.u., dotted: P =
0.7 p.u.).

92

5.2. Comparison of power-synchronization control and vector current control
static synchronous compensator (STATCOM) mode. For simplicity, only the magnitudes
of the transfer functions are plotted.
As it was mentioned before, one typical feature of high-impedance ac systems is
that the filter-bus voltage is more sensitive to power variations of the VSC-HVDC link.
This effect is reflected in transfer functions JUf Id (s) and JUf Iq (s) where their magnitudes
are getting larger with decreased SCRs of the ac system. Meanwhile, the magnitudes of
the off-diagonal transfer functions, increase with lower SCRs, which imply difficulties
for the outer-loop controllers, i.e., more interactions between the active-power control
loop and the alternating-voltage control loop are expected. However, even more severe
difficulties are experienced with increased loadings of the VSC-HVDC link that are shown
below.
Fig. 5.2 shows the Bode plots of the transfer functions of the Jacobian transfer matrix regarding the variations of the loadings of a VSC-HVDC link connected to an ac
system with SCR = 1.0. It is easily observed that the magnitudes of the off-diagonal
transfer functions increase dramatically with increased loadings, while the magnitude of
JPId (s) decreases. Beyond 50% of the VSC-HVDC loading (the dashed curves), the magnitudes of the off-diagonal transfer functions are greater than those of the diagonal transfer
functions. In control theory, such a process is called ill-conditioned process which implies
particular difficulties for the outer-loop controllers [82]. Of all the transfer functions, the
variation of JUf Id (s) plays the most critical role. From Fig. 4.22c in Chapter 4, it can be
found that JUf Id (s) has −180◦ phase shift at low frequencies which means that the increase of iref always results in a decreased filter-bus voltage Uf , or vise versa.2 This effect
d
is greatly amplified with higher-loading conditions for a VSC-HVDC converter connected
to a weak ac system. This is also the reason to that the magnitude of JPId (s) decreases
with high-loading conditions as will be shown by the time-simulation results in below.
Figs. 5.3-5.5 show the time-simulation results from PSCAD/EMTDC by applying a
0.1 p.u. step to iref at 0.1 s. The pre-conditions of the three figures correspond to the three
d
operating points of the Bode plots in Fig. 5.2 respectively. The time-simulation results
agree with the conclusion drawn from the frequency-domain analysis. For example, the
time-simulation results in Fig. 5.5 show that, with 70% loading of the VSC-HVDC link,
the 0.1 p.u. step of iref does not give any increase of the active-power output due to the
d
decreased filter-bus voltage Uf , although the converter current ic still follows the current
cd
ref
reference id . The same conclusion can be drawn by the frequency-domain analysis.
In the following, similar frequency-domain analysis and time simulations are performed for power-synchronization control. The network configurations and operating
points are chosen identical as those for the analysis of vector current control.
Fig. 5.6 shows the Bode plots of the transfer functions of the Jacobian transfer
matrix for power-synchronization control regarding variations of SCRs of the ac system.
2

JPIq (s) and JUf Iq (s) also have −180◦ phase shift at low frequencies, but they are in the same direction
for iref .
q

93

iref
d

0.1

ic

cd

0.05

P (p.u.)

d

cd

iref, ic (p.u.)

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

0
0

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1
0.05

Uf (p.u.)

0
0

1
0.8
0

time (sec)

Fig. 5.3 Step response of id for vector current control at P = 0.0 p.u. with SCR = 1.0.

ref

id

0.6

ic

cd

0.55

d

cd

iref, ic (p.u.)

0.65

0.5
0

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

P (p.u.)

0.65
0.6
0.55

Uf (p.u.)

0.5
0

1
0.8
0

time (sec)

Fig. 5.4 Step response of id for vector current control at P = 0.5 p.u. with SCR = 1.0.

94

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

iref, ic (p.u.)

ref

id
ic

cd

0.75

d

cd

0.8

P (p.u.)

0.7
0

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.8
0.75
0.7

Uf (p.u.)

0.65
0

1
0.8
0

time (sec)

Fig. 5.5 Step response of id for vector current control at P = 0.7 p.u. with SCR = 1.0.

0

10

10

-2

0

10

-2

|JPθ(jω)|

|JPV(jω)|

10

10

10

4

-2

10

0

10

2

10
ω (rad/sec)

10

10
ω (rad/sec)

2

10

4

0

10
ω (rad/sec)

2

10

4

0

10

0

-2

f

f

|JU θ(jω)|

|JU V(jω)|

10

0

10

0

2

10
ω (rad/sec)

10

4

10

Fig. 5.6 Bode plots of the transfer functions of the Jacobian transfer matrix for power-synchronization control with P = 0.0 p.u. (solid: SCR = 5.0, dashed: SCR = 2.0, dotted:
SCR = 1.0).

95

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
It is interesting to note that the magnitudes of the off-diagonal transfer functions decrease
slightly with decreased SCRs of the ac system. However, the variations of the magnitudes
of JPθ (s) are larger than JPId (s) in Fig. 5.1. Similar to vector current control, Fig. 5.7
shows that the magnitudes of the off-diagonal transfer functions increase with higher
loadings of the VSC-HVDC system for power-synchronization control, which means that
the outer-loop controllers of power-synchronization control also experience difficulties
with increased loadings of the VSC-HVDC system. However, the magnitude of JUf θ (s)
is much lower than JUf Id (s) with the same operating conditions. Besides, the magnitude
of JPθ (s) does not decrease as dramatically as JPId (s) with increased loadings of the
VSC-HVDC system.
The above difference between JUf θ (s) and JUf Id (s) is the key factor that affects
the dynamic performance of power-synchronization control and vector current control
for VSC-HVDC links connected to weak ac systems. With power-synchronization control, the active power is controlled by the phase angle of the VSC voltage, meanwhile
the magnitude of the VSC voltage is held constant. Such a control strategy minimizes
dynamic variations of the filter-bus voltage. However, with vector current control, the
inner-current control loop only controls the currents, while control of the filter-bus voltage is left to the outer-loop controller. Accordingly, dynamic variations of the filter-bus
voltage are much higher for vector current control than for power-synchronization control. Such dynamic variations of the filter-bus voltage create substantial difficulties for the
outer-loop controllers.
Figs. 5.8-5.10 show the simulation results from PSCAD/EMTDC when applying
a 0.1 rad step to θv for power-synchronization control. Comparing with the simulation
results with vector current control, the variations of the filter-bus voltage are much smaller
for power-synchronization control in response to the same power changes. Besides, the
dynamic responses of the open-loop transfer functions for power-synchronization control
are more stable than those for vector current control since the Jacobian transfer matrix
for vector current control includes more feedback loops, especially the PLL introduces a
delay. The difference of the dynamic response can also be explained by the Bode plots of
the transfer functions, where the magnitudes are “flatter” at low frequencies for powersynchronization control than for vector current control.
Generally speaking, with power-synchronization control, the control system can
maintain stable operation with load angle up to θu0 = 60◦ , i.e., approximately 86% of the
rated power, for VSC-HVDC links connected to ac systems with SCR = 1.0, whereas
it is very difficult for vector current control to operate even with 50% loading. Ref. [45]
gives more simulation results for such comparisons.
However, if the connected ac system is strong, vector current control seems to be
more advantageous than power-synchronization control. Figs. 5.11-5.12 show the Bode
plots of the transfer functions of the Jacobian transfer matrix for vector current control
and power-synchronization control with SCR variations from 3 to 10. In Fig. 5.11 vector
96

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

0

10

10

-2

0

10

-2

|JPθ(jω)|

|JPV(jω)|

10

10

10

4

-2

10

0

10

2

10
ω (rad/sec)

10

10
ω (rad/sec)

2

10

4

0

10
ω (rad/sec)

2

10

4

0

10

0

-2

f

f

|JU θ(jω)|

|JU V(jω)|

10

0

10

0

2

10
ω (rad/sec)

10

4

10

θv (rad)

Fig. 5.7 Bode plots of the transfer functions of the Jacobian transfer matrix for power-synchronization control with SCR =1.0 p.u. (solid: P = 0.0 p.u., dashed: P = 0.5 p.u., dotted:
P = 0.7 p.u.).

0.1
0.05

P (p.u.)

0
0

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1
0.05
0
0

1

f

U (p.u.)

0.1

0.8
0

time (sec)

Fig. 5.8 Step response of θv for power-synchronization control at P = 0.0 p.u. with SCR = 1.0.

97

θv (rad)

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

0.7
0.65
0.6
0

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

P (p.u.)

0.65
0.6
0.55

U (p.u.)

0.5
0

f

1
0.8
0

time (sec)

Fig. 5.9 Step response of θv for power-synchronization control at P = 0.5 p.u. with SCR = 1.0.

θv (rad)

1.05
1
0.95

P (p.u.)

0.9
0

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.8
0.75
0.7
0.65
0

1

f

U (p.u.)

0.1

0.8
0

time (sec)

Fig. 5.10 Step response of θv for power-synchronization control at P = 0.7 p.u. with SCR = 1.0.

98

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

10

10

-2

0

10

-2

q

d

|JPI (jω)|

0

|JPI (jω)|

10

10

10

4

10

10

-2

10
ω (rad/sec)

2

10

4

0

10
ω (rad/sec)

2

10

4

0

10

0

-2

q

d

|JQI (jω)|

0

10

2

10
ω (rad/sec)

|JQI (jω)|

10

0

10

0

2

10
ω (rad/sec)

10

4

10

Fig. 5.11 Bode plots of the transfer functions of the Jacobian transfer matrix for vector current
control with P = 0.0 p.u. (solid: SCR =10.0 p.u., dashed: SCR =5.0 p.u., dotted:
SCR =3.0 p.u.).

0

10

10

-2

0

10

-2

|JPθ(jω)|

|JPV(jω)|

10

10

10

4

-2

10

0

10

2

10
ω (rad/sec)

10

10
ω (rad/sec)

2

10

4

0

10
ω (rad/sec)

2

10

4

0

10

0

-2

f

f

|JU θ(jω)|

|JU V(jω)|

10

0

10

0

2

10
ω (rad/sec)

10

4

10

Fig. 5.12 Bode plots of the transfer functions of the Jacobian transfer matrix for power-synchronization control with P = 0.0 p.u. (solid: SCR =10.0 p.u., dashed: SCR =5.0 p.u.,
dotted: SCR =3.0 p.u.).

99

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
current control controls the reactive power instead of controlling the alternating voltage,
which is acceptable for strong ac-system connections.
By comparing Fig. 5.11 and Fig. 5.12, it can be seen that the magnitude of the offdiagonal elements of the Jacobian transfer matrix for vector current control are generally
less than those for power-synchronization control, and more importantly, the magnitudes
of the diagonal elements of the Jacobian transfer matrix for vector current control are
much less sensitive to SCR variations than those for power-synchronization control. In
other words, vector current control is more robust than power-synchronization control if
the connected ac system is guaranteed to be strong.
Fig. 5.13 shows the time-simulation results of vector current control from PSCAD/EMTDC by applying 0.1 p.u. steps with iref and iref respectively where the connected ac
q
d
system has an SCR of 10. Vector current control shows excellent reference tracking and
decoupling capabilities.
It should be noted that the Jacobian transfer matrices include various control functions for both power-synchronization control and vector current control. The parameters
of those control functions can also affect the above frequency-domain analysis and timesimulation results. However, the selection of control parameters will not affect the major conclusions drawn in this section since the difference between vector current control
and power-synchronization control is more related to their fundamental control strategies.
Therefore, such an analysis is not given in the thesis but typical values for those control
parameters are selected.
Among the two control methods compared in this section, power-synchronization
control is the most suitable control system for VSC-HVDC links connected to weak ac
systems. Consequently, only power-synchronization control will be applied for the remaining work of the thesis.

5.3 Multivariable feedback designs
As shown in Section 5.2, the magnitudes of the off-diagonal transfer functions of the Jacobian transfer matrix become larger for weak-ac-system connections. To achieve high
control bandwidth and to decouple the interactions between the active-power control loop
and the alternating-voltage control loop, multivariable feedback control design should be
applied. Fig. 5.14 shows the general control block diagram for multivariable feedback designs, i.e., a controller transfer matrix K(s) with four controllers K11 (s), K12 (s), K21 (s),
and K22 (s) is applied for active-power and alternating-voltage control. In this section,
two multivariable feedback design approaches are investigated: Internal model control
(IMC) and H∞ control. For simplicity, the impedance-source system shown in Fig. 4.4 is
assumed as the ac-network configuration.
100

P (p.u.)

iref, ic (p.u.)
d
cd

5.3. Multivariable feedback designs

0.1

iref

0.05

icd

d
c

0
0

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1
0.05
0
0

iref, ic (p.u.)
q
cq

0.1

iref

0.1

0.2

0

Q (p.u.)

0.1
0.05
0
-0.05
0
0.05
0
-0.05
-0.1

0.1

0.2

q

ic
cq
0.3

0.4

0.5

0.3

0.4

0.5

time (sec)

Fig. 5.13 Step response of iref and iref for vector current control at P = 0.0 p.u. with ac-system
q
d
SCR = 10.0.

K11 (s)
Pref + -

K12 (s)

U ref

K 21 (s)
+-

+
+

+

∆θ v

∆P
AC-Jacobian
transfer
∆V
matrix
∆U f
V0
J ( s)

+

K 22 (s)

Fig. 5.14 Multlivariable feedback control of grid-connected VSCs using power-synchronization
control

101

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
r

+

K (s)

u

J (s)

y

~
J ( s)

~ -+
y

Fig. 5.15 Block diagram of the internal model control system.

5.3.1 Internal model control
The concept of IMC design is based on the idea that the transfer functions of the closedloop system can be directly shaped by explicitly including a process model in the controller [88]. Fig. 5.15 shows the the block diagram of the IMC design, where r = [Pref Uref ]T ,
u = [∆θv ∆V /V0 ]T , and y = [P Uf ]T . From Fig. 5.15, it can be easily observed that, if
the process J (s) is stable and the process model J (s) is equal to J (s), the whole system
is internally stable if and only if K (s) is stable. Thus, if K (s) is chosen as
K (s) = J−1 (s) F (s)

(5.4)

y = J (s) J−1 (s) F (s) r = F (s) r.

(5.5)

the output response would be

Therefore, the bandwidth of the closed-loop system would be solely determined by the
filter F (s), which is often defined as a first-order filter with the desired bandwidth. However, for VSC-HVDC applications, the above philosophy cannot work directly, since the
Jacobian transfer matrix J (s) contains an RHP transmission zero as shown in Chapter 4.
A direct inverse of J (s) would end up with an unstable controller.
In such a situation, a factorization technique needs to be applied [89]. Consequently,
the process model can be factorized into two parts
J (s) = Jp (s) Jn (s)

(5.6)

where Jn (s) contains the invertible elements and Jp (s) contains the non-invertible elements.
An easy factorization method is to place the RHP transmission zero at the diagonal
part of Jp (s) as
ZRHP −s
0
ZRHP +s
Jp (s) =
(5.7)
ZRHP −s
0
ZRHP +s
where ZRHP is the RHP transmission zero of J (s). The invertible matrix Jn (s) can be
solved by (5.6) and (5.7). A low-pass filter matrix F (s) is chosen to specify the closed102

5.3. Multivariable feedback designs
K ' (s)

r

+

+

-

K (s)

u

J (s)

y

+
~
J ( s)

Fig. 5.16 IMC in classical feedback-control structure.

loop bandwidth
F (s) =

αp
s+αp

0

0

αu
s+αu

(5.8)

where αp and αu are the desired bandwidths of the active-power controller and alternatingvoltage controller. In specifying the bandwidth αp and αu , the RHP transmission zero of
J (s) has to be considered. As a rule of thumb, the bandwidth of the closed-loop system
should be chosen at least lower than half of the location of the RHP zero [82].
Thus, the controller is designed as
K (s) = J−1 (s) F (s) .
n

(5.9)

If the model is assumed to be perfect, i.e., J (s) J−1 (s) = Jp (s), then the output response
n
would be
y = Jp (s) F (s) r.
(5.10)
Because both Jp (s) and F (s) are diagonal, the resulting closed-loop response is also
diagonal.
The IMC controller can easily be formulated into the classical feedback-control

structure, as shown by the block diagram in Fig. 5.16. The controller K (s), therefore,
can be written as

K (s) = K (s) [I − J (s) K (s)]−1
(5.11)
where I represents the identity matrix. Substituting (5.6) and (5.9) into (5.11) yields


K (s) = J−1 (s) F (s) [I − Jp (s) F (s)]−1 .
n

(5.12)

Due to its feedback-control nature, IMC is able to compensate for load disturbances and
model uncertainties. However, for controller design, a “nominal” model needs to be defined. The “nominal” model should be chosen in the “center” of the possible process
models such that the model uncertainties are minimized within all the possible ac-network
configurations and operating conditions.
103

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
For VSC-HVDC applications, this is not a simple task, since there are a few factors
that affect the model uncertainties of the Jacobian transfer matrix. The following is just a
simple rule to give an initial guidance. The nominal model has to be re-adjusted based on
the results of the robustness tests as shown later.
For power-synchronization control, the power-angle relationship has the dominant
role in the controller design. From Fig. 5.6, it can be observed that the magnitude of
JPθ (s) varies with the SCR of the ac system. The uncertainty of JPθ (s) should be given
higher priority in the controller design since it affects the effective gain of the active-power
controller. If the ac system is simplified as an impedance source and the ac capacitor at
the filter bus is neglected, it is known from Table 4.1 that the static magnitude of JPθ (s) is
inversely proportional to the inductance L, which is the sum of the ac-system inductance
Ln and the phase-reactor inductance Lc . If the short-circuit ratios of the weakest and
strongest ac-system scenarios are SCRmin and SCRmax respectively, it is reasonable to
choose the short-circuit ratio SCRnom of the nominal ac system such that
1
1
=
1
2
ω1 Lc + SCRnom

1
1
+
1
1
ω1 Lc + SCRmin
ω1 Lc + SCRmax

.

(5.13)

2ω1 Lc SCRmin SCRmax + (SCRmin + SCRmax )
.
2 + ω1 Lc (SCRmin + SCRmax )

(5.14)

The solution to (5.13) is found to be
SCRnom =

If there is no other preference, the operating point of the nominal model should be chosen
with the full loading of the VSC-HVDC link.
Since the process model is included in the IMC controller, the order of the controller is high if the ac-system configuration is complex. Thus, a simple process model
should be chosen as the nominal model to reduce the order of the controller. For this particular application, the ac capacitor at the filter bus is neglected in the nominal model, but
included in the robustness tests as a model uncertainty.
Fig. 5.17 shows the step response of the active-power and alternating-voltage of
the closed-loop system of the nominal model. The parameters and initial conditions of the
nominal model are given in Table 5.1. As shown by the figure, the IMC controller successfully decouples the cross-coupling between the two control channels. However, the test
result shows that the controller has poor robustness with power-direction uncertainty. The
reason is that the transfer function JPV (s) is power-direction dependent (cf. Table 4.1).
Since the power direction is a known information for a VSC-HVDC link, it is feasible to
design two different controllers for the rectifier and the inverter operation modes respectively.
Fig. 5.18 and Fig. 5.19 show the robustness-test results of the IMC controllers regarding variations of the SCRs of the ac system and the power levels for rectifier and
inverter operation respectively. With an SCR = 1.0 p.u., the maximum loading is chosen
as P = 0.86 p.u. instead of P = 1.0 p.u.
104

5.3. Multivariable feedback designs

From: Pref To: P

From: Uref To: P
1
P (p.u.)

P (p.u.)

1
0.5
0
0

0.5
0

0.1
0.2
time (sec)

0.3

0

From: Pref To: Uf
1
Uf (p.u.)

Uf (p.u.)

0.3

From: Uref To: Uf

1
0.5
0
0

0.1
0.2
time (sec)

0.5
0

0.1
0.2
time (sec)

0.3

0

0.1
0.2
time (sec)

0.3

Fig. 5.17 Active-power and alternating-voltage step responses of the IMC controller with the
nominal model. Control parameters: αp = 100 rad/sec, αu = 100 rad/sec.

Table 5.1 Parameters of the nominal model used for IMC and H∞ control design. Per unit based
on 350 MVA and 195 kV.

Parameters
Rc
ω 1 Cf

0.17 p.u.
0.01 p.u.

ω1 Ln

0.667 p.u.

E0

1.0 p.u.

V0

1.08 p.u.

θu0 (θv0 )

40.6◦ (51◦ )

kv
High-pass current control

0.2 p.u.

Rn

Initial conditions

0.01 p.u.

ω1 Lc
Main-circuit parameters

Values

0.45 p.u.

αv

40 rad/s

105

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

From: Pref To: P

From: Uref To: P
1.5

1

1

P (p.u.)

P (p.u.)

1.5

0.5
0
-0.5
0

0.5
0
-0.5

0.1
0.2
time (sec)

0.3

0

1
0.5
0
0

0.3

From: Uref To: Uf

Uf (p.u.)

Uf (p.u.)

From: Pref To: Uf

0.1
0.2
time (sec)

1
0.5
0

0.1
0.2
time (sec)

0.3

0

0.1
0.2
time (sec)

0.3

Fig. 5.18 Robustness tests by active-power and alternating-voltage step responses with the IMC
controller for the VSC-HVDC converter operating in inverter mode. Model variations:
SCR = 1, 2, 5. P = 0.0, 0.5, 1.0 p.u.

From: Pref To: P

From: Uref To: P
1.5

1

1

P (p.u.)

P (p.u.)

1.5

0.5
0

0

-0.5
0

0.5

-0.5
0.1
0.2
time (sec)

0.3

0

From: Pref To: Uf
1
Uf (p.u.)

Uf (p.u.)

0.3

From: Uref To: Uf

1
0.5
0
0

0.1
0.2
time (sec)

0.5
0

0.1
0.2
time (sec)

0.3

0

0.1
0.2
time (sec)

0.3

Fig. 5.19 Robustness tests by active-power and alternating-voltage step responses with the IMC
controller for the VSC-HVDC converter operating in rectifier mode. Model variations:
SCR = 1, 2, 5. P = 0.0, −0.5, −1.0 p.u.

106

5.3. Multivariable feedback designs

+

K (s)

u

J (s)

(s )

I+

¡

 
r

A

M

(s )

+
+

y

-

Fig. 5.20 Additive and multiplicative uncertainties.

For the real system, the measuring signals are usually low-pass filtered, and the
controller computation and PWM switching can also introduce a delay. Thus, a transfer
matrix with the expression
∆J (s) =

1
e−Td s
1+τ s

0

0

1
e−Td s
1+τ s

(5.15)

is cascade-connected to the Jacobian transfer matrix J (s) for the robustness tests. The
parameters of ∆J (s) are chosen as τ = 3 ms and Td = 0.5 ms.
The design of the IMC controller is an iterative process. If the robustness is not
satisfied, the controller has to be detuned, i.e., choosing lower values of αp and αu , to meet
the robustness requirement. In the next subsection, another multivariable feedback design
approach, i.e., H∞ control, is introduced. With H∞ control, the robustness requirement
on the controller can be explicitly specified in the initial controller design stage.

5.3.2 H∞ control
H∞ control is another methodology for multivariable controller design. The main feature of H∞ control is its explicit way in dealing with model uncertainties, i.e., an H∞
controller can achieve closed-loop stability with satisfactory performance under process
variations as well as in the presence of other uncertainties such as disturbances and errors
in the sensors [82, 90].
Fig. 5.20 shows a standard multivariable feedback-control block diagram including process uncertainties. ∆A (s) and ∆M (s) represent the additive and multiplicative
uncertainties respectively. The transfer matrices S(s), R(s) and T(s) are defined as
S (s) = [I + J (s) K (s)]−1
R (s) = K (s) [I + J (s) K (s)]−1
T (s) = J (s) K (s) [I + J (s) K (s)]−1

(5.16)

where S(s) and T(s) are known as sensitivity and complementary sensitivity functions
respectively. The matrix R(s) does not have a name yet.
107

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
The maximum singular value of the sensitivity function σ [S (jω)] determines the
¯
disturbance attenuation, because S (s) is, in fact, the closed-loop transfer function from
load disturbances to process output. Thus, a disturbance attenuation performance specification can be written as
−1
σ [S (jω)] ≤ |W1 (jω) |.
¯
(5.17)
−1
Allowing the weighting function |W1 (jω) | to depend on frequency ω enables to specify
a different attenuation factor for each frequency ω. With H∞ control, the performance can
be achieved by finding the controller K (s) through solving the problem

||W1 (jω) S (jω) ||∞ < γ

(5.18)

where ∞ denotes the infinite norm. The value γ is a constant which indicates the accuracy
to which the optimal loop matches the desired loop shape. W1 (s) is a diagonal transfer
matrix that has W1 (s) as the diagonal elements.
The maximum singular value of the complementary sensitivity function σ [T (jω)]
¯
is used to measure the stability margins with respect to ∆M , as shown in Fig. 5.20.
Assuming the additive uncertainty ∆A (jω) = 0, taking σ [∆M (jω)] to be the
¯
definition of the size of ∆M (jω), the size of the smallest multiplicative destabilizing
uncertainty ∆M (jω) is
1
.
(5.19)
σ [∆M (jω)] =
¯
σ [T (jω)]
¯
The smaller is σ [T (jω)], the greater will be the size of the smallest destabilizing multi¯
plicative perturbation, and hence the greater will be the stability margin.
A similar result is available for relating the stability margin with respect to the
additive plant perturbations
σ [∆A (jω)] =
¯

1
.
σ [R (jω)]
¯

(5.20)

As a consequence of (5.19) and (5.20), the stability margins of control systems are specified via singular-value inequalities such as
−1
σ [R (jω)] ≤ |W2 (jω) |
¯

−1
σ [T (jω)] ≤ |W3 (jω) |.
¯

(5.21)

It is common practice to lump the effects of all plant uncertainties into a single ficti−1
tious multiplicative perturbation ∆M , i.e., allowing the weighting function |W3 (jω) |
to depend on frequency ω to specify a different attenuation factor for each frequency ω.
The stability margin can be achieved by finding the controller K (s) through solving the
problem
||W3 (jω) T (jω) ||∞ < γ
(5.22)
where W3 (s) is also a diagonal transfer matrix that has W3 (s) as the diagonal elements.
In order to guarantee closed-loop stability and at the same time to achieve desired control
108

5.3. Multivariable feedback designs
performance under process uncertainties, the objectives of performance and robust stability can be simultaneously achieved by finding a controller K (s) that satisfies both (5.18)
and (5.22). The solution to the optimal problem is often called H∞ controller based on
mixed performance and robustness objectives. The numerical methods for solving H∞
optimization problem are usually complex. Thus, commercial softwares, such as MATLAB, are commonly used to ease such tasks. The major work for the designer is then to
specify W1 (jω) and W3 (jω) to meet the design requirement of the control performance
and robustness.
Following the definition in (5.16), the sensitivity function S (s) and complementary
sensitivity function T (s) have the relation
S (s) + T (s) = I

(5.23)

i.e., σ [S (jω)] and σ [T (jω)] cannot both be small at the same frequency. The relationship
¯
¯
between S (s) and T (s) reflects the inherent conflict between control performance and
robustness. Usually, this conflict can be resolved by requiring σ [S (jω)] to be small at low
¯
frequencies, and σ [T (jω)] to be small at high frequencies, due to the fact that the control
¯
performance is more important in the low-frequency range, while measurement noise and
process uncertainties are more often of high-frequency nature.3
With H∞ design, the closed-loop system performance is basically defined by the
weighting functions W1 (jω) and W3 (jω). Optimization algorithms are used to synthesize the controller with the bandwidth between the crossover frequency of W1 (jω) and
W3 (jω). Similar to IMC design, the RHP transmission zero of the Jacobian transfer matrix has to be considered in specifying the bandwidth of the closed-loop system, i.e., the
desired bandwidth should be lower than at least half of the location of the RHP zero. The
basic principle for selecting weighting functions are to give W1 (jω) a low-pass property
and W3 (jω) a high-pass property. It is necessary to ensure that the crossover frequency
for the Bode plot of W1 (jω) is below the crossover frequency of W3 (jω), such that there
is a gap for the desired loop shape to pass. Fig. 5.21 shows the Bode plots of W1 (jω) and
W3 (jω), which have the following transfer functions
s
M

+ ωS
s + AS ωS
s + ωT
M
W3 (s) =
AT s + ωT
W1 (s) =

(5.24)

where M represents the desired bounds on ||S(jω)||∞ and ||T(jω)||∞, AS and AT are
the desired disturbance attenuation inside bandwidth for S(jω) and T(jω), and ωS and
ωT are the crossover frequencies of W1 (jω) and W3 (jω) respectively.
Fig. 5.22 shows the active-power and alternating-voltage step responses of the linear closed-loop system of the nominal model where the nominal model is also chosen
3

Unfortunately, this is not fully true for VSC-HVDC applications, since the variations of SCRs of the ac
system and power levels of the VSC-HVDC link create process uncertainties in the low-frequency range.

109

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

2

10

1

10

0

10

|W1 (jω)|, |W3 (jω)|

10

-1

|W1 (jω)|

10

0

10

1

|W3 (jω)|

2

10
ω (rad/sec)

10

3

10

4

Fig. 5.21 Bode plots (magnitude) of wighting functions W1 (jω) and W3 (jω). Parameters: M =
2.0, AS = 0.0005, AT = 0.03, ωS = 70 rad/sec, ωT = 130 rad/sec [solid: weighting
function |W1 (jω) |, dashed: weighting function |W3 (jω) |].

From: Pref To: P

From: Uref To: P
1
P (p.u.)

P (p.u.)

1
0.5
0
0

0.5
0

0.1
0.2
time (sec)

0.3

0

From: Pref To: Uf
1
Uf (p.u.)

Uf (p.u.)

0.3

From: Uref To: Uf

1
0.5
0
0

0.1
0.2
time (sec)

0.5
0

0.1
0.2
time (sec)

0.3

0

0.1
0.2
time (sec)

0.3

Fig. 5.22 Active-power and alternating-voltage step responses of the H∞ controller with the nominal model.

110

5.3. Multivariable feedback designs
as defined in Table 5.1. Similar to the IMC controller, the H∞ controller also successfully decouples the cross-coupling between the two control channels. However, different
from IMC, where a transparent algorithm was used, H∞ control achieved the decoupling through optimization. Consequently, a slight cross-coupling can still be observed in
Fig. 5.22. The parameters of the weighting functions W1 (s) and W3 (s) have been chosen
in such a way that the H∞ controller has the same response time as the IMC controller in
order to compare the results from the two controllers.
Fig. 5.23 and Fig. 5.24 show the robustness-test results of the H∞ controller regarding variations of SCRs of the ac system and power levels of the VSC-HVDC link for
rectifier and inverter operations respectively.
In these two sub-sections, two multivariable feedback design approaches, i.e., IMC
and H∞ control, have been applied for a VSC-HVDC link connected to a weak ac system. Time simulations are performed with the linear models to verify the robustness of
the controllers regarding variations of SCRs of the ac system and power levels of the
VSC-HVDC link. From the time-simulation results, no significant difference is observed
between the two controllers. In the next subsection, a comparison in the frequency domain
is given.

5.3.3 Performance and robustness comparison
To show the advantages of the multivariable feedback designs proposed in the previous
subsections, the diagonal controller proposed in Chapter 3 is used as a reference design,
i.e.,
kp
0
s
.
(5.25)
K (s) =
0 ksu
Fig. 5.25 shows the active-power and alternating-voltage step responses of the VSCHVDC converter using the diagonal controller. The nominal model is chosen to be the
same as that is used for the IMC and the H∞ control design, i.e., with the parameters
given in Table 5.1. The control parameters kp and ku are tuned so as to give similar response times as the other two controllers.
Fig. 5.26 and Fig. 5.27 show the robustness tests for the VSC-HVDC converter
operating in rectifier mode and inverter mode respectively. Generally speaking, the timedomain responses of the diagonal controller are more oscillatory than those of the other
two controllers since the diagonal controller disregards the cross-coupling between the
two control loops.
Fig. 5.28 shows the comparison of the above three controllers by plotting the maximum singular values of their sensitivity functions σ [S (jω)] together with the magnitude
¯
−1
of the weighting function W1 (s) that was used to synthesize the H∞ controller. The
values of σ [S (jω)] for the IMC and H∞ controllers are almost identical, which is in
¯
111

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

From: Pref To: P

From: Uref To: P
1.5

1

1

P (p.u.)

P (p.u.)

1.5

0.5
0
-0.5
0

0.5
0
-0.5

0.1
0.2
time (sec)

0.3

0

1
0.5
0
0

0.3

From: Uref To: Uf

Uf (p.u.)

Uf (p.u.)

From: Pref To: Uf

0.1
0.2
time (sec)

1
0.5
0

0.1
0.2
time (sec)

0.3

0

0.1
0.2
time (sec)

0.3

Fig. 5.23 Robustness tests by active-power and alternating-voltage step responses with the H∞
controller for the VSC-HVDC converter operating in inverter mode. Model variations:
SCR = 1, 2, 5. P = 0.0, 0.5, 1.0 p.u.

From: Pref To: P

From: Uref To: P
1.5

1

1

P (p.u.)

P (p.u.)

1.5

0.5
0

0

-0.5
0

0.5

-0.5
0.1
0.2
time (sec)

0.3

0

From: Pref To: Uf
1
Uf (p.u.)

Uf (p.u.)

0.3

From: Uref To: Uf

1
0.5
0
0

0.1
0.2
time (sec)

0.5
0

0.1
0.2
time (sec)

0.3

0

0.1
0.2
time (sec)

0.3

Fig. 5.24 Robustness tests by active-power and alternating-voltage step responses with the H∞
controller for the VSC-HVDC converter operating in rectifier mode. Model variations:
SCR = 1, 2, 5. P = 0.0, −0.5, −1.0 p.u.

112

5.3. Multivariable feedback designs

From: Pref To: P

From: Uref To: P
1
P (p.u.)

P (p.u.)

1
0.5
0
0

0.5
0

0.1
0.2
time (sec)

0.3

0

From: Pref To: Uf
1
Uf (p.u.)

Uf (p.u.)

0.3

From: Uref To: Uf

1
0.5
0
0

0.1
0.2
time (sec)

0.5
0

0.1
0.2
time (sec)

0.3

0

0.1
0.2
time (sec)

0.3

Fig. 5.25 Active-power and alternating-voltage step responses of the diagonal controller with the
nominal model. Control parameters: kp = 50 rad/s, ku = 60.

From: Pref To: P

From: Uref To: P
1.5

1

1

P (p.u.)

P (p.u.)

1.5

0.5

0.5

0

0

-0.5

-0.5

0

0.2
0.4
time (sec)

0.6

0

From: Pref To: Uf
1
Uf (p.u.)

Uf (p.u.)

0.6

From: Uref To: Uf

1
0.5
0
0

0.2
0.4
time (sec)

0.5
0

0.2
0.4
time (sec)

0.6

0

0.2
0.4
time (sec)

0.6

Fig. 5.26 Robustness tests of the diagonal controller for the VSC-HVDC converter operating in
rectifier mode. Model variations: SCR = 1, 2, 5. P = 0.0, −0.5, −1.0 p.u.

113

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
From: Pref To: P

From: Uref To: P
1.5
P (p.u.)

P (p.u.)

1.5
1
0.5

1
0.5

0

0

-0.5

-0.5

0

0.2
0.4
time (sec)

0.6

0

From: Pref To: Uf
1
Uf (p.u.)

Uf (p.u.)

0.6

From: Uref To: Uf

1
0.5
0
0

0.2
0.4
time (sec)

0.5
0

0.2
0.4
time (sec)

0.6

0

0.2
0.4
time (sec)

0.6

Fig. 5.27 Robustness tests of the diagonal controller for the VSC-HVDC converter operating in
inverter mode. Model variations: SCR = 1, 2, 5. P = 0.0, 0.5, 1.0 p.u.

agreement with the step responses shown in Fig. 5.17 and Fig. 5.22. However, the value
of σ [S (jω)] of the diagonal controller is not bounded by W −1 (s), which implies that
¯
the performance of the diagonal controller is worse than the performance requirement
specified by W −1 (s), especially with the peak at around ω = 80 rad/s. This also generally agrees with the time-domain performance. The step response shown in Fig. 5.25 for
the diagonal controller, is more oscillatory than the step responses shown in Fig. 5.17 and
Fig. 5.22 for the IMC and H∞ controllers. The oscillation mode of the diagonal controller
is at around ω = 80 rad/s.
Fig. 5.29 shows the comparison of the above three controllers by plotting the maximum singular values of their complementary sensitivity functions σ [T (jω)] together
¯
−1
with the magnitude of the weighting function W3 (s) that was used to synthesize the
H∞ controller. The values of σ [T (jω)] for the IMC and H∞ controllers are almost iden¯
tical, but the IMC controller, in fact, shows even slightly lower values at frequencies above
ω = 400 rad/s. Of course, this difference cannot be reflected by the robustness tests in
the time domain since a τ = 3 ms (ω = 333.3 rad/s) low-pass filter has been applied in
the test model [cf. (5.15)]. The frequency-domain comparison confirms the similarity of
robustness of the two controllers by the time-domain tests. In Fig. 5.29, it shows that the
value of σ [T (jω)] of the diagonal controller is not bounded by the weighting function
¯
−1
W3 (s), which implies that the robustness of the diagonal controller is worse than the
−1
robustness requirement specified by W3 (s), mainly at the peak around ω = 80 rad/s.
By comparing Fig. 5.28 with Fig. 5.29, both σ [S (jω)] and σ [T (jω)] for the diagonal
¯
¯
controller have a high peak around ω = 80 rad/s, which seem to be contradictory to the
114

5.3. Multivariable feedback designs

-1
1

|W (jω)|

Diagonal controller

10

0

H controller
σ [S(jω)]



IMC controller
10

-1

10

-2

10

0

10

1

2

10
ω (rad/sec)

10

3

10

4

Fig. 5.28 Maximum singular values of the sensitivity functions for nominal performance comparison [solid: H∞ controller, dashed: IMC controller, dotted: diagonal controller, thick
−1
line: weighting function |W1 (jω) |].

|W-1(jω)|
3

10

Diagonal controller

0

σ [ T(jω)]

H∞ controller

10

-1

IMC controller

10

-2

10

0

10

1

2

10
ω (rad/sec)

10

3

10

4

Fig. 5.29 Maximum singular values of the complementary sensitivity functions for robustness
comparison [solid: H∞ controller, dashed: IMC controller, dotted: diagonal controller,
−1
thick line: weighting function |W3 (jω) |].

115

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
constraint in (5.23). However, since S(s) and T(s) are transfer matrices, they also differ
by the phases of the transfer functions. Thus, the real meaning of the constraint in (5.23)
is that σ [S (jω)] and σ [T (jω)] cannot both be small at the same frequency, but they can
¯
¯
certainly both be large at the same frequency. In fact, it can be mathematically proven
that σ [S (jω)] and σ [T (jω)] differ at most by 1 [91], i.e., a very high σ [S (jω)] always
¯
¯
¯
implies a very high σ [T (jω)]. In practice, for a controller that yields very poor perfor¯
mance with the nominal model, it certainly implies that the controller would also have
poor robustness for model variations!
Remark 1: It is remarkable that the IMC controller solved by the simple algorithm
achieves similar performance and robustness (or even slightly better robustness in the
high-frequency range) as the more advanced optimization-based H∞ control. Generally
speaking, IMC design might yield poor controllers for processes that have poorly damped
poles, since IMC simply inverses the process while the poorly damped poles, in fact,
still remain in the system. However, with power-synchronization control, this problem is
avoided, since the high-pass current control function HHP (s) has shifted all the resonant
poles of the Jacobian transfer matrix towards the left-half plane (cf. Fig. 4.11). On the
other hand, compared to H∞ control, the order of the controller solved by IMC is lower.
With the nominal model used in this section, IMC design yields a controller only of 4th
order while H∞ design yields a controller of 8th order.
Remark 2: The sensitivity function S(s) is a good indicator of the performance
of the closed-loop system. However, it is only valid for the performance of the nominal
model. Meanwhile, the bounded complementary sensitivity function T(s) only ensures
robust stability, i.e., the closed-loop system is only stable within the specified range of
model uncertainties. Thus, these two functions cannot be used as indicators of the performance of the worst-case scenario, i.e., robust performance. To synthesize controllers
that fulfil robust performance, the structured singular value, i.e., the µ, needs to be applied [92]. However, controllers synthesized by µ can be conservative for this application
since the variations of SCRs of the ac system and power levels of the VSC-HVDC link are
real-number uncertainties, while µ is only tight for complex-number perturbations [93],
e.g., model uncertainties in the high-frequency range. On the other hand, in this section, it
has been demonstrated that the robustness tests performed in the time domain provides a
good overview of the performance of the closed-loop system with all the possible scenarios. Accordingly, the frequency-domain approach and the time-domain approach should
be used as complementary tools for controller design and evaluation.
Remark 3: In this section, the controllers are designed and tested based on the simple impedance-source representation of the ac system. In reality, the topology of the ac
system is, of course, much more complex. However, given the same SCR and the same
operating point of the VSC-HVDC converter, such variations with the process model appear mainly in the high-frequency range. These types of uncertainties can easily be dealt
116

5.4. Direct-voltage control
with by feedback control, especially with the high-pass current control function applied.
Therefore, for VSC-HVDC applications, the major challenges for robustness of the controller are the variations of SCRs of the ac system and power levels of the VSC-HVDC
converter. One exception is that, if there is a series-compensated ac line feeding into the
converter station, the series-compensated ac line should be included in the nominal model
for the controller design instead of being treated as a model uncertainty, since the characteristics of a series-compensated ac line can be special as shown in Chapter 4. Another
situation is that, if there are other large input devices in the vicinity of the converter stations and their dynamic interactions with the VSC-HVDC link are of concern, then an
equivalent system is necessary to be developed to properly reflect the topology of the ac
system.

5.4 Direct-voltage control
For VSC-HVDC applications, at least one of the converter stations has to control the direct
voltage, while the other converter station controls the active power. The active power is
thus automatically balanced between the two converter stations. In this section, various
aspects of direct-voltage control are discussed.

5.4.1 Controller design
The block diagram of the direct-voltage controller is shown in Fig. 5.30, where the multivariable power-synchronization controller is used as an inner loop. For the direct-voltage
controller, a PI controller is proposed with the control law
Pref = − Kpd +

Kid
s

(uref )2 − u2 .
dc
dc

(5.26)

Fdc (s)

If the power-synchronization controller is assumed to be sufficiently fast, i.e., Pref =
PVSC , and the dc link is assumed to be a single capacitor, i.e., Gdc (s) = 2/(sCdc )
[cf. (4.75)], then the closed-loop system is expressed as
u2 =
dc

s2 C

2(Kpds + Kid )
(uref )2 .
+ 2Kpd s + 2Kid dc
dc

(5.27)

The control parameters are selected as
2
Kpd = αd Cdc and Kid = αd Cdc /2

(5.28)

which places two real poles of the closed-loop system at s = −αd . It is generally difficult
to achieve reference tracking and disturbance reduction by a single feedback controller.
117

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
Power-synchronization
control inner loop

K 22 (s)
U ref + f1

K 21 (s)

P ref
1

K12 (s)
+-

+

+

∆V
V0

∆θ v
+

∆U f 1
AC-Jacobian
transfer
− ∆PVSC1
matrix
∆P
1

J (s )

+

DC-Jacobian − ∆PVSC2
transfer
matrix
G dc ( s)
2
∆udc1

K11 (s)

Fdc1 (s)

2
∆udc2

ref
(udc1 ) 2

+
K r (s)

Fig. 5.30 Control block diagram of the direct-voltage controller.

The solution is to use a two degree-of-freedom controller, where the reference (uref )2
dc
is connected to a prefilter to improve the reference tracking while the controller Fdc (s)
is only tuned for disturbance reduction. A convenient practical choice of prefilter is the
lead-lag network
1 + sT1
Kr (s) =
(5.29)
1 + sT2
where T1 > T2 is chosen to speed up the response and T1 < T2 is chosen to slow down
the response. The effect of the prefilter is demonstrated in Fig. 5.31, which shows that the
overshoot of the step response of the direct-voltage controller is removed by the prefilter.
As it has been shown in Chapter 4, for long overhead lines, the dc-line inductance
and the dc capacitors may create a resonance peak in the low-frequency range. If the
resonance appears within the bandwidth where the direct-voltage controller is active, an
effective way to mitigate its impact is to add a notch filter. A notch filter commonly has
the expression
2
s2 + 2ξ1 ωn s + ωn
Fn (s) = 2
(5.30)
2
s + 2ξ2 ωn s + ωn
where the three adjustable parameters are ξ1 , ξ2 , and ωn . The ratio of ξ2 /ξ1 sets the depth
118

5.4. Direct-voltage control

0.5

u

dc

(p.u.)

1

0
0

0.05

0.1
time (sec)

0.15

0.2

Fig. 5.31 Step response of the direct voltage. The overshoot of the direct voltage is removed by a
prefilter (solid: without prefilter, dashed: with prefilter). Direct-voltage controller: αd =
40 rad/sec. Prefilter: T1 = 0.075 sec, T2 = 0.0925 sec.

of the notch, and ωn is the resonance frequency. Fig. 5.32 shows the effect of the notch
filter by the open-loop transfer function of the direct-voltage control, which has the expression
Hdc (s) = −Fdc (s)Fn (s)Gdc11 (s).

(5.31)

where the parameters of Gdc11 (s), i.e., Gdc (s) are given in Table 4.4. It should be noted
that the notch filter may adversely affect the phase margin of the direct-voltage controller,
as shown in Fig. 5.32.

5.4.2 DC-capacitance requirement
In Section 5.4.1, it has been shown that, by explicitly including the dc capacitance in the
parameters of the direct-voltage controller, one can freely place the poles of the closedloop system. This implies that, as long as its size is known, the dc-capacitance does not
affect the linear stability of the direct-voltage control. However, for disturbance reduction,
the dc capacitance has an important role. In this section, the dc-capacitance requirement
for weak-ac-system connections is discussed. The applied method follows the procedure
introduced by [85]. To simplify the analysis, the dc link is assumed to be the single
dc-capacitor model, i.e., Gdc (s) = 2/(sCdc ). If the power-synchronization loop has a
bandwidth that is considerably higher than that of the direct-voltage control, the transfer
function from the load disturbance to the error signal becomes
Gpe (s) =

1+

2
sCdc
α
s+ 2d
αd Cdc
s

2
sCdc

=

2s
.
Cdc (s + αd )2

(5.32)

If the worst scenario is considered, i.e., the active-power output to the other converter
station is changed stepwise from the maximum active power Pm to 0, e.g., the converter
is suddenly blocked. The step response of the error signal edc (t) in the time domain
119

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
2

10

0

10

|Hdc(jω)|

10

-2

10

1

10

2

10

3

10

2

10

3

arg Hdc(jω) (deg.)

100
0
-100
-200
-300 1
10

ω (rad/sec)

Fig. 5.32 Bode plots of Hdc (s) to show the effect of the notch filter for reducing the dc resonance
peak (solid: without notch filter, dashed: with notch filter). Direct-voltage controller:
αd = 40 rad/sec. Notch filter: ωn = 322 rad/sec, ξ1 = 0.2, ξ2 = 0.8.

becomes
edc (t) = L−1 Gpe (s)
= L−1

Pm
s

2Pm
Cdc (s + αd )2

=

2Pm − αd
te t .
Cdc

(5.33)

The time derivative of edc (t) is
dedc
2Pm
=
(1 − αd t) e−αd t
dt
Cdc

(5.34)

which has a local maximum for t = 1/αd . By substituting this into (5.33), the maximum
error is found to be
2Pm −1
edc,max =
e .
(5.35)
αd Cdc
By considering udc,max the maximum direct-voltage allowed, then edc,max = u2
dc,max −
ref 2
(udc ) , the required dc capacitance is
Cdc >

2Pm e−1
.
ref 2
[u2
dc,max − (udc ) ]αd

(5.36)

A common expression for dc capacitance is using its time constant Tdc , which is defined
as
2
Cdc0 UdcN
Cdc
Tdc =
=
(5.37)
2PdcN
2
120

5.4. Direct-voltage control
where UdcN is the rated (base) direct voltage (kV) and PdcN is the rated (base) power
(MW) of the VSC-HVDC link, and Cdc0 is the dc capacitance (µF). Substituting (5.37)
into (5.36) yields
Pm e−1
Tdc > 2
.
(5.38)
[udc,max − (uref )2 ]αd
dc

It should be noted, for the dc-capacitance requirement, only per unit values of the direct voltage and the dc power are applicable for (5.38), while either per unit values or
real values can be applied in (5.36). For weak-ac-system connections, the VSC-HVDC
link needs to operate with large load angles, where the RHP transmission zero of the ac
Jacobian transfer matrix J(s) moves closer to the origin. The RHP zero limits the bandwidth of the power-synchronization controller, and eventually limits the bandwidth of the
direct-voltage control. A rule of thumb is that the bandwidth of the power-synchronization
controller should be lower than half of the location of the RHP zero [82]. If the voltages
of the ac source and the filter bus are assumed to be nominal, i.e., E0 ≈ 1.0 p.u. and
Uf 0 ≈ 1.0 p.u., the location of the RHP zero can be simplified from (4.17) as
s = ±ω1

E0 cos θu0
≈ ±ω1
Uf 0 − E0 cos θu0

cos θu0
.
1 − cos θu0

(5.39)

Based on the definition of SCR in (5.3), the well-known power-angle equation between
the ac source and the filter bus can be expressed as
P =

E0 Uf 0
sin θu0 ≈ SCR sin θu0 .
ω1 Ln

(5.40)

If it is further assumed that the direct-voltage control is four times slower than the powersynchronization loop, then αd can be solved by (5.39) and (5.40). Accordingly,
1
αd < ω1
8

cos θu0
1
= ω1
1 − cos θu0
8

Pm
1 − ( SCR )2

1−

Pm
1 − ( SCR )2

.

(5.41)

Another issue that needs to be taken into account is that, if the bandwidth of the directvoltage controller is chosen four times slower than the power-synchronization loop, the
inner loop will affect the maximum voltage variation. Fig. 5.33 shows the ratio of bandwidth reduction by the inner loop, which is approximated by a first-order filter with bandwidth αp . Considering this effect, (5.38) is adjusted as
Tdc >

1.3Pm e−1
.
ref 2
[u2
dc,max − (udc ) ]αd

(5.42)

Given the maximum loading Pm and maximum allowed direct voltage udc,max , the relationship between dc-capacitance requirement and the SCR of the ac system can be established by (5.41) and (5.42). If the worst scenario is considered, the maximum loading
121

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

2
1.5

e

dc,dmax

/e

°
dc,max

2.5

1
1

2

3

4

5

6

7

8

9

10

α /α
p

d

Fig. 5.33 Bandwidth reduction by the inner loop.
30

T

dc

(ms)

25
20
15
10
5
1

1.2

1.4

1.6

1.8

2

SCR

Fig. 5.34 DC-capacitance requirements for weak-ac-system connections with uref = 1.0 p.u.
dc
(solid: udc,max = 1.2 p.u., dashed: udc,max = 1.3 p.u., dotted: udc,max = 1.4 p.u.).

should be chosen as Pm = 1.0 p.u. However, as it was mentioned before, for very weakac-system connections, it is recommended that the load angle shall not be above 60◦ to
maintain a reasonable stability margin. For instance, if the SCR of the ac system is 1.0,
then the maximum loading is Pm = SCR sin θu0 = 0.86 p.u. Fig. 5.34 shows the plots
of dc-capacitance requirements for ac systems with SCR ≤ 2.0 with different allowed
udc,max .

5.5 Interconnection of two very weak ac systems
In this section, interconnection of two very weak ac systems by a VSC-HVDC link is
investigated by one design example and followed by linear-model verification and ac-fault
studies simulated by PSCAD/EMTDC. In the design example, a 350 MW VSC-HVDC
link is applied to interconnect two very weak ac systems, where the ac system connected
to the inverter side (station 1, power receiving end) has network strength SCR1 = 1.0,
while the ac system connected to the rectifier side (station 2, power sending end) has
network strength SCR2 = 1.2. The detailed parameters of the converter stations are given
in Appendix C.
From Fig. 5.30, it can be easily observed that, if one converter station controls
the direct voltage, while the other converter station controls the active power, the two
122

5.5. Interconnection of two very weak ac systems
U ref
f1
P ref
1

DC-Jacobian
PowerPower− ∆PVSC1
transfer − ∆PVSC2 synchronization
synchronization
matrix
inner loop
∆P
inner loop
1
G dc ( s)
2
∆udc1

Fdc1 (s)
+
K r1 (s)

ord
1

P +

Fpc (s)

+

P2ref

2
∆udc2

-

-

U ref
f2

Fdc2 (s)
+
K r2 (s)

ref 2
dc1

ref
(udc2 )2

+ (u )

ref
∆(udc1 )2

Fig. 5.35 Control block diagram of a VSC-HVDC link interconnecting two weak ac systems.

converter stations are, in fact, linearly independent. This implies that the stability of one
converter station does not affect the stability of the other converter station. One might
consequently conclude that there is nothing more special for a VSC-HVDC link connected
to two weak systems than it is only connected to one weak system. This might be true
if only linear effects are considered. However, the real system is non-linear. For VSCHVDC operations, the direct voltage has to be carefully maintained around its nominal
value. For instance, a big direct-voltage drop might temporarily limit the capability of the
alternating-voltage controller and negatively affect the linear stability of the closed-loop
system.
The proposed control structure for weak-ac-system interconnection is shown in
Fig. 5.35, where the dashed block represents the power-synchronization control inner
loop in Fig. 5.30. The basic idea of the design is that both of the two converter stations
have direct-voltage controllers, while the active-power controlling station controls the
active power by adding an additional contribution to the reference of the direct-voltage
controller and its output shall be limited. With the proposed control structure the linear
independence between the rectifier station and the inverter station is lost. However, the
bandwidth of the direct-voltage controllers is much higher than the bandwidth of the acsystem dynamics. This implies that the “firewall” effect of the VSC-HVDC link is still in
force.
In the following, the detailed procedure for controller design and parameter settings
are given. The linear model is compared to the simulation results from PSCAD/EMTDC
for each major design step.
1. Power-synchronization inner loop. Due to the very low SCR of the inverter ac
system, the maximum allowed power is Pm1 = 0.86 p.u., which corresponds to approximately θu10 = 60◦ load angle. The ac system at the rectifier side has a slightly
123

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
higher SCR, but considering the losses of the converter station and of the dc cable,
it has a maximum loading Pm2 = −0.91 p.u., which corresponds approximately to
a load angle of θu20 = −50◦ .
The power-synchronization controller is used as the inner loop of the direct-voltage
controller as previously shown in Fig. 5.30. For this application, the IMC controller
investigated in Section 5.3 is applied to parameterize the power-synchronization
controller. According to the discussion in Section 5.4.2, the bandwidth of the powersynchronization controller should be lower than half of the RHP-zero location of
the ac Jacobian transfer matrix. By using (5.39), the bandwidths of the activepower controllers of power-synchronization control at the two converter stations
can be calculated as αp1 = 160 rad/s and αp2 = 210 rad/s. The bandwidths of the
alternating-voltage controllers, however, do not need to be very high. Thus, they
have been chosen as αu1 = αu2 = 50 rad/s.
2. Direct-voltage controller. As mentioned in Section 5.4.1, the bandwidth of the
direct-voltage controllers should be chosen at least four times slower than the activepower controller of the power-synchronization control. Alternatively, the bandwidth
of the direct-voltage controllers can be calculated directly by substituting the SCR
and maximum loading into (5.41), which yields αd1 = 39 rad/s and αd2 = 53 rad/s.
However, the linear analysis shows that the direct-voltage controller at the rectifier
side (station 2) tends to have less phase margin for the same load angle than that at
the inverter (station 1) side. Therefore, the bandwidths of both of the direct-voltage
controllers have been chosen as αd1 = αd2 = 40 rad/s. Fig. 5.36 shows the step
response of the direct voltages at the two converter stations with the comparison
between the linear model and the nonlinear model in PSCAD/EMTDC.
The maximum allowed direct voltage is 1.3 p.u. Based on the curves of the dccapacitance requirement in Fig. 5.34, the dc capacitor has been chosen to make the
total dc capacitance Tdc = 15 ms. Fig. 5.37 shows the plots of disturbance reduction of the direct-voltage controllers at the two converter stations with the comparison between the linear model and the nonlinear model in PSCAD/EMTDC. The
maximum direct-voltage peak in Fig. 5.37(b) is below the specification udc,max =
1.3 p.u. The discrepancies between the results of the linear models and the nonlinear simulations are due to that the results of the linear model are obtained at
one operating point, while the disturbance reduction in PSCAD/EMTDC involves
operating-point changes. The tests in PSCAD/EMTDC are performed by blocking
the converter at the other station with maximum loading.
3. Active-power controller. In principle, either of the VSC-HVDC converter stations
can be the power-controlling station. In this example, the inverter station (station 1)
controls the active power by a proportional-integral (PI) controller with the control
124

5.5. Interconnection of two very weak ac systems
1.15
udc1 (p.u.)

1.1
1.05
1
0.95
0

0.05

0.1
time (sec)

0.15

0.2

0.15

0.2

(a)

1.1
1.05

u

dc2

(p.u.)

1.15

1
0.95
0

0.05

0.1
time (sec)

(b)

Fig. 5.36 Step response of the direct-voltage controllers (solid: nonlinear simulation, dashed: linear model). (a) Direct-voltage step at station 1. (b) Direct-voltage step at station 2.

udc1 (p.u.)

1
0.8
0.6
0

0.1

0.2

0.3

0.4

0.5

time (sec)

(a)
1.4
udc2 (p.u.)

1.3
1.2
1.1
1
0

0.1

0.2

0.3

0.4

0.5

time (sec)

(b)

Fig. 5.37 Disturbance reductions of the direct-voltage controllers with maximum load changes
(solid: nonlinear simulation, dashed: linear model). (a) Converter block at station 2. (b)
Converter block at station 1.

125

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
law
∆(uref )2 = −Kpp (1 +
d1

Kip
ord
) P1 − P1 .
s

(5.43)

Fpc (s)

The parameters of the active-power controller are tuned by the root-locus technique
in two steps:
• Step 1: Start with tuning the proportional gain Kpp by applying P control, i.e.,
Fpc (s) = −Kpp .
• Step 2: The integral gain Kip is tuned by applying the full PI controller.
Fig. 5.38 shows the root-loci of the closed-loop system by applying the proportional controller. By varying Kpp from 0.0 to 1.0, three dominant pole pairs are
affected. The pole pair p5,6 (p5 = −10.8 rad/s, p6 = −0.296 rad/s) is shifted towards the left-half plane, which can be viewed as the stabilizing effect of feedback
control. However, both the pole pairs p1,2 (p1,2 = −63.7 ± j232 rad/s) and p3,4
(p3 = −37.1 rad/s, p4 = −50 rad/s) are shifted towards the right-half plane. The
frequency-domain analysis shows that p1,2 is related to the gain margin, while p3,4
is related to the phase margin of the active-power control. Kpp = 0.6 is chosen to
get a balance of stability and response time, which places the three pole pairs at
p1,2 = −53 ± j227 rad/s, p3,4 = −27 ± j24 rad/s, and p5,6 = −15.1 ± j3 rad/s.
Fig. 5.39 shows the root-loci of the closed-loop system by applying the PI-type
active-power controller, where p0 is a pole introduced by the integral controller. By
varying Kip from 0.0 to 60, p0 moves quickly towards the left-half plane. The pole
pair p1,2 is rather insensitive to the variation of Kip , and p5,6 move towards two lefthalf plane zeros on the real axis. However, p3,4 is negatively affected. Frequencydomain analysis also shows that the phase margin of the active-power control is
reduced by increased integral gain Kip . However, a larger integral gain is necessary in reducing the power-recovery time after ac-system faults and minimizing the
steady-state error. Finally, Kip = 30 is chosen which places the three dominant pole
pairs at p1,2 = −54 ± j225 rad/s, p3,4 = −13.2 ± j29.4 rad/s, p5 = −28 rad/s,
and p6 = −10 rad/s.
Fig. 5.40 shows the step response of the active-power control at low and high power
levels respectively. The step response at the high power level corresponds to the
operating point applied for the root-locus tuning.
It should be noted that the output of the active-power controller should be limited
to avoid too large direct-voltage variations. In this example, the limitation of the
output of the active-power controller is chosen as ±0.25, which corresponds to
approximately +12% and −13% direct-voltage variations.
126

5.5. Interconnection of two very weak ac systems

250
p

200

1

150

ω (rad/sec)

100
p

50

3

p

5

0

p6

-50

p

4

-100
-150
p

-200
-250
-80

-70

2

-60

-50

-40
-30
ω (rad/sec)

-20

-10

0

10

Fig. 5.38 Root-loci of the dominant poles of the closed-loop system by applying P-type activepower control. Kip = 0.0, variations of Kpp from 0 to 1.0.

250
p

200

1

150

ω (rad/sec)

100
p

50

3

p

0

0
-50

p

4

-100
-150
p

-200
-250
-80

-70

-60

2

-50

-40
-30
ω (rad/sec)

-20

-10

0

10

Fig. 5.39 Root-loci of the dominant poles of the closed-loop system by applying PI-type activepower control. Kpp = 0.6, variations of Kip from 0 to 60.

127

P1 (p.u.)

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

0.8
0.75
0.7
0

0.1

0.2

0.3

0.4

0.5

0.3

0.4

0.5

time (sec)

P1 (p.u.)

(a)

0.1
0.05
0
0

0.1

0.2
time (sec)

(b)

Fig. 5.40 Step response of the active-power controller at low and high power levels at converter
station 1 (solid: nonlinear simulation, dashed: linear model). Active-power controller:
Kpp = 0.6, Kip = 30. (a) Step from P1 = 0.7 p.u. to P1 = 0.8 p.u. (b) Step from
P1 = 0.0 p.u. to P1 = 0.1 p.u.

The fault ride-through capability of the VSC-HVDC link is tested in PSCAD/EMTDC by applying three-phase ac-system faults at both of the converter stations. The
VSC-HVDC link initially operates with the maximum loading, i.e., P1 = 0.86 p.u. The
VSC-HVDC link is supposed to ride through ac-system faults without relying on telecommunications between the two converter stations.
In Fig. 5.41, a three-phase ac fault with 0.2 sec duration is applied at the inverter
station (station 1), i.e., the power-controlling station in this example, at 0.1 sec. One consequence of the ac fault is that the direct voltage increases to approximately 1.3 p.u. due
to the loss of power output. The rectifier station (station 2) brings down the direct voltage
to its nominal value after an initial overshooting. Another consequence of the ac fault is
the increase of the modulus of the valve current |ic1 |. After detecting the fault, the current
limiter reduces the converter current to half of the maximum load current Imax (or any
other desired value) except a very short current spike at the fault inception.
Fig. 5.42 shows a three-phase ac fault with 0.2 sec duration applied at the rectifier
station, i.e., the direct-voltage controlling station in this example, at 0.1 sec. During the
ac-system fault, the power-controlling station controls the direct voltage to a lower voltage
level which is 13% less than the nominal value. The 13% is a result of the limitation of the
active-power controller. Since the ac fault is applied at the rectifier side, there is no over128

5.5. Interconnection of two very weak ac systems

P1 (p.u.)

1
0.5
0
0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

udc1 (p.u.)

1.5
1
0.5
0
0
| ic1| (p.u.)

2
1
0
0

time(sec)

Fig. 5.41 Fault ride-through capability of the VSC-HVDC link during a three-phase ac-system
fault at the inverter station (station 1). Upper plot: active power from the VSC-HVDC
link at station 1. Middle plot: direct voltage at station 1. Lower plot: valve current at
station 1.

P1 (p.u.)

1
0.5
0
0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

udc1 (p.u.)

1.5
1
0.5
0
0
| ic1| (p.u.)

2
1
0
0

time (sec)

Fig. 5.42 Fault ride-through capability of the VSC-HVDC link during a three-phase ac-system
fault at the rectifier station (station 2). Upper plot: active power from the VSC-HVDC
link at station 1. Middle plot: direct voltage at station 1. Lower plot: valve current at
station 1.

129

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems
current problem at the inverter station (station 1). A specific amount of active power flows
in the reverse direction during the fault period. However, the current limitation controller
at the rectifier (station 2) limits the fault current to half of the maximum load current.

5.6 Summary
In this chapter, the control and modeling issues for VSC-HVDC links connected to highimpedance ac systems are investigated. Power-synchronization control and vector current
control are compared for VSC-HVDC links connected to weak ac systems. It is concluded that, for weak-ac-system connections, the voltage variations at the filter bus are
much less for power-synchronization control than for vector current control. Thus, powersynchronization control is the most suitable controller for VSC-HVDC links connected
to weak ac systems. Two multivariable feedback designs, i.e., internal model control and
H∞ control, are applied and compared with the simple diagonal controller. The multivariable feedback design show clear advantages in control performance and robustness.
The IMC controller is simple yet achieves similar results as the more advanced H∞ controller. A two-degree-of-freedom direct-voltage control with a prefilter to reduce the overshoot is proposed. For VSC-HVDC links connected to weak ac systems, a higher value
of dc capacitance is required. A control structure for interconnecting two very weak ac
systems is proposed. As an example, it is shown that a VSC-HVDC link using powersynchronization 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.

130

Chapter 6
Control of VSC-HVDC Links
Connected to Low-Inertia AC Systems
In this chapter, power-synchronization control is applied to control VSC-HVDC links connected to low-inertia ac systems, or island systems. In Section 6.1, general aspects of
low-inertia ac systems are described. In Section 6.2, a frequency droop controller and an
alternating-voltage droop controller are proposed. In Section 6.3, a typical island system
which includes four common power components, i.e., a synchronous generator, an induction motor, some passive RLC loads, and a VSC-HVDC link is modeled by the Jacobian
transfer matrix modeling concept. In addition, the linear model is applied for controlparameter tuning using root-locus techniques. In Section 6.4, simulation studies are performed with PSCAD/EMTDC to demonstrate the flexibility of power-synchronization control for various network conditions. In Section 6.5, the Jacobian transfer matrices for a
synchronous generator and an induction motor are developed, and their characteristics
are analyzed. Finally, the subsynchronous characteristics of a VSC-HVDC converter are
analyzed by the frequency-scanning method in Section 6.6. The major results of this chapter are summarized in Section 6.7. Some results of this chapter are included in [94].

6.1 General aspects of low-inertia ac systems
In this chapter, various aspects of a VSC-HVDC link connected to another type of weak
ac systems, i.e., low-inertia or island systems, are investigated. Low-inertia systems are
considered to have limited number of rotating machines in the system, or no rotating machine 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. It may also be the case when an
HVDC link initially is connected to a large ac system, but comes into island operation
due to tripping of critical ac lines to the large system, or black start after a blackout.
For the conventional LCC-HVDC system, the commutations of the converter valves
depend on the ability of the ac system to maintain the required voltage and frequency.
131

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
Therefore, the conventional LCC-HVDC system has a requirement on the minimum inertia of the connected ac system. Similar to the definition of SCR, it was proposed in [5] to
use the effective inertia constant Hdc as a measure of the relative rotational inertia, which
is defined as
Hac
Hdc =
(6.1)
PdcN
where Hac (MW · s) is the total rotational inertia of the ac system and PdcN is the rated dc
power. For a conventional LCC-HVDC link connected to an island system, an effective
inertia constant, Hdc , of at least 2.0 s to 3.0 s is required for satisfactory operation [32].
Synchronous condensers (SCs) have been the traditional means to increase the inertia of an island system [95]. Since SCs also increase the short-circuit capacity of the
ac system, the weak-ac-system problem of the conventional LCC-HVDC system, either
low SCR or low inertia, can generally be solved by installation of SCs. However, SCs can
substantially increase the investment and maintenance costs of an HVDC project.
In this respect, the VSC-HVDC technology has a clear advantage over the conventional LCC-HVDC technology. Different from the thyristor valves, where relatively
stiff voltage and frequency of the ac system are pre-conditions for valve commutations,
the VSC can produce its own alternating-voltage waveform independent of the ac system.
Thus, a VSC-HVDC link can even be connected to a passive network with no other power
source at all [96, 97].
However, control of VSC-HVDC links connected to an island system is technically
different from connected to a large ac system for the following reasons:
1. Due to the relatively limited number of power generating units in a typical island
system, the VSC-HVDC link should participate in frequency control of the island
system instead of applying constant power control.
2. An island system with low inertia is often of small geographic extent and the generators and loads are found electrically close to the converter station. More interactions are expected between the VSC-HVDC converter and other input devices, e.g.,
the risks of voltage-control hunting and SSTI with local synchronous generators are
higher.
3. In some situations, island operation is a consequence of ac-system failures by losing critical ac lines to a large ac system. The control system of the VSC-HVDC link
is required to be able to handle the transition of operation modes seamlessly.
In the following sections, the above issues related to low-inertia ac systems are investigated for a VSC-HVDC link using power-synchronization control.
132

6.2. Controller design

ω ref

Kf
+-

∆Pref

1 + sT f

+-

∆ω

Pref
+

kp

∆θ v

s

Island
system

P

s
1 + sTm

Fig. 6.1 Frequency droop controller.

6.2 Controller design
In this chapter, it is assumed that the VSC-HVDC link is connected to a strong system
at the other converter station, which controls the direct voltage of the dc link constant
disregarding the power variations of the converter connected to the island system. Thus,
only the controllers of the converter station connected to the island system are discussed.
In Chapter 5, multivariable feedback designs have been applied for VSC-HVDC
links connected to high-impedance ac systems. However, such designs are not considered
necessary for low-inertia system connections, since the bandwidth requirement for the
control system is generally not as critical for island operation. Thus, to simplify the control
design discussions, the simple diagonal controller in (5.25) is applied. The tuning of the
control parameters kp and ku will be further discussed in Section 6.3 using the root-locus
techniques.

6.2.1 Frequency droop control
For a VSC-HVDC link connected to an island system, the VSC-HVDC converter shall
not have constant power control but rather participate in frequency control of the island
system. A frequency controller of the following form is proposed
∆Pref =

Kf
1 + Tf s

[ωref − ω]

(6.2)

where ωref is the reference and ω is the measured value of the angular frequency. The
output of the frequency controller is added to the power reference of the active-power
controller, as shown in Fig. 6.1. In the figure, the frequency measurement is taken by
the derivative of the angle output of the active-power controller with a measuring time
constant Tm , typically in the range of 10 − 20 ms. Kf and Tf are the gain and time
constant of the frequency controller.
Such a design gives the VSC-HVDC link a frequency droop characteristic. If there
are any other power generating units in the island system, the load sharing is determined
133

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
PCC

Rtg + jX tg

SG
ig

VSC
iv

+
eg
-

Rtv + jX tv

P , Q1
1

+
uf
-

P2 , Q2

P, Q

Fig. 6.2 Voltage droop control for parallel connected voltage-control units.

by the frequency droop of each power generating unit, which is defined as R = ∆ω/∆P .
For two power generating units with frequency droops R1 and R2 , the following relationship is established for the power outputs ∆P1 and ∆P2
∆P1
R2
=
.
∆P2
R1

(6.3)

Accordingly, the power generating unit with smaller frequency droop shares more loads
in the island system. The frequency droop of the VSC-HVDC link with the proposed
frequency controller in (6.2) can be expressed as
∆ω =

1
kp

1
(Pref − P ).
+ Kf

(6.4)

Rvsc

Thus, the frequency droop of the VSC-HVDC link depends on both Kf and kp . The time
constant Tf should be chosen similar to the time constant of the turbines of the local
generators in the island system.

6.2.2 Alternating-voltage droop control
If two or more alternating-voltage controlling units are connected to a common bus, the
alternating-voltage controls should be coordinated to avoid voltage hunting between the
units. Fig. 6.2 shows a VSC-HVDC link connected in parallel with a synchronous generator with step-up transformers. In Fig. 6.2, eg , uf are the terminal/filter-bus voltage
vectors of the generator, the VSC-HVDC converter. P and Q are the active power and
reactive power to the island system. P1 and Q1 are the active power and reactive power
from the synchronous generator, while P2 and Q2 are the active power and reactive power
from the VSC-HVDC link. Rtg and Xtg are the resistance and leakage reactance of the
step-up transformer of the synchronous generator, while Rtv and Xtv are the resistance
and leakage reactance of the converter transformer of the VSC-HVDC converter. The solution to solve the voltage-control confliction between the synchronous generator and the
134

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

SG
ig

+

Large ac
system

eg
-

Ltv

Ltm

IM
im

VSC

+

+

em
-

ev
Ll

Rl

Cl +

iL

Passive
load

iv

AC network

u pcc
-

Fig. 6.3 A typical island system.

VSC-HVDC converter is using the so-called “load compensation”, which is a common
solution for synchronous generators terminated at the same bus [32]. That is, instead of
controlling the alternating voltage of the point-of-common-coupling (PCC) bus, the VSCHVDC converter and the synchronous generator both control voltages between their own
terminal/filter-bus voltages and the voltage of the PCC to give droop characteristics to
their alternating-voltage controls. With load compensation, the magnitudes of the resulting compensated voltages Vcg of the synchronous generator and Vcv of the VSC-HVDC
converter are
Vcg =|eg + kg (Rtg + jXtg )ig |

Vcv =|uf + kc (Rtv + jXtv )iv |

(6.5)

where the compensation ratios kg and kc are typically chosen between 50% and 80%. By
adjusting the compensating ratios, the reactive-power sharings between the VSC-HVDC
converter and the synchronous generator are re-distributed.

6.3 Dynamic modeling and linear analysis of a typical island system
Fig. 6.3 shows an island system introduced by [98], which includes four common components in a typical island system, i.e., a synchronous generator (SG), an induction motor
(IM), some passive RLC loads, and the VSC-HVDC link. The induction-motor loads are
135

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
included in the studied system since they are quite common in industrial systems [99].
The space vectors eg , em and ev are the terminal voltage vectors of the SG, the IM, and
the VSC-HVDC converter, while ig , im and iv are the corresponding injecting current
vectors. Ltg , Ltm , and Ltv are the leakage inductances of the step-up transformers of the
SG, the IM, and the converter transformer of the VSC-HVDC converter. Rl , Ll and Cl are
the resistance, inductance, and capacitance of the passive loads. iL is the current vector
flowing through Ll , and upcc is the voltage vector of the PCC. An ac line is connected to
a large ac system at the PCC bus. This connection is intentional added to demonstrate the
mode shift of the VSC-HVDC link in Section 6.4.

6.3.1 Jacobian transfer matrix
In Chapter 4, the concept of Jacobian transfer matrix was proposed as a unified dynamic
modeling technique for ac/dc systems. For the network configurations applied in Chapter 4, the ac Jacobian transfer matrices were derived directly from the state-space representation of the combined ac network and the main circuit of the VSC. Such a procedure
is practical if only one input device is connected to a simple ac network. However, for a
larger ac network connected to several input devices, it is necessary to develop the statespace representation of the ac network and the electrical part of each individual input
device separately as several “modules”, while the ac Jacobian transfer matrix is derived
by connecting all the “modules” together. Such an approach not only eases the task for
model development, but more importantly, the developed model is easier to be modified
if the topology of the ac network and/or the input devices are changed.
In the following subsections, such an approach is applied for the model development of the island system in Fig. 6.3. All the three input devices, i.e., the SG, the IM,
and the VSC-HVDC converter are modeled as current sources which inject currents to
the ac network, while the ac network supplies the terminal voltage to each input device [100, 101].
Modeling of the ac network
The ac network is defined as the electrical circuit inside the dashed box in Fig. 6.3. In
Chapter 4, a constant-frequency stiff voltage source E was assumed, while the d component of the synchronous dq frame was chosen aligned with the real part of E. Such a
constant-frequency stiff voltage source cannot be assumed in an island system. However,
for model development this is not really a problem. In fact, any electrical node in the island system can principally be chosen as a reference point, as long as the reference frame
is not “disturbed” by the dynamics of the system. For this particular application, the PCC
of the ac-network is chosen as the reference point, a common synchronous R − I reference frame is defined which has the R axis aligned with the real axis of the voltage vector
upcc . In the defined synchronous R − I reference frame, the dynamic equations of the ac
136

6.3. Dynamic modeling and linear analysis of a typical island system
network can be expressed as
diL
= upcc − jω1 Ll iL
dt
dupcc
upcc
Cl
= ig + im + iv − iL −
− jω1 Cl upcc
dt
Rl
dig
eg = upcc + jω1 Ltg ig + Ltg
dt
dim
em = upcc + jω1 Ltm im + Ltm
dt
div
ev = upcc + jω1 Ltv iv + Ltv
.
dt
Ll

(6.6)

By linearization and writing in component form, (6.6) can be expressed in state-space
form
dxn
= An xn + Bn un
dt
yn = Cn xn + Dn un + Dn1

dun
dt

(6.7)

where the matrices An and Bn are


0

ω1

1
Ll

0


 −ω
1
0
0

1
Ll

An = 
1
0 − Cl1Rl
ω1
 − Cl

1
0 − Cl −ω1 − Cl1Rl





0




 0

 , Bn = 
 1



 Cl

0

0

0

0

0

0

0

0

0

0

1
Cl

0

1
Cl

1
Cl

0

1
Cl

0

0




0 


0 


1
Cl

(6.8)

and the matrices Cn and Dn1 are



0 0 0 0 0 0

T



 0 0 0 0 0 0 


Cn = 
 , Dn1
 1 0 1 0 1 0 


0 1 0 1 0 1









=






Ltg

0

0

0

0

0

Ltg

0

0

0

0

0

Ltm

0

0

0

0

0

Ltm

0

0

0

0

0

Ltv

0

0

0

0

0

0




0 


0 


0 


0 


Ltv

(6.9)
137

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
Ra

-

ωrgψ q

+

Ll

isd

+

L1d

i1d

esd

L fd

i fd

Lad

R fd

+

R1d

-

e fd

Ra

+

+

ωrgψ d
-

Ll

isq

L1q

i1q

esq

L2 q

i2 q

Laq

R1q

-

R2 q

Fig. 6.4 Equivalent circuit of the synchronous generator.

and the matrix Dn is


0


 ω1 Ltg


 0

Dn = 
 0


 0

0

−ω1 Ltg

0

0

0

0

0

0

0

0

0

0

0

−ω1 Ltm

0

0

0

ω1 Ltm

0

0

0

0

0

0

0

−ω1 Ltv

0

0

0

ω1 Ltv

0

and the inputs, outputs, and state variables are
un =

∆igR ∆igI ∆imR ∆imI ∆ivR ∆ivI

yn =

∆egR ∆egI ∆emR ∆emI ∆evR ∆evI

xn =

∆iLR ∆iLI ∆upccR ∆upccI

T

.















(6.10)

T
T

(6.11)

The subscript R represents the real component, while I represents the imaginary component in the common ac-network R − I frame.
Modeling of the electrical part of the synchronous generator
Fig. 6.4 shows the equivalent circuit of a salient pole synchronous machine based on the
two-axis theory where the three-phase windings of the synchronous machine are repre138

6.3. Dynamic modeling and linear analysis of a typical island system
sented by two orthogonal windings along the d and q axes [32]. In a dq frame chosen along
the rotor DQ axis, the dynamic equations of the stator and the rotor can be expressed as
esd =
esq =
ef d =
0=
0=
0=

dψd
g
− ψq ωr − Ra isd
dt
dψq
g
+ ψd ωr − Ra isq
dt
dψf d
+ Rf d if d
dt
dψ1d
+ R1d i1d
dt
dψ1q
+ R1q i1q
dt
dψ2q
+ R2q i2q
dt

(6.12)

where esd and esq are the stator voltage components in the d and q directions respectively.
The quantity ef d is the field voltage. The quantities isd and isq are the stator current components in the d and q directions respectively. The quantity if d is the field current. The
quantities i1d , i1q , and i2q are the currents of the damping circuits (two damping circuits
in q axis). Ra is the stator resistance. R1d , R1q , and R2q are the resistances of the damping
circuits. Rf d is the resistance of the field circuit (d axis). ψd and ψq are the flux linkages of
the stator in the d and q directions. ψf d is the flux linkage of the field circuit. ψ1d , ψ1q , and
g
ψ2q are the rotor flux linkages. ωr is the rotor angular speed. The flux linkages in (6.12)
are expressed as
ψd = −(Lad + Ll )isd + Lad if d + Lad i1d
Lsd

ψq = −(Laq + Ll )isq + Laq i1q + Laq i2q
Lsq

ψf d = −Lad isd + (Lad + Lf d )if d + Lad i1d
Lf md

ψ1d = −Lad isd + Lad if d + (Lad + L1d )i1d
L1md

ψ1q = −Laq isq + (Laq + L1q )i1q + Laq i2q
L1mq

ψ2q = −Laq isq + Laq i1q + (Laq + L2q )i2q

(6.13)

L2mq

where Ll is the stator leakage inductance. Lad and Laq are the d and q axis mutual inductances. Lf d is the field leakage inductance. L1d , L1q and L2q are the leakage inductances
of the damping circuits. By linearization, the dynamic equations of the synchronous gen139

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
erator can be expressed in state-space form
dxg
= −L−1 R xg + L−1 Bu ug +
dt
Ag

yg =

L−1 Bω L−1 Bf

Bg

ug1

Bg1

1 0 0 0 0 0
0 1 0 0 0 0

xg

(6.14)

Cg

where the inductance matrix L is

−Lsd
0
Lad
Lad
0
0

 0
−Lsq
0
0
Laq
Laq


 −Lad
0
Lf md Lad
0
0

L=
 −Lad
0
Lad L1md
0
0


 0
−Laq
0
0
L1mq Laq

0
−Laq
0
0
Laq L2mq















and the matrix R is

g
g
g
−Ra
ωr0 Lsq
0
0
−ωr0 Laq −ωr0 Laq

g
g
g
 −ωr0 Lsd −Ra ωr0 Lad ωr0 Lad
0
0



0
0
Rf d
0
0
0

R=

0
0
0
R1d
0
0



0
0
0
0
R1q
0

0
0
0
0
0
R2q
and the matrices Bu , Bω , and Bf are
Bu =
Bω =

1 0 0 0 0 0
0 1 0 0 0 0

(6.15)















(6.16)

T

, Bf =

0 0

Rf d
g
ωr0 Lad

−isq0 Lsq (isd0 Lsd − if d0 Lad ) 0 0 0 0

0 0 0

T

T

(6.17)

and the inputs, outputs, and state variables are
ug =

∆esd ∆esq

yg =

∆isd ∆isq

T
T

, ug1 =
, xg =

g
∆ωr ∆ef d

T

∆isd ∆isq ∆if d ∆i1d ∆i1q ∆i2q

T

(6.18)

where the subscript 0 denotes the value of the operating point. In (6.18), the rotor speed
g
∆ωr is connected to the transfer function of the rotor, while the field voltage ∆ef d is
140

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

q

es

esI

ωrg
esq

d
esd

δ

ω1

esR

R

Fig. 6.5 AC-network R − I reference frame vs. synchronous-generator dq frame.

connected to the excitation control. The dynamic equations described by (6.12) is only
valid if the reference frame is chosen along the DQ axes of the rotor. In order to connect
the synchronous generator to the ac network, the state-space description in (6.14) has
to be transformed to the common ac-network R − I frame, as shown in Fig. 6.5. The
angle δ represents the electrical angle between the two reference frames. Accordingly, the
terminal voltage of the synchronous generator in the two reference frames are related as
edq = eRI e−jδ
s
s

(6.19)

in linearized component form
∆esd

=

∆esq

cos δ0

sin δ0

∆esR

− sin δ0 cos δ0

∆esI

+

−esR0 sin δ0 + esI0 cos δ0
−esR0 cos δ0 − esI0 sin δ0

PE

∆δ.

PE1

(6.20)
Similarly, the stator current of the synchronous generator in the two reference frames are
related as
iRI = idq ejδ
(6.21)
s
s
in linearized component form
∆isR
∆isI

=

cos δ0 − sin δ0

∆isd

sin δ0

∆isq

cos δ0

+

−isd0 sin δ0 − isq0 cos δ0
isd0 cos δ0 − isq0 sin δ0

PI

∆δ. (6.22)

PI1

Substituting (6.20) and (6.22) into (6.14), yields
dxg




= Ag xg + Bg u g + Bg1 u g1
dt



y g = Cg xg + Dg1 u g1

(6.23)
141

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
Lsλ

Rs

+

Lrλ

is

Rr
ir

es

+
jωrmψ r
-

Lm

-

Fig. 6.6 Equivalent circuit of the induction motor.

where


ug =


u g1 =

∆esR ∆esI

T

g
∆δ ∆ωr ∆ef d



Cg = PI Cg , Dg1 =



, yg=
T

T

∆isR ∆isI




, Bg = Bg PE , Bg1 =

PI1 zeros (2, 2) .

Bg PE1 Bg1
(6.24)

That is, the input and output variables have been transformed to the common ac-network

R − I frame. After reference-frame transformation, ug1 has one additional input variable
∆δ, which is connected to the rotor transfer function. It should be noted that the state
variables of (6.23) are still in the synchronous-generator dq frame.
Modeling of the electrical part of the induction motor
Fig. 6.6 shows the equivalent circuit of a single cage induction motor based on the twoaxis theory [32]. Different from the synchronous generator, the two axes of the induction
motor are symmetrical. Rs and Rr are the resistances of the stator and rotor respectively.
Lsλ and Lrλ are the inductances of the stator and rotor respectively. Lm is the mutual
inductance. The vectors is and ir are the stator and rotor current vectors respectively. ψr
is the rotor flux. The vector es is the stator voltage vector. The reference direction of the
stator current vector is chosen outwards to match the current direction defined in Fig. 6.3.
In a synchronous reference frame, the dynamic equations of the stator and rotor can be
expressed as
dψs
+ jψs ω1 − Rs is
dt
dψr
m
0=
+ jψr (ω1 − ωr ) + Rr ir
dt

es =

(6.25)

m
where ωr is the rotor angular speed. ψs and ψr are the stator and rotor fluxes, which are
defined as

ψs = −Ls is + Lm ir , ψr = Lr ir − Lm is
142

(6.26)

6.3. Dynamic modeling and linear analysis of a typical island system
where Ls = Lsλ + Lm and Lr = Lrλ + Lm . The dynamic equations of the induction motor
thus can be written in state-space form
dxm
= −L−1 R xm + L−1 Bu um + L−1 Bω um1
dt
Am

ym =

Bm

1 0 0 0
0 1 0 0

Bm1

xm

(6.27)

Cm

where the matrices L and R are

−Ls
0
Lm 0

 0
−Ls 0 Lm

L=
 −Lm
0
Lr 0






R=



0

−Lm

0









Lr

−Rs

ω1 Ls

0

−ω1 Ls

−Rs

ω1 Lm

0

m
(ω1 − ωr0 )Lm

Rr

m
−(ω1 − ωr0 )Lm

0

m
(ω1 − ωr0 )Lr

and the matrices Bu and Bω are
Bu =
Bω =

1 0 0 0



−ω1 Lm





m
−(ω1 − ωr0 )Lr 

Rr
0

(6.28)

T

0 1 0 0
0 0 (−Lm isq0 + Lr irq0 ) (Lm isd0 − Lr ird0 )

T

(6.29)

and the inputs, outputs, and state variables are
um =

∆esd ∆esq

ym =

∆isd ∆isq

T
T

m
, um1 = ∆ωr

, xm =

∆isd ∆isq ∆ird ∆irq

T

.

(6.30)

m
The angular speed ∆ωr of the induction motor is connected to the transfer function of the
rotor. For the induction motor, the common ac-network R−I frame can be directly chosen
as the reference dq frame. Therefore, there is no need for reference-frame transformation.

Modeling of the electrical part of the VSC-HVDC converter
Fig. 6.7 shows the main circuit of the VSC-HVDC converter, where Lv represents a small
fictitious inductance for the convenience of model development. In a synchronous reference frame, by taking into account HHP (s), the dynamic equations of the main circuit of
143

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
VSC
Lv

Rc

Lc

iv

V0∠θ v 0
ic

+

+

ev

uf

-

-

+
Cf

v

-

Fig. 6.7 Main circuit of the VSC-HVDC converter.

the VSC-HVDC converter can be expressed as [cf. (4.45)]
dic
= (V0 + ∆V )ejθv − HHP (s) ic − uf − Rc ic − jω1 Lc ic
dt
duf
Cf
= ic − iv − jω1 Cf uf
dt
div
Lv
= uf − ev − jω1 Lv iv .
dt
Lc

(6.31)

The Laplace transform variable s of HHP (s) can be eliminated in the same way as shown
in (4.47). By linearization and writing in component form, the dynamic equation of the
VSC-HVDC converter can be written in state-space form
dxv
= Av xv + Bv uv + Bv1 uv1
dt
yv = Cv xv

(6.32)

where the Av matrix is











Av = 









144

−αv −

Rc +kv
Lc

−ω1

ω1
−αv −

Rc +kv
Lc

1
1 0 − Lc

0 1

0

0 0 − αv
Lc

0

0

0

1
− Lc

0

0

0

0

0

− αv Rc
Lc

ω1 αv

−ω1 αv

− αv Rc
Lc

0 0

0

− αv
Lc

0

0

1
Cf

0

0 0

0

ω1

1
− Cf

0

0

1
Cf

0

0

1
− Cf

0

0

0 0

1
Lv

0

0

ω1

0

0

0 0

0

1
Lv

−ω1

0

0 0 −ω1






















(6.33)

6.3. Dynamic modeling and linear analysis of a typical island system
and the Bv , Bv1 , and Cv matrices are
Bv =

Bv1 =
Cv =

1
0 0 0 0 0 0 − Lv

T

0
1
− Lv

0 0 0 0 0 0

0

− V0 sin θv0
Lc

V0 cos θv0
Lc

sin
− αv V0Lc θv0

αv V0 cos θv0
Lc

0 0 0 0

V0 cos θv0
Lc

V0 sin θv0
Lc

αv V0 cos θv0
Lc

αv V0 sin θv0
Lc

0 0 0 0

T

0 0 0 0 0 0 1 0

(6.34)

0 0 0 0 0 0 0 1

and the inputs, outputs, and state variables are
T

, uv1 =

∆θv

∆V
V0

T

uv =

∆evd ∆evq

, yv =

xv =

∆icd ∆icq ∆ρcd ∆ρcq ∆uf d ∆uf q ∆ivd ∆ivq

∆ivd ∆ivq
T

.

T

(6.35)

The input ∆θv is connected to the active-power controller, while ∆V /V0 is connected to
the alternating-voltage controller. For the VSC-HVDC converter, the common ac-network
R − I frame can also be directly chosen as the synchronous dq frame to avoid referenceframe transformation.
Jacobian transfer matrix of the island system
Before connecting to the ac network, the state-space representations of the electrical parts
of the three input devices, i.e., the synchronous generator, the induction motor and the
VSC-HVDC converter are lumped into one state-space model as
dxz
= Az xz + Bz uz + Bz1 udz
dt
yz = Cz xz + Dz1 uz1


(6.36)


T
T
where xz = [xT xT xT ]T , uz = [(ug )T uT uT ]T , yz = [(yg )T ym yv ]T , and uz1 =
g
m
v
m
v

[(u g1 )T uT uT ]T . Az , Bz , Bz1 , Cz , Dz1 are the block diagonal matrices composed of
m1
v1
the corresponding input-device matrices in (6.23), (6.27) and (6.32), i.e.,


 ′

 ′

Ag
Bg
Bg1






 , Bz = 
 , Bz1 = 

Az = 
Am
Bm
Bm1






Av
Bv
Bv1
 ′



Cg
Dg1




 , Dz1 = 
.
Cz = 
Cm
zeros (1, 1)




Cv
zeros (2, 2)

145

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
Note that each input and output of the state-space model of the input devices in (6.36)
and the state-space model of the ac network in (6.7) are reciprocal, i.e., each output of
one system is the input of the other system. Consequently, the state-space model of the
combined electrical systems can be solved as
N(Az + Bz Dn Cz ) NBz Cn

dxE
=
dt

Bn Cz

An

xE

AE

+

N(Bz Dn Dz1 + Bz1 )
Bn Dz1

uE +

NBz Dn1 Dz1
zeros (4, 6)

duE
dt

(6.37)

BE ·uE

where N = (I − Bz Dn1 Cz )−1 , xE = [xT xT ]T , and uE = uz1 . In (6.37), the derivaz
n
tive of the input duE /dt appears on the right side of the state-space equation. However,
considering the structure of Dz1 , the differentiation is, in fact, only effective on the rotor
g
g
angle ∆δ of the synchronous generator, which yields ∆ωr = d∆δ/dt. Since ∆ωr belongs
to uE , (6.37) can still be fit in standard state-space form.
Assuming per unit quantities, the output variables of the electrical system can be
expressed as
Teg = Im {is conj(ψ)} = isq ψd − isd ψq
Eg =

e2 + e2
sq
sd

Tem = Im {is conj(ψs )} = isq ψsd − isd ψsq

P = Re {uf conj(iv )} = uf d ivd + uf q ivq

Uf =

u2 d + u2 q
f
f

(6.38)

where Teg is the electrical torque of the synchronous generator. Eg is the terminal voltage
of the synchronous generator. Tem is the electrical torque of the induction motor. P is the
active power from the VSC-HVDC converter. Uf is the filter-bus voltage of the VSCHVDC converter. In (6.38), Teg , Tem , P and Uf are expressed by state variables, but Eg
is expressed by esd and esq , which are neither state variables nor inputs. Therefore, the
terminal voltages ( of all the input devices) are solved by (6.36) and (6.7), yields
uz =

Nt (Dn Cz + Dn1 Cz Az ) Nt Cn

xE

+ Nt (Dn Dz1 + Dn1 Cz Bz1 )uE + Nt Dn1 Dz1

duE
dt

(6.39)

where Nt = (I − Dn1 Cz Bz )−1 . By having all the output variables expressed by state
variables and inputs, the outputs of the electrical systems can be expressed in state-space
form after linearizing (6.38)
yE = CE xE + DE uE .
(6.40)
146

6.3. Dynamic modeling and linear analysis of a typical island system
The state-space representation (6.37) and (6.40) can be further written in input-output
transfer matrix form
yE = CE (sI − AE )−1 BE + DE uE

(6.41)

J(s)

where the Jacobian transfer matrix J(s) is the linear description of the electrical part of
the island system. That is, J(s) is a 5x6 transfer matrix which has
uE =

g
m
∆δ ∆ωr ∆ef d ∆ωr ∆θv

T

∆V
V0

(6.42)

as the inputs and
yE =

∆Teg ∆Eg ∆Tem ∆P ∆Uf

T

(6.43)

as the outputs.

6.3.2 Integrated linear model
Fig. 6.8 shows the integrated linear model of the island system. The central part is the
ac Jacobian transfer matrix J(s), i.e., the controlled process of the island system. The
non-electrical parts, i.e., the controllers, are described in the following:

Excitation system of the synchronous generator
The input signals of the exciter are the alternating-voltage reference Uref and the compensated terminal voltage Vcg . The output signal is the field voltage ef d . KA and TA are the
gain and time constant of the exciter.

Rotor dynamics of the synchronous generator
Considering a single mass rotor, the dynamics of the rotor are determined by the equation
of motion:
g
1
d∆ωr
g
=
(T g − Teg − KD ∆ωr ).
(6.44)
dt
2H m
By applying the Laplace transform, the corresponding control law can be expressed as
g
∆ωr =

1
(T g − Teg ).
2Hs + KD m

(6.45)

HRotor (s)

147

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

Induction motor

m
Tm

+

1
2Hs + K D

-

Rotor

U ref +

∆V
V0

ku
s

-

AC-voltage
control
v
c

V =| u f + kc jX tv i v |
Load compensation

∆Tem

∆ωrm

∆e fd

-

+

∆U f
∆P

∆E g
AC-Jacobian
transfer matrix
J(s)

∆θ v

kp

Vcg =| e g + k g jX tg i g |
Load compensation

∆Teg
1
s

∆ωrg

1
2Hs + K D

s

-

Powersynchronization
loop

∆Pref

Kf
1 + sT f
Frequency
controller

-

Exciter

∆δ
Pref

+

KA
1 + sTA

-

+
Rotor ∆T g
m

∆ω
s
1 + sTm
Frequency
measurement

1
R

-

+

1
1 + sTG

∆Y

Governor

1
1 + sTCH
Turbine

Pref
VSC-HVDC

Synchronous generator

Fig. 6.8 A complete linear model of the example island system.

148

ref
Eg

6.3. Dynamic modeling and linear analysis of a typical island system
Turbine and governing systems of the synchronous generator
In the thesis, a steam turbine model is assumed. A simplified steam turbine model can be
represented as
1 + sFHP TRH
g
∆Tm =
∆Y
(6.46)
(1 + sTCH )(1 + sTRH )
with the typical values FHP = 0.3, TRH = 7.0 s, TCH = 0.3 s [32]. For a non-reheat steam
turbine model [32], the transfer function is
g
∆Tm =

1
∆Y.
1 + sTCH

(6.47)

The steam flow in the turbine is controlled by the governing system. The governor senses
the frequency error and adjusts the steam into the turbine. The governor has the speed
g
deviation ∆ωr and the load-reference set-point Pref as input variables. The output of the
governor is the steam gate ∆Y , which supplies the input to the steam turbine. The frequency droop R determines the power-sharing proportion of the synchronous generator.
Rotor dynamics of the induction motor
The rotor dynamics of the induction motor are similar to the synchronous generator. The
input signals are the mechanical torque from the load and the electrical torque of the
motor. Similar to the synchronous generator, the rotor dynamics of the induction motor
can be expressed as
1
m
∆ωr =
(T m − Tem ).
(6.48)
2Hs + KD m
It should be noted that the generator convention is applied in (6.48). In other words, if the
m
mechanical load is TL in motor operation, then it gives Tm = −TL .
Active-power and alternating-voltage control of the VSC-HVDC link
The control laws are given by (5.25).
Frequency and alternating-voltage droop control of the VSC-HVDC link
The control laws are given by (6.2) and (6.5). The frequency measurement is obtained by
differentiating angle output of the active-power controller with a measuring time constant
Tm , typically in the range of 10 − 20 ms. It should be noted that the load compensation of
the VSC-HVDC converter and the synchronous generator in Fig. 6.8 are in their nonlinear
form. It requires linearization if their effects are to be considered in the linear model.
The technical data of the VSC-HVDC link, the synchronous generator and the induction motor are given in Appendix C. The rest parameters of the linear model for the
island system are given in Table 6.1.
149

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
Table 6.1 Parameters of the linear model for the island system. Loads: per unit based on 350 MVA
and 400 kV. VSC-HVDC: per unit based on 350 MVA and 195 kV.

Parameters

200, 0.02 s

R, TG , TCH ,

0.067, 0.02 s, 0.3 s
100 rad/s, 60

kv , αv
VSC-HVDC

1.75 p.u., 16.8 p.u., 0.01 p.u.

kp , ku

Synchronous generator

Rl , ω1 Ll , ω1 Cl
KA , TA

Loads

Values

0.3 p.u., 40 rad/s

Kf , Tf , Tm

3.72, 0.3 s, 0.02 s

6.3.3 Linear analysis
As it was discussed in Chapter 5, power-synchronization control of VSC-HVDC link is
fundamentally of MIMO nature. However, with the simple diagonal controller proposed
for island operation, the control parameters can be selected based on single-input singleoutput (SISO) technique by tuning one loop at a time. Accordingly, the tuning process is
performed in two steps:
• Step 1: Starting with the integral gain of the alternating-voltage control ku = 0.0,
plot the root-loci of the dominant poles by varying the integral gain kp of the activepower controller. Find the suitable parameter range of kp .
• Step 2: Using kp chosen at step 1, plot the root-loci of the dominant poles by varying
ku to find a suitable parameter range of ku .
Fig. 6.9 shows the root-loci of the dominant poles of the closed-loop system by
varying kp from 0 to 140 rad/s. Two pairs of dominant poles of the open-loop system are
affected by varying kp . The critical poles are the pole pair p1,2 (p1,2 = −127 ± j298),
which move towards the right-half plane with increased kp . The pole pair p1,2 imposes
an upper limitation on kp . Thus, kp should be chosen between 80 − 100 rad/s to get
a trade-off between dynamic-response time and stability margin. The pole pair at p3,4
(p3,4 = −36.4 ± j17.1) becomes more damped by increasing kp .
Fig. 6.10 shows the root-loci of the dominant poles of the closed-loop system by
varying ku with kp = 100 rad/s. The pole p0 is a pole at the origin which is introduced
by the integrator. By increasing ku from 0 to 120, p0 moves towards the left-half plane.
The two dominant pole pairs are also affected by variations of ku . The pole pair p3,4
moves towards the left-half plane, however, the pole pair at p1,2 slightly moves towards
the right-half plane instead. Therefore, the above tuning steps should be iterative to find a
compromised combination of kp and ku to place the two dominant pole pairs. Finally, the
150

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

300
p

1

ω (rad/sec)

200
100

p

3

0
p

-100

4

-200
p

2

-300
-150

-100

-50

0

ω (rad/sec)

Fig. 6.9 Root-loci of the dominant poles of the island system. ku = 0.0, variations of kp from
0 rad/s to 140 rad/s.

300
p

ω (rad/sec)

200

1

p3

100

p

0

0
-100

p4

-200

p

2

-300
-150

-100

-50

0

ω (rad/sec)

Fig. 6.10 Root-loci of the dominant poles of the island system. kp = 100.0 rad/s, variations of
ku from 0 to 120.

151

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
control parameters are chosen with kp = 100 rad/s and ku = 60, which end up with the
dominant pole pairs at p1,2 = −77.8 ± j269 rad/s and p3,4 = −74.3 ± j56 rad/s.

6.4 Simulation studies
To demonstrate the dynamic performance of the VSC-HVDC link, the island system in
Fig. 6.3 is simulated with PSCAD/EMTDC. The other end of the VSC-HVDC link is
assumed to be connected to a strong ac system.
One of the features of power-synchronization control is its flexibility with different network conditions, i.e., the proposed control structure can deal with various possible
operating conditions, as well as handle the transitions between the operation modes seamlessly. In this section, this feature of power-synchronization control is demonstrated.
Initially, the island system is connected to a large ac system with constant frequency. In this operation mode, the VSC-HVDC link is possible to apply constant power
control. The initial condition is as following. The total loads in the island system is
320 MW whereas 120 MW from induction-motor loads (4 units) and 200 MW from
RLC passive loads. The synchronous generator supplies 100 MW, while the VSC-HVDC
link supplies 150 MW with constant power control. The rest of the loads are balanced by
the large system, i.e., 70 MW power transmitted to the island system.
Fig. 6.11 shows the simulation results when the line connection to the large ac
system is lost at 0.1 sec. By detecting this event (by transient frequency measurement
or the state of the ac breaker of the tripped ac line), the VSC-HVDC link enables the
frequency droop control. The power that is lost from the large ac system is shared by
the VSC-HVDC link and the local synchronous generator. The proportion of the power
sharing is determined by the frequency droops of the synchronous generator and the VSCHVDC link where it is equally shared in this case. After disconnection from the large
ac system, the frequency of the island system deviates slightly from the nominal value
due to the frequency droop characteristics of the VSC-HVDC link and the synchronous
generator.
Fig. 6.12 is a continuation simulation from Fig. 6.11. The local synchronous generator is also tripped at 0.1 sec (new time frame). The island system then becomes a passive
network with induction-motor loads and RLC passive loads. The VSC-HVDC link supplies all the loads in the island system. Due to the frequency-droop characteristics of the
VSC-HVDC link, the frequency of the island system deviates much from the nominal
value. At 1.5 sec, the power order of the VSC-HVDC link is adjusted to 0.9 p.u. to match
the loads in the island system. Consequently, the frequency in the island system is back
to the nominal value.
Fig. 6.13 and Fig. 6.14 demonstrate the fault ride-through capability of the VSCHVDC link in two different operating conditions, i.e., large-ac-system connection and
island operation. With both of the operating conditions, the VSC-HVDC link manages
152

6.4. Simulation studies

P dc (MW)

200
150
100
0

0.5

1

1.5

2

2.5

3

0.5

1

1.5

2

2.5

3

0.5

1

1.5
time (sec)

2

2.5

3

P g (MW)

200
150
100
50
0

f1 (p.u.)

1.1
1
0.9
0

Fig. 6.11 Trip of the ac line to the large ac system (entering island operation). Upper plot: active power from the VSC-HVDC link. Middle plot: active power from the synchronous
generator. Lower plot: network frequency.

P dc (MW)

400
300
200
100
0

0.5

1

1.5

2

2.5

3

0.5

1

1.5

2

2.5

3

0.5

1

1.5
time (sec)

2

2.5

3

P g (MW)

200
100
0
0

f1 (p.u.)

1.1
1
0.9
0.8
0

Fig. 6.12 Trip of the local synchronous generator (entering passive network operation). At 1.5 sec,
the power order of the VSC-HVDC link is adjusted to 0.9 p.u.

153

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

U

pcc

(p.u.)

1
0
-1
0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

P dc (p.u.)

1
0.5
0
0

1

c

| i | (p.u.)

2

0
0

time(sec)

Fig. 6.13 Fault ride-through capability of the VSC-HVDC with large-ac-system connection. Upper plot: alternating voltage at the PCC. Middle plot: active power from the VSCHVDC. Lower plot: valve current of the VSC-HVDC.

U

pcc

(p.u.)

1
0
-1
0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

P dc (p.u.)

1
0.5
0
0

1

c

| i | (p.u.)

2

0
0

time(sec)

Fig. 6.14 Fault ride-through capability of the VSC-HVDC in island operation.

154

6.5. Jacobian transfer matrix for other input devices
E g 0∠θ g 0
SG

Ln

Rn

E0∠00

+

+

eg

E

-

ig

-

Fig. 6.15 A synchronous generator connected to an impedance ac source.

recovery from the severe three-phase ac-system faults applied at the PCC. After detecting the faults, the current limiter reduces the fault currents to half of the maximum load
current Imax or any other desired values.
The above simulation studies demonstrate the flexibility of power-synchronization
control for various operating conditions. It should be noted that black start of the island
system is just the reverse procedure of the above. However, the synchronous generator
needs to be synchronized to the VSC-HVDC link before it can be re-connected to the
island system.

6.5 Jacobian transfer matrix for other input devices
In Chapter 4, the poles and transmission zeros of the ac Jacobian transfer matrices are analyzed for grid-connected VSCs connected to several simplified ac-network configurations.
It has been found that the poles and transmission zeros of the Jacobian transfer matrix are
informative in understanding the basic characteristics of the ac system. The major findings are that, the resonant (complex) poles of the Jacobian transfer matrices generally
correspond to the resonances in the ac system (except the poles at the grid frequency),
while the operating-point-dependent zeros have a close relationship with angle stability
and voltage stability phenomena in ac systems. By moving to the origin, the zeros impose
a fundamental limitation on power transmission in the ac network. In addition, the RHP
zero of the Jacobian transfer matrix imposes a fundamental limitation on the achievable
bandwidth of the VSC.
In this section, the characteristics of the Jacobian transfer matrix are further analyzed for two additional conventional input devices in power systems, i.e., the synchronous generator and the induction motor. The goal of the analysis is to gain more
insights of the Jacobian transfer matrix modeling concept.

6.5.1 Synchronous generator
Fig. 6.15 shows a synchronous generator connected to an impedance ac source, which
is the classical one-machine infinite-bus (OMIB) system. Rn and Ln are the resistance
and inductance of the ac system. The bold letter symbols E and eg represent the voltage vectors of the ac source and the terminal voltage of the generator. E0 and Eg0 are
155

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

Tmg +

-

1
2Hs + K D

Jacobian
transfer
matrix

Rotor
ref
Eg +

-

KA
1 + TA s
Exciter

∆Teg

∆ωrg

∆e fd

J (s )

∆E g

Fig. 6.16 Closed-loop system of the synchronous generator.

their corresponding voltage magnitudes. The ac source is used as the voltage reference,
and the phase angle of eg in the steady-state operating point is θg0 . The quantity ig is
the current vector with the reference direction from the synchronous generator to the ac
system. The technical data of the synchronous generator are given in Appendix C (no
step-up transformer connected). The resistance and reactance of the ac system are chosen
as Rn = 0.05 p.u. and ω1 Ln = 1.0 p.u. (per unit based on 150 MVA and 20 kV). The
reason for using such a long line with a comparably low voltage is to facilitate studies on
the zeros of the Jacobian transfer matrix.
By applying the Jacobian transfer matrix modeling concept, Fig. 6.16 shows the
closed-loop system of the synchronous generator connected to the ac system. For simplicity, the governing system of the synchronous generator is neglected, i.e., a constant
mechanical torque is assumed. As it was shown in Fig. 6.8, the synchronous generator
g
has three input variables to the Jacobian transfer matrix, i.e, the rotor speed ∆ωr , the rotor angle ∆δ, and the field voltage ∆ef d . This is rather inconvenient for analysis since
g
∆ωr and ∆δ are not independent. Therefore, in Fig. 6.16, the boundary between the Jag
cobian transfer matrix and the controllers are re-defined by only having ∆ωr and ∆ef d
g
as inputs, while the integrator between ∆ωr and ∆δ is formed into the Jacobian transfer
matrix, i.e., the modified Jacobian transfer matrix has the expression


JTe δ (s)
JTe Efd (s) 
 JTe ω (s) +
s





JTe ω (s)



.
J(s) = 
(6.49)

JEg δ (s)
 J (s) +

JEg Efd (s) 
 Eg ω
s




JEg ω (s)

Table 6.2 lists the poles and the transmission zeros of J(s) for the synchronous generator
at no-load operation, and Fig. 6.17 shows the pole-zero map of J(s) by varying the load
angle of the synchronous generator.
156

6.5. Jacobian transfer matrix for other input devices

Table 6.2 Locations of poles and transmission zeros of the Jacobian transfer matrix J(s) for the
synchronous generator with θg0 = 0◦ (no load operation).

Poles

rad/sec

Zeros

rad/sec

p1,2

−2.56 ± j314.1

z1,2

−0.002 ± j314.1

−25.6

z4

−0.44

z6

p3

−40.4

−2.13

p4
p5
p6
p7

z3

−243.7

z5

−24.3
−0.9
INF

0

p

300

1

z1

ω (rad/sec)

200
100
z

3

z -z
4

z

6

5

0
p3-p7
θg0=90

-100

°

θg0=45

°

θg0=27

°

θg0=17

°

-200
-300

z

p2
-200

0

2

200

400
600
ω (rad/sec)

800

1000

1200

Fig. 6.17 Loci of the transmission zeros of the Jacobian transfer matrix for the synchronous generator with increased load angles.

157

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
It is interesting to note that, similar to that for the grid-connected VSCs, the Jacobian transfer matrix for the synchronous generator also has a resonant pole pair at the
grid frequency (p1,2 ). However, a zero pair z1,2 appears very close to the locations of the
resonant pole pair. The zero pair z1,2 is affected only slightly by the load-angle variations,
i.e., the resonant poles at the grid frequency are canceled by the resonant zeros. Although
the pole-zero cancelation is basically not affected by operating-point variations, the balance can be broken by series-compensated ac lines. As it was shown in Section 4.2.3, a
series capacitor splits the resonant pole pair at the grid frequency into two other resonant
pole pairs, one at the subsynchronous frequency and the other at the supersynchronous
frequency [cf. (4.56)]. The SSR problem of the synchronous generator is related to the
pole pair at the subsynchronous frequency.
The integrating process between the rotor speed and the rotor angle has the advantage to reduce the peaks of various resonances in the ac system. However, the integrator also reduces the phase margin of the rotor loop. The small-signal stability or
low-frequency oscillations in power systems are largely due to the poor phase margin of
the rotor loop of the synchronous generators, which is worsened by fast excitation control.
It is believed by the author that the PSS solution can, in some sense, be viewed upon as
a multivariable feedback-control design which aims to decouple the interaction between
the rotor loop and the excitation control.
Similar to the VSC, the Jacobian transfer matrix for the synchronous generator also
has but one operating-point-dependent zero z6 , which moves from infinity towards the
origin with increased load angles. By θg0 = 90◦ , z6 is very close to the origin. However,
the RHP zero z6 usually does not create any problem for the synchronous generator, since
a synchronous generator normally operates with a load angle lower than 40◦ due to the
transient stability limitation [64]. The RHP zero z6 at the operating point θg0 = 40◦ will
not impose any bandwidth limitation on the synchronous generator that usually has a
bandwidth lower than 10 rad/s.

Fig. 6.18 shows the four transfer functions of the Jacobian transfer matrix, JTe ω (s),

JTe Efd (s), JEg ω (s), and JEg Efd (s) overlapped with the frequency-scanning results from
PSCAD/EMTDC. The synchronous generator is assumed to operate with a load angle
θg0 = 30◦ . Due to limitations of the applied frequency-scanning technique in PSCAD/EMTDC, only the results with frequencies higher than 6.28 rad/s (1 Hz) are shown.

The transfer functions of JTe ω (s), JTe Efd (s) show the pole-zero cancelation phenomena of the Jacobian transfer matrix at the grid frequency which are also confirmed by
the frequency-scanning results from PSCAD/EMTDC.

From the magnitude and phase plot of JTe ω (s), it can be observed that the integrator
between the rotor speed and the rotor angle gives a negative slope to the magnitude, which
suppresses the high-frequency resonances. However, it also introduces a −90◦ phase shift
which limits the phase margin of the rotor loop.

158

6.5. Jacobian transfer matrix for other input devices
0

5

10
|JT E (jω)|

|JT ω(jω)|

10

e fd

´

e

0

10

-5

10

-5

0

10

0

e

-50
-100
-150 -2
10

0

10
ω (rad/sec)

-2

2

10

10

2

10

ARG J E (jω) (deg.)
T
e fd

´
ARG JT ω(jω) (deg.)

10 -2
10

-100

-200 -2
10



10
|JE E (jω)|

g fd

0

10

´

g

|JE ω(jω)|

2

10

0

5

-5

-200

-300

0

10
ω (rad/sec)


(c) JEg ω (s).

2

10

-5

10

-2

2

10

10

ARG J E (jω) (deg.)
E
g fd

0

10

g

´
ARG JE ω(jω) (deg.)

0

10
ω (rad/sec)

(b) JTe Efd (s).

10

-400 -2
10

2

10

0

(a) JTe ω (s).

10 -2
10

0

10

0

10

2

10

0

-100

-200 -2
10

0

10
ω (rad/sec)

2

10

(d) JEg Efd (s).

Fig. 6.18 Bode plots of the transfer functions of the Jacobian transfer matrix for the synchronous generator (solid: linear models, dashed: frequency-scanning results from
PSCAD/EMTDC).

6.5.2 Induction motor
Fig. 6.19 shows an induction motor connected to a constant-frequency stiff voltage source.
The quantity im is the stator-current vector of the induction motor. The reference direction
of the stator current vector is chosen outwards to be consistent with other input devices.
The technical data of the induction motor are given in Appendix C (no step-up transformer
connected).
By applying the Jacobian transfer matrix modeling concept, Fig. 6.20 shows the
closed-loop system of the induction motor connected to the ac system. The induction
m
motor only has the rotor speed ∆ωr connected to the Jacobian transfer matrix, while the
electrical torque Tem is the output. Thus, the induction motor is modeled as a single-input159

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
E0∠00
IM

+
im

E

Fig. 6.19 An induction motor connected to a stiff voltage source.
m
Tm +

1
2Hs + K D

-

∆ωrm

Rotor

Jacobian
∆Tem
transfer matrix
J (s)

Fig. 6.20 Closed-loop system of the induction motor.

single-output (SISO) feedback control system, where the rotor is treated as the controller,
similar to the synchronous generator. In practice, there are also rotor-speed or squaredrotor-speed dependent contribution to the mechanical torque Tm , which can be modeled as
part of the controller. However, since the major subject of this section is about the Jacobian
transfer matrix, a constant mechanical torque is assumed for simplicity. Accordingly, the
Jacobian transfer matrix has the expression
J(s) = JTe ω (s)

(6.50)

i.e., the Jacobian transfer matrix is a 1x1 matrix with JTe ω (s) as the only element. Table 6.3 lists the poles and zeros of the Jacobian transfer matrix for the induction motor
at no-load operation. Fig. 6.21 shows the pole-zero map by increasing the loading of the
induction motor.
Table 6.3 Locations of poles and zeros of the Jacobian transfer matrix of the induction motor at
no-load operation (slip = 0.0).

Poles

rad/sec

Zeros

rad/sec

p1,2

−9.4 ± j314.16

z1,2

−5.1 ± j314.1

p3

−13.3

z3

−13.3

p4

−13.3

Similar to the synchronous generator, the resonant pole pair p1,2 at the grid frequency is also canceled by the resonant zero pair z1,2 . This implies that the induction
motor might also suffer from SSR problems if a series-compensated ac line is connected
in the vicinity. Of course, it is exceptional rare for induction motors to have sufficiently
long rotor shafts to excite such oscillation modes.
160

6.5. Jacobian transfer matrix for other input devices

p1

300

z1

ω (rad/sec)

200
100
p

3

z

3

0
p4
slip=0

-100

slip=0.02

slip=0.032

slip=0.042

-200
-300

p
-15

z

2

-10

2

-5
ω (rad/sec)

0

5

Fig. 6.21 Locus of z3 with increased slips.

3
z3=-5.6 rad/s

z3=0 rad/s
2.5

z3=-10.3 rad/s

1.5

m

Te (p.u.)

2

1

0.5
z =-13.3 rad/s
3

0
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1-slip

Fig. 6.22 Locus of z3 on the torque-slip curve of the induction motor .

161

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

e

|JT ω(jω)|

10

0

10

-2

-4

10 -1
10

10

0

10

0

10

1

10

2

10

3

1

10

2

10

3

(jω) (deg.)

50

-50

ARG J

e

T ω

0

-100
-150 -1
10

10
ω (rad/sec)

Fig. 6.23 Bode plot of JTe ω (s) (solid: linear model, dashed: frequency-scanning result from
PSCAD/EMTDC).

With slip = 0.0, two poles p3 and p4 , as well as a real LHP zero z3 have exactly the
same location. By increasing the slip of the induction motor, p3 and p4 move slightly apart
along the imaginary-axis direction. However, z3 moves towards the origin, and it reaches
the origin with slip = 0.042.
Fig. 6.22 shows the well-known torque-slip curve of the induction motor. It is interesting to note that z3 reaches the origin exactly at the pull-out slip of the induction
motor!
Fig. 6.23 shows the Bode plot of the transfer function JTe ω (s) overlapped with the
frequency scanning result from PSCAD/EMTDC with frequencies higher than 6.28 rad/s.
The pole-zero cancelation phenomenon of the Jacobian transfer matrix at the grid frequency can be observed. The frequency-scanning result from PSCAD/EMTDC confirms
the same phenomenon.

6.6 Subsynchronous characteristics
In the previous sections, the rotor of the synchronous generator was assumed to be made
up of a single mass. With such a representation of the rotor, the bandwidth of a synchronous generator is usually in the frequency range of 0.2 to 2 Hz. In reality, however, the
rotor of a steam-turbine generator has a complex mechanical structure consisting of several predominant masses connected by shafts of finite stiffness. Therefore, when the generator is perturbed, torsional oscillations occur between sections of the turbine-generator
162

6.6. Subsynchronous characteristics

E

+

Rn

Rl

Ln

400 kV

400 kV
PCC

Ltg

Ltg

Ltv

20 kV

195 kV

SG

PCC

VSC

150 MVA

Ltv

20 kV

195 kV

SG

VSC

150 MVA

350 MW

350 MW
(a)

(b)

Fig. 6.24 AC-network configurations for investigation of the subsynchronous characteristics of a
VSC-HVDC link. (a) Large ac-system connection. (b) Island operation.

rotor. These torsional oscillation modes, typically in the subsynchronous frequency range,
could interact with the electrical system in an adverse manner. There are predominantly
two types of subsynchronous torsional interactions (SSTIs) of concern:
1. Subsynchronous resonances with series-compensated transmission lines.
2. Torsional interactions with large power-electronic devices, such as HVDC systems
and FACTS devices.
In this section, the subsynchronous characteristics of a VSC-HVDC link located in the
vicinity of a steam-turbine synchronous generator are analyzed. The two ac-network configurations to be investigated are shown in Fig. 6.24. With the ac-network configuration
shown in Fig. 6.24(a), a VSC-HVDC link and a synchronous generator are connected
through an ac transmission line to a constant-frequency stiff voltage source E, which
represents a large ac system. With the ac-network configuration shown in Fig. 6.24(b),
a VSC-HVDC link and a synchronous generator are connected in an island system. The
resistive load Rl is used to adjust the load flow in the island system.
163

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

Tmg +

Gm (s )

∆ωrg

∆Teg
Ge (s )
Fig. 6.25 Closed-loop system formed by the linearized electrical and mechanical dynamics.

The technical data of the synchronous generator and the VSC-HVDC link are given
in Appendix C.

6.6.1 Frequency-scanning method
As it was mentioned in Chapter 4, there are predominantly two methods for SSTI analysis:
eigenvalue analysis and frequency scanning. The eigenvalue-analysis method is a rigorous tool. However, it requires detailed representation of the mechanical system of the
synchronous generator. The frequency-scanning method is based on the complex-torque
theory [81]. With the closed-loop system shown in Fig. 6.25, the transfer function between
g
∆ωr and ∆Teg can be expressed as
Ge (jω) = De (jω) − jKe (jω)

(6.51)

where De and Ke are defined as the damping torque and the synchronizing torque. Without the electrical system, the damping torque of the mechanical system is always positive
due to friction. Thus, if De (jω) > 0, unstable subsynchronous oscillations (SSOs) cannot
occur.
It is a common practice to use the electrical damping curve of the synchronous
generator to evaluate the subsynchronous characteristics of an HVDC link [102, 103]. If
the characteristics of the HVDC link are taken into consideration a modified electrical


damping curve De (jω) is obtained. Now, if De (jω) is larger than the original electrical
damping curve De (jω) without the HVDC link, i.e.,

De (jω) > De (jω)

(6.52)

the negative influence of the HVDC link on SSOs can be eliminated.
As a screening tool to evaluate the subsynchronous characteristics of an HVDC
link, the frequency-scanning method has the following merits:
1. There is no need to model the mechanical system of the synchronous generator.
2. It can be performed by both frequency-domain analysis and time-domain simulations.
164

6.6. Subsynchronous characteristics
Ge (s)

U ref +

-

∆V
V0

ku
s

∆ωrg
AC-Jacobian
transfer
matrix

AVC
Pref +

∆Teg

∆U f

∆θ v

kp

J (s)

s

-

∆P

PSL

− ∆PVSC1
DC-Jacobian − ∆PVSC2
transfer
matrix
G dc ( s)
2
∆udc1

− (α d Cdc +

2
α d Cdc

DVC

2s

)
+

2
∆udc2

-

ref
(udc1 ) 2

Fig. 6.26 Jacobian transfer matrix formulation of the large ac-system connection configuration in
Fig. 6.24(a).

6.6.2 Large ac-system connection
System definition
Fig. 6.26 shows the Jacobian transfer matrix formulation of the large ac-system connection configuration in Fig. 6.24(a). The transfer function Ge (s) is defined as the transfer
g
function between ∆ωr to ∆Teg with all the feedback loops of the VSC-HVDC link closed
(inside the dashed box). In principle, Ge (s) should also take into account the effects of
the exciter and possibly the power-system stabilizer (PSS) of the synchronous generator.
However, since the goal of the investigation is to study the subsynchronous characteris′
tics of the VSC-HVDC link, where the difference between De (jω) and De (jω) is of more
concern, the excitation system of the synchronous generator is neglected.
The VSC-HVDC converter has two possible control modes: active-power control
and direct-voltage control. As it was discussed in Chapter 5, if the direct voltage is controlled by only one converter station, the two converter stations are linearly independent.
Consequently, in either of the operation modes, it is enough to have the control system
165

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
of one converter station modeled. Also for the sake of simplicity, the dc-link circuit is
assumed to be a single capacitor, i.e., Gdc (s) = 2/(sCdc). Since the dc capacitance Cdc is
explicitly included into the control parameters of the direct-voltage controller, the value
of Cdc has no impact on Ge (s).
Table 6.4 gives the base control parameters selected for the VSC-HVDC converter.
The controllers of the VSC-HVDC converter have been tuned relatively conservative to
operate the VSC-HVDC link for wider ac-network conditions. The impact of the control
parameters on the subsynchronous characteristics of the VSC-HVDC converter will be
further discussed after presenting the initial results based on the control parameters given
in Table 6.4.
Table 6.4 Control parameters of the VSC-HVDC converter with large ac-system connection. Per
unit based on 350 MVA and 195 kV.

Controller

Parameters

Values

Power-synchronization loop

kp

60 rad/s

Alternating-voltage control

ku

60

Direct-voltage control

αd

40 rad/s

High-pass current control

kv , αv

0.2 p.u., 40 rad/s

For analyzing SSTI between a synchronous generator and a nearby HVDC link, the
unit interaction factor (UIF) is commonly used [104, 105], which is defined as
UIF =

MWdc
MVAg

1−

SCg
SCtot

2

(6.53)

where MWdc is the rating of the dc system, MVAg is the rating of the generator, SCtot is
the short-circuit capacity at the PCC including the synchronous generator unit, and SCg
is the short-circuit capacity at the PCC excluding the generator.
Based on the parameters of the synchronous generator and its step-up transformer
in Appendix C, the short-circuit capacity contribution from the synchronous generator at
′′
the PCC is MVAg /(Ld + Ltg ) = 545.3 MVA. By adjusting the parameters of the ac line
to the large system, ac-network configurations with different UIF values are obtained. As
shown by Table 6.5, the UIF value is directly related to the SCR of the ac system1, i.e., the
UIF values are higher with weak-ac-system connections. The resistances of the ac lines
are chosen such that the ac lines have 85◦ impedance angles.
In the following analysis, the synchronous generator is assumed to supply Pg =
0.1 p.u. power, while the VSC-HVDC link transmits 90% of the rated power, i.e., Pdc =
±0.9 p.u. for inverter and rectifier operation respectively.
1

The synchronous generator is not included in the calculation of the SCR.

166

6.6. Subsynchronous characteristics
Model validation
Fig. 6.27 shows the model validation by comparing the frequency-scanning results from
the linear model and the time-domain simulation. The ac-network is chosen to have
UIF = 0.3. The VSC-HVDC converter operates as an inverter in power control mode. The
electrical damping curves are obtained by both the linear model and the time simulations
in PSCAD/EMTDC. The agreement between the linear model and the time simulation is
good.
Results from the linear analysis
In the following, the electrical damping curves of the synchronous generator are plotted
using the linear model to show the subsynchronous characteristics of the VSC-HVDC
converter for ac-network configurations with the four UIF values defined in Table. 6.5.
The VSC-HVDC converter operates in the following four modes:
1. Rectifier operation, power control mode.
2. Inverter operation, power control mode.
3. Rectifier operation, direct-voltage control mode.
4. Inverter operation, direct-voltage control mode.
Figs. 6.28-6.31 show the electrical damping curves of the synchronous generator affected
by the VSC-HVDC converter. The figures clearly show that the VSC-HVDC converter can
have a significant impact on the electrical damping of the generator, and the impact increases with higher UIF values (weaker ac systems). In addition, the impact also depends
on the operation mode of the VSC-HVDC converter. The following can be observed from
the linear results:
1. In rectifier operation, power control mode, the impact of the VSC-HVDC converter is positive in the entire subsynchronous frequency range. The reason could
be that the VSC-HVDC converter resembles the behavior of a resistive load in this
operation mode.
2. In inverter operation, power control mode, the impact of the VSC-HVDC converter is positive in the lower subsynchronous frequency range. However, it gives
negative damping contributions in the higher subsynchronous frequency range for
approximately UIF > 0.1.
3. With direct-voltage control, for either rectifier or inverter operation, the VSCHVDC converter tends to give negative damping in the lower subsynchronous frequency range for approximately UIF > 0.1. However, the negative contribution is
rarely above 10 Hz (62.8 rad/s) even with the highest UIF = 0.7.
167

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

Table 6.5 Parameters of the ac line for different UIF values. Per unit based on 350 MVA and
400 kV.

UIF

ω1 Ln

SCR

0.02

0.0056 p.u.

0.0655 p.u.

15.2

0.1

0.0147 p.u.

0.168 p.u.

5.9

0.3

0.0315 p.u.

0.3581 p.u.

2.8

0.7

5

Rn

0.0665 p.u.

0.7746 p.u.

1.3

5

Linear model
Time simulation

4

4
3
D (jω)

3
D (jω)

e

2

e

Linear model
Time simulation

2

1

1

0

0

-1
0

50

100

150
200
ω (rad/sec)

(a) Without VSC-HVDC.

250

300

-1
0

50

100

150
200
ω (rad/sec)

250

300

(b) With VSC-HVDC.

Fig. 6.27 Model validation with UIF = 0.3 network configuration (solid: linear models, dashed:
time simulations). (a) Electrical damping curves of the generator (Pg = 0.1 p.u.) without the VSC-HVDC link in the system. (b) Electrical damping curves of the generator
(Pg = 0.1 p.u.) with the VSC-HVDC link in the system (inverter operation, power
control mode, Pdc = 0.9 p.u.).

168

6.6. Subsynchronous characteristics
5

3

4
De (jω)

De (jω)

4

5

Without HVDC
With HVDC

2
1
0
-1
0

3
2
1
0

100
200
ω (rad/sec)

-1
0

300

(a) UIF = 0.02.
5
4

5

Without HVDC
With HVDC

4

300

2
1

Without HVDC
With HVDC

3

De (jω)

De (jω)

100
200
ω (rad/sec)

(b) UIF = 0.1.

3

2
1

0
-1
0

Without HVDC
With HVDC

0
100
200
ω (rad/sec)

-1
0

300

(c) UIF = 0.3.

100
200
ω (rad/sec)

300

(d) UIF = 0.7.

Fig. 6.28 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the VSC-HVDC
converter (rectifier operation, power control mode, Pdc = −0.9 p.u.).
5

5

Without HVDC
With HVDC

3

4
De (jω)

De (jω)

4

2
1
0
-1
0

3
2
1
0

100
200
ω (rad/sec)

-1
0

300

(a) UIF = 0.02.
5

5

Without HVDC
With HVDC

3
2
1
0
-1
0

100
200
ω (rad/sec)

300

(b) UIF = 0.1.

4
De (jω)

De (jω)

4

Without HVDC
With HVDC

Without HVDC
With HVDC

3
2
1
0

100
200
ω (rad/sec)

(c) UIF = 0.3.

300

-1
0

100
200
ω (rad/sec)

300

(d) UIF = 0.7.

Fig. 6.29 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the VSC-HVDC
converter (inverter operation, power control mode, Pdc = 0.9 p.u.).

169

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
5

3

4
De (jω)

De (jω)

4

5

Without HVDC
With HVDC

2
1
0
-1
0

3
2
1
0

100
200
ω (rad/sec)

-1
0

300

(a) UIF = 0.02.
5
4

5

Without HVDC
With HVDC

4

300

2
1

Without HVDC
With HVDC

3

De (jω)

De (jω)

100
200
ω (rad/sec)

(b) UIF = 0.1.

3

2
1

0
-1
0

Without HVDC
With HVDC

0
100
200
ω (rad/sec)

-1
0

300

(c) UIF = 0.3.

100
200
ω (rad/sec)

300

(d) UIF = 0.7.

Fig. 6.30 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the VSC-HVDC
converter (rectifier operation, direct-voltage control mode, Pdc = −0.9 p.u.).
5

5

Without HVDC
With HVDC

3

4
De (jω)

De (jω)

4

2
1
0
-1
0

3
2
1
0

100
200
ω (rad/sec)

-1
0

300

(a) UIF = 0.02.
5

100
200
ω (rad/sec)

300

(b) UIF = 0.1.

Without HVDC
With HVDC

3

4
De (jω)

De (jω)

4

Without HVDC
With HVDC

2
1

Without HVDC
With HVDC

2
0

0
-1
0

100
200
ω (rad/sec)

(c) UIF = 0.3.

300

-2
0

100
200
ω (rad/sec)

300

(d) UIF = 0.7.

Fig. 6.31 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the VSC-HVDC
converter (inverter operation, direct-voltage control mode, Pdc = 0.9 p.u.).

170

6.6. Subsynchronous characteristics
Impact of the control parameters of the VSC-HVDC converter
Besides the operation modes, the control parameters of the VSC-HVDC converter can
also have a strong impact on the electrical damping of the generator. The most critical
parameters are the gain kv of the high-pass current control HHP (s) and the gain kp of the
active-power control.
To evaluate the effect of HHP (s) on the electrical damping, an ideal operation mode
of the VSC-HVDC converter is studied. The active-power control and the alternatingvoltage control loops are open, i.e., the VSC-HVDC converter operates as a constantfrequency stiff voltage source. The results from the linear analysis in Fig. 6.32(a) show
that the high-pass current control HHP (s), in fact, gives negative damping to the generator.
This seems to be contradictory to the role of HHP (s), which is introduced for the purpose
of damping in the first place.
To understand this phenomenon, the damping-torque concept of the synchronous
generator has to be interpreted by feedback-control theory. As shown by Fig. 6.26, the
electrical damping curve De is essentially the real part of the transfer function JTe ω of
the Jacobian transfer matrix with the controllers of the VSC-HVDC converter closed.
Assuming constant mechanical torque input, the stability margin of the rotor loop of the
generator can be evaluated by its open-loop transfer function HGen , which is expressed as
HGen (s) = Ge (s)HRotor (s)

(6.54)

where HRotor (s) is the transfer function of the rotor, which is expressed in (6.45) for a
single-mass rotor. The concept of damping torque (either electrical or mechanical) has
a close relationship with the stability margin of the rotor loop of the synchronous generator. The higher the damping torque, the more stability margin the synchronous generator has. However, such a stability margin is manifested by two aspects: phase margin
and gain margin. The reduction of either of the stability margins will result in reduced
damping torque. Based on this interpretation of the damping-torque concept, it would be
easier to understand the negative damping effect of HHP(s). As shown in Fig. 6.33(a),
with HHP (s) the gain of HGen (s) becomes steeper in the higher frequency range (approximately 15 − 250 rad/s), which is useful for suppressing other resonances in the ac
system. However, the reduction of gain HGen (s) also results in the reduction of the phase
of HGen . For this particular case, however, the phase margin of HGen (s) is more critical
for the stability of the generator. Consequently, HHP(s) yields negative damping to the
synchronous generator since it reduces the phase margin of the synchronous generator in
the subsynchronous frequency range. On the other hand, as shown in Fig. 6.33(b), a higher
gain kp of the power-synchronization control loop of the VSC-HVDC converter makes the
slope of HGen less steep in the higher frequency range (approximately 13 − 280 rad/s),
which substantially increases the phase margin of HGen . This effect is reflected as positive
damping torque in Fig. 6.32(b).
171

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

5

5

k =0.0

k =0.0

v

k =0.5
v

p

2

e

e

p

k =100

3
D (jω)

2

k =60

4

v

3
D (jω)

p

k =0.2

4

1

1

0

0

-1

-1

-2
0

50

100

150
200
ω (rad/sec)

250

-2
0

300

(a) kp = 0.0, ku = 0.0, variations of kv .

50

100

150
200
ω (rad/sec)

250

300

(b) kv = 0.2 p.u., ku = 0.0, variations of kp .

Fig. 6.32 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the control parameters of the VSC-HVDC converter (inverter operation, power control mode,
Pdc = 0.9 p.u.). UIF = 0.7.

0

kv=0.0

-4
1

kp=0.0
kp=100

10

10

2

-2

-140
-160
-180
-200
10

1

10

2

ω (rad/sec)

(a) kp = 0.0, ku = 0.0, variations of kv .

-4

10

(jω) (deg.)

10

1

10

2

-140
-160

Gen

Gen

(jω) (deg.)

10

arg H

10

arg H

0

|H

|H

10

-2

Gen

Gen

kv=0.5

10

(jω)|

(jω)|

10

-180
-200
10

1

10

2

ω (rad/sec)

(b) kv = 0.2 p.u., ku = 0.0, variations of kp .

Fig. 6.33 Phase margin of the generator (Pg = 0.1 p.u.) affected by the control parameters of
the VSC-HVDC converter (inverter operation, power control mode, Pdc = 0.9 p.u.).
UIF = 0.7.

172

6.6. Subsynchronous characteristics
5

5

k =0.0

k =0.0

u

k =100
u

u

2

e

e

u

k =100

3
D (jω)

2

k =60

4

u

3
D (jω)

u

k =60

4

1

1

0

0

-1

-1

-2
0

50

100

150
200
ω (rad/sec)

250

300

(a) Rectifier operation, power control.

-2
0

50

100

150
200
ω (rad/sec)

250

300

(b) Inverter operation, power control.

Fig. 6.34 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the alternatingvoltage control of the VSC-HVDC converter (power control mode, Pdc = ±0.9 p.u.).
Variations of ku . UIF = 0.7.

The alternating-voltage controller of the VSC-HVDC converter also affects the
electrical damping of the synchronous generator. As shown in Fig. 6.34, the alternatingvoltage controller has the effect to bring the synchronous generator and the VSC-HVDC
converter electrically “closer” for both rectifier and inverter operation. Consequently, with
higher gain ku , the impact of the VSC-HVDC converter on the electrical damping curve
of the synchronous generator increases, either positive or negative.

6.6.3 Island operation
System definition
Fig. 6.35 shows the Jacobian transfer matrix formulation of the island system shown
g
in Fig. 6.24(b), where Ge (s) is the transfer function between ∆ωr to ∆Teg with all the
feedback-control loops of the VSC-HVDC converter closed (inside the dashed box). Table 6.6 shows the base control parameters selected for the VSC-HVDC converter. In island operation, more interactions between the synchronous generator and the VSC-HVDC
converter in the subsynchronous frequency range are expected. According to the definition in (6.53), the UIF with the island-operation configuration in Fig. 6.24(b) is
UIF =

MWdc
= 2.33
MVAg

(6.55)

simply because SCg = 0, i.e., the short-circuit power of the ac system is zero if the
synchronous generator is not included.
173

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

Ge (s )

U ref + -

∆V
V0

ku
s

AVC

kp

AC-Jacobian
transfer
matrix

∆θ v

-

∆ωrg

∆P

Pref + -

∆Teg

∆U f

s

J (s )

PSL
s
1 + sTm
∆Pref

∆ω

Frequency
measurement

Kf
1 + sT f
Frequency
controller

Fig. 6.35 Jacobian transfer matrix formulation of the island system shown in Fig. 6.24(b).

Table 6.6 Control parameters of the VSC-HVDC converter in island operation. Per unit based on
350 MVA and 195 kV.

Controller

Values

Active-power control

kp

60 rad/s

Alternating-voltage control

ku

60

Frequency control

Kf , Tf , Tm ,

3.72, 0.3 s, 0.02 s

High-pass current control

174

Parameters

kv , αv

0.2 p.u., 40 rad/s

6.6. Subsynchronous characteristics
Rectifier operation
To operate the VSC-HVDC converter as a rectifier, the synchronous generator generates
85% of its rated power, i.e., Pg = 0.85 p.u. (150 MVA power base). The generated power
is then transmitted by the VSC-HVDC link to the ac system connected to the other converter station, i.e., the VSC-HVDC converter operates with Pdc = −0.36 p.u. (350 MVA
power base). No load is connected in the island system, i.e., Rl = 0.0.
Fig. 6.36 shows the electrical damping curves of the generator. In Fig. 6.36(a),
the HVDC converter is represented by a constant-frequency stiff voltage source. Similar
to the large ac-system connection configuration, the VSC-HVDC converter operating in
rectifier mode in the island system contributes with only positive damping to the generator
in the entire subsynchronous frequency range. The damping effects are, however, much
higher due to the high UIF value. The frequency-scanning results from PSCAD/EMTDC
(dashed curve) confirm the same effect. The two plots produced by the linear model and
the time simulation match fairly well.
Inverter operation
To operate the VSC-HVDC converter as an inverter, the synchronous generator generates
10% of its rated power, i.e., Pg = 0.1 p.u., while the VSC-HVDC link operates with
Pdc = 0.9 p.u. Resistive loads (330 MW) are connected in the island system to consume
the power from the generator and the VSC-HVDC link.
Fig. 6.37 shows the electrical damping curves of the generator. In Fig. 6.37(a), the
HVDC converter is represented by a constant-frequency stiff voltage source. Compared
to Fig. 6.36(a), the electrical damping of the synchronous generator in Fig. 6.37(a) is
apparently higher, which is due to the damping effects contributed by the resistive loads
connected in the island system. As shown in Fig. 6.37(b), the influence on the damping
curve of the generator by the VSC-HVDC converter operating in inverter mode is also
similar to the large ac-system connection configuration but with a higher effect due to the
high UIF value. The VSC-HVDC converter contributes with positive damping in the lower
frequency range but with negative damping in the higher frequency range. The frequencyscanning results from PSCAD/EMTDC (dashed curve) confirm the same effect. The two
plots in Fig. 6.37 produced by the linear model and the time simulation also match fairly
well.
The influence of the control parameters kp and kv of the VSC-HVDC converter
on the electrical damping is also similar to the large ac-system connection configuration.
As shown in Fig. 6.38, the high-pass current control contributes with negative damping,
while the active-power control contributes with positive damping. Therefore, from subsynchronous characteristics point of view, a high gain kp of the active-power control is
favorable even if it is not necessary from a power-control point of view.
The control parameters of the frequency controller of the VSC-HVDC converter
175

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

5

5

Linear model
Time simulation

4

4
3
D (jω)

3
D (jω)

e

2

e

Linear model
Time simulation

2

1

1

0

0

-1
0

50

100

150
200
ω (rad/sec)

250

-1
0

300

(a) With constant-frequency stiff voltage source.

50

100

150
200
ω (rad/sec)

250

300

(b) With VSC-HVDC converter.

Fig. 6.36 Electrical damping curves of the generator (Pg = 0.85 p.u.) affected by the VSCHVDC converter (rectifier operation, Pdc = −0.36 p.u.).

5

5

Linear model
Time simulation

4

3

2

2

e

e

D (jω)

3
D (jω)

Linear model
Time simulation

4

1

1

0

0

-1

-1

-2
0

50

100

150
200
ω (rad/sec)

250

300

(a) With constant-frequency stiff voltage source.

-2
0

50

100

150
200
ω (rad/sec)

250

300

(b) With VSC-HVDC converter.

Fig. 6.37 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the VSC-HVDC
converter [inverter operation, Pdc = 0.9 p.u., Rl = 1.06 p.u. (330 MW resistive
loads)].

176

6.6. Subsynchronous characteristics
5

5

k =0.0

k =0.0

v

k =0.5
v

p

2

e

e

p

k =100

3
D (jω)

2

k =60

4

v

3
D (jω)

p

k =0.2

4

1

1

0

0

-1

-1

-2
0

50

100

150
200
ω (rad/sec)

250

300

(a) kp = 0.0 rad/s, ku = 0.0, variation of kv .

-2
0

50

100

150
200
ω (rad/sec)

250

300

(b) kv = 0.2 p.u., ku = 0.0, variation of kp .

Fig. 6.38 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the control parameters of the VSC-HVDC converter (inverter operation, power control mode,
Pdc = 0.9 p.u.).

might also affect its subsynchronous characteristics. However, since a large time constant
Tf is usually applied, such an influence mainly appears in the low-frequency range where
no risk of SSTI is expected. Therefore, no further discussion of the sensitivity of the
subsynchronous characteristics of the VSC-HVDC converter to those control parameters
is given in the thesis.

6.6.4 Summary of the subsynchronous characteristics
In this section, the subsynchronous characteristics of a VSC-HVDC converter using powersynchronization control are investigated by the frequency-scanning method. Two types of
ac-network configurations, i.e., large ac-system connection and island operation, are investigated. The following conclusions are drawn for the large ac-system connection:
1. A VSC-HVDC converter can have considerable impact on the electrical damping curve of a synchronous generator for network configurations with UIF values
greater than 0.1. The impact increases with higher UIF values.
2. The VSC-HVDC converter might have different subsynchronous characteristics
depending on its operation mode. The safest mode is when the converter operates
as a rectifier in the power control mode. In this operation mode, the VSC contributes
with only positive damping in the entire subsynchronous frequency range, similar
to resistive loads. By operating as an inverter, the VSC contributes with positive
damping in the lower frequency range. However, it contributes with slightly negative damping in the higher frequency range. In the direct-voltage control mode, in
either rectifier or inverter operation, the VSC might contribute with negative damp177

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems
ing in the lower frequency range. However, this frequency range is rarely above
62.8 rad/s (10 Hz) even with the highest UIF value (UIF = 0.7).
3. The control parameters of the VSC-HVDC converter might also affect the subsynchronous characteristics. Surprisingly, the high-pass current control HHP (s), which
is introduced for the purpose of damping, negatively affects the electrical damping
of a synchronous generator. Further frequency-domain analysis shows that the reason is that HHP (s) reduces the phase margin of the synchronous generator. On the
other hand, the active-power control of the VSC-HVDC converter provides positive
damping. In other words, the negative-damping frequency range in the inverter operation mode can be reduced by selecting a higher gain of the active-power control.
The alternating-voltage control has the effect of shortening the electrical distance
between the VSC-HVDC converter and the synchronous generator. That is, with
higher gain of the alternating-voltage control, higher subsynchronous impact from
the VSC-HVDC converter is expected, either positive or negative.
In island operation, stronger interactions between the VSC-HVDC converter and the synchronous generator in the subsynchronous frequency range are expected. The impact of
operation modes and control parameters on the subsynchronous characteristics of the
VSC-HVDC converter is similar to large ac-system connections, but with higher magnitudes. From an SSTI point of view, a higher gain of active-power control is more favorable
if the stability margin of the VSC-HVDC link can be ensured.
With the two types of investigated network configurations, the only possible risk
of SSTI is when the VSC-HVDC converter operates as an inverter with network configurations having high UIF values, where the VSC-HVDC converter contributes with
negative damping in the higher subsynchronous frequency range. The negative-damping
frequency range, however, can be reduced by proper tuning of the control parameters.
Thus, a subsynchronous damping controller (SSDC) for a VSC-HVDC converter using
power-synchronization control is not considered necessary.

6.7 Summary
In this chapter, the control and modeling issues for VSC-HVDC links connected to lowinertia ac systems are investigated. For VSC-HVDC links connected to low-inertia ac
systems, a frequency droop controller and an alternating-voltage droop controller are normally required to coordinate frequency and voltage control with local power/voltage controlling units. The Jacobian transfer matrix modeling concept is applied to model a typical
island system which includes a synchronous generator, an induction motor, a VSC-HVDC
link and some RLC loads. The linear model has been used for tuning the control parameters of the VSC-HVDC link. The simulation studies show that power-synchronization
control is flexible for various operating modes related to island operation, while the mode
178

6.7. Summary
shifts are handled seamlessly. The subsynchronous characteristics of a VSC-HVDC link
using power-synchronization control are analyzed by the frequency-scanning method. A
VSC-HVDC converter might have different subsynchronous characteristics depending on
its operation mode. For some particular operation modes, a VSC-HVDC converter might
contribute with negative damping. However, the negative-damping frequency range can
be reduced by proper tuning of the control parameters.

179

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

180

Chapter 7
Conclusions and Future Work
This chapter presents the conclusions of the thesis and suggests future work.

7.1 Conclusions
PWM-based VSC-HVDC systems show many advantages compared to the thyristorbased LCC-HVDC system. One prominent feature is that the VSC-HVDC system has
the potential to be connected to very weak ac systems where the LCC-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, for grid-connected VSCs is proposed. A grid-connected VSC using power-synchronization control
basically resembles the dynamic behavior of a synchronous machine. However, due to
the technical requirements for a VSC-HVDC link and various limitations of VSC valves,
additional control functions are required to deal with various practical issues during operation. Such control functions include:
• A high-pass current control function to damp various resonances in ac systems.
• A current limitation function to ride through ac-system faults.
• A bumpless-transfer scheme for switching the synchronization input of the VSC,
and an anti-windup scheme for alternating-voltage control.
• A negative-sequence current controller to mitigate unbalance valve currents in the
steady state or during unbalanced ac-system faults.
By the comparison performed in the thesis, it is shown that power-synchronization control
is superior to the traditional vector current control for VSC-HVDC links connected to
weak ac systems.
181

Chapter 7. Conclusions and Future Work
The Jacobian transfer matrix is proposed as a unified modeling technique for dynamic modeling of ac/dc systems. With the ac Jacobian transfer matrix concept, a synchronous ac system is viewed upon as one multivariable feedback-control system where
the feedback controllers and the controlled process, i.e., the Jacobian transfer matrix, are
explicitly defined. Thus, the stability of a power system is uniquely defined as the stability
of the closed-loop system formed by the controllers and the Jacobian transfer matrix.
One interesting finding is that the transmission zeros of the ac Jacobian transfer matrix reflect several power-system stability phenomena. By using a simplified ac-network
configuration, the thesis shows that the transmission zeros of the ac Jacobian transfer
matrix have close relationships with angle and voltage stability defined by the classical
power-system theory. Two additional common power-system components, i.e., the synchronous generator and induction motor, are also modeled by the Jacobian transfer matrix
concept. It is discovered that the transmission zeros of the ac Jacobian transfer matrices
are useful to interpret some classical power-system concepts, such as the synchronizing
torque for the synchronous generator and the pull-out slip for the induction motor, from a
feedback-control point of view.
The complex poles of the ac Jacobian transfer matrix represent the various resonances in the ac system. The high-pass current control of power-synchronization control
and the inner current controller of vector current control are all shown to have general
damping effects on the resonances regardless of the frequency of the resonance.
The similar concept is extended to model dc systems constructed by multiple VSC
terminals. By using a π-link model, it is proven that the dc Jacobian transfer matrix is an
inherently unstable process, where the dc resistance has a destabilizing effect. Accordingly, the feedback controller, i.e., the direct-voltage controller, has to be applied at least
at one of the converter stations to stabilize the dc system. Similar to the complex poles
of the ac Jacobian transfer matrix, the complex poles of the dc Jacobian transfer matrix
also represent various resonances in the dc system. Long overhead dc-transmission line
may create resonance peaks in the low frequency range where the control system of the
VSC-HVDC link is active. A notch filter in the direct-voltage control loop is proposed to
reduce the resonance peak.
For VSC-HVDC links using power-synchronization control, the SCR of the ac system is no longer a critical limiting factor, i.e., VSC-HVDC links are feasible to be connected to ac systems with any short-circuit capacity. In addition, the VSC-HVDC link
contributes with short-circuit capacity to the ac system at the PCC, however, without increasing the short-circuit current thanks to its current limiting capability during ac-system
faults. The major consideration is the RHP zero of the ac Jacobian transfer matrix, which
moves closer to the origin with larger load angles. The RHP zero imposes a fundamental
limitation on the achievable bandwidth of the VSC, which implies that the VSC-HVDC
link shall not operate with a too large load angle for maintaining a reasonable stability
margin. For weak-ac-system connections, higher values of dc capacitance are also nec182

7.2. Future work
essary to reduce the direct-voltage variations. A control structure for interconnecting two
very weak ac systems is proposed in the thesis. This control structure 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 VSC-HVDC links connected to low-inertia ac systems, a frequency droop controller and an alternating-voltage droop controller are normally required to coordinate
frequency and voltage control with local power/voltage controlling units. In the thesis,
the Jacobian transfer matrix modeling concept is applied to model a typical island system which includes a synchronous generator, an induction motor, a VSC-HVDC link and
some RLC loads. The linear model has been used for tuning the control parameters of
the VSC-HVDC link. The simulation studies show that power-synchronization control is
flexible for various operation modes related to island operation, while the mode shifts are
handled seamlessly.
The subsynchronous characteristics of a VSC-HVDC link using power-synchronization control are analyzed by the frequency-scanning method. Two ac-network configurations, i.e., large ac-system connection and island operation, are investigated. A VSCHVDC converter might have different subsynchronous characteristics depending on its
operation mode. For most of the operation modes, the VSC-HVDC converter contributes
with positive damping to a synchronous generator located in the vicinity. The only possible risk of SSTI is when the VSC-HVDC converter operates as an inverter in a high-UIF
system. The negative-damping frequency range, however, can be reduced by proper tuning
of the control parameters. Thus, a subsynchronous damping controller for a VSC-HVDC
link using power-synchronization control is not considered necessary.

7.2 Future work
The following is a list of possible future work:
• Power-synchronization control is a feasible control system for any grid-connected
VSC. Another possible application is control of STATCOMs with energy storage.
By using power-synchronization control, the STATCOM is able to operate with
various challenging operating conditions.
• By using power-synchronization control, the SCR of the ac system is no longer a
limiting factor for VSC-HVDC links connected to weak ac systems. However, the
variation of SCRs, i.e., model uncertainty, is still a challenge. Besides the robust
control methods proposed in the thesis, other possible solutions to deal with model
uncertainties are 1. adaptive control. 2. online detection of the SCR of the ac system.
• The losses of the VSC valves have been neglected in the linear models in the thesis.
However, it was shown by frequency-scanning results that the representation of the
183

Chapter 7. Conclusions and Future Work
valve losses in linear models has an importance if dc resonances are of concern.
To properly represent the valve losses, the linear model should take into account
the topology of the converter as well as the applied PWM technique, which are of
nonlinear nature.
• The structure proposed for interconnecting two very weak ac systems has a possibility to be simplified. A possible simplification is to remove the power-synchronization control loop but to use the direct-voltage control to supply the synchronization input to the VSC instead. Such a modification has the potential to improve the
response time of the VSC-HVDC link and reduce the dc-capacitance requirement.
• The modeling concept of the ac Jacobian transfer matrix is useful for studying the
interactions of high power-electronics devices located in the vicinity, e.g., multiinfeed HVDC links, interactions between HVDC links and FACTS devices, etc.
• The modeling concept of the dc Jacobian transfer matrix is useful for studying
the potential future dc systems. In this thesis, only a π-link dc model with two
VSC terminals is analyzed. Larger dc systems with more VSC terminals should be
analyzed to gain a thorough understanding of the property of a dc system.
• In analyzing the subsynchronous characteristics of a VSC-HVDC converter, two
ac-network configurations, i.e., large ac-system connection and island operation,
are investigated. Another scenario to be investigated is when a series-compensated
ac line is connected in the vicinity.

184

References
[1] “VSC transmission,” Cigr´ Working Group B4.37-269, Tech. Rep., 2005.
e
[2] G. Asplund, K. Eriksson, and H. Jiang, “DC transmission based on voltage source
converters,” in Cigr´ Colloquium on HVDC and FACTS, South Africa, 1997.
e
[3] S. Johansson, G. Asplund, E. Jansson, and R. Rudervall, “Power system stability
benefits with VSC DC-transmission systems,” in Cigr´ Conference B4-204, Paris,
e
France, 2004.
[4] L. Zhang, L. Harnefors, and P. Rey, “Power system reliability and transfer capability improvement by VSC-HVDC (HVDC light),” in Cigr´ Regional Meeting,
e
Tallin, Estonia, 2007.
[5] “IEEE guide for planning DC links terminating at AC locations having low shortcircuit capacities,” IEEE Std 1204-1997, Tech. Rep., 1997.
[6] B. Franken and G. Andersson, “Analysis of HVDC converters connected to weak
systems,” IEEE Trans. Power Syst., vol. 5, no. 1, pp. 235–242, February 1990.
[7] A. Gavrilovic, “AC/DC system strength as indicated by short circuit ratios,” in
AC/DC Power Transmission International Conference, London, 1991.
[8] O. B. Nayak, A. M. Gole, D. G. Chapman, and J. B. Davies, “Dynamic performance of static and synchronous compensators at an HVDC inverter bus in a very
weak ac system,” IEEE Trans. Power Syst., vol. 9, no. 3, pp. 1350–1358, August
1994.
[9] D. Jovcic, N. Pahalawaththa, and M. Zavahir, “Investigation of the use of inverter control strategy instead of synchronous condensers at inverter terminal of
an HVDC system,” IEEE Trans. Power Delivery, vol. 15, no. 2, pp. 704–709, April
2000.
[10] D. Jovcic, L. A. Lamont, and L. Xu, “VSC transmission model for analytical studies,” in IEEE Power Engineering Society General Meeting, Toronto, Canada, 2003.
185

References
[11] M. Durrant, H. Werner, and K. Abbott, “Model of a VSC HVDC terminal attached
to a weak ac system,” in IEEE Conference on Control Applications, Istanbul, 2003.
[12] P. Fischer, “Modelling and control of a line-commutated HVDC transmission system interacting with a VSC STATCOM,” Ph.D. dissertation, Royal Institute of
Technology, Stockholm, Sweden, 2007.
[13] L. Harnefors, M. Bongiornos, and S. Lundberg, “Input-admittance calculation
and shaping for controlled voltage-source converters,” IEEE Trans. Ind. Electron.,
vol. 54, no. 6, pp. 3323–3334, December 2007.
[14] “On voltage and power stability in ac/dc systems,” Cigr´ Working Group 14.05e
222, Tech. Rep., 2003.
[15] P. Kovacs, Transient Phenomena in Electrical Machines. Elsevier, 1984.
[16] P. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machinery and
Drive Systems, 2nd Edition. Piscataway, NJ: IEEE Press, 2002.
[17] M. P. Kazmierkowski, R. Krishnan, and F. Blaabjerg, Control in Power Electronics.
Academic Press., 2002.
[18] J. Holtz, “The representation of ac machines dynamics by complex signal flow
graphs,” IEEE Trans. Ind. Electron., vol. 42, no. 3, pp. 263–271, June 1995.
[19] D. N. Zmood, D. G. Holmes, and G. H. Bode, “Frequency-domain analysis of three
phase linear current regulators,” IEEE Trans. Ind. Applicat., vol. 37, no. 2, pp. 601–
610, March/April 2001.
[20] L. Harnefors, “Modeling of three-phase dynamic systems using complex transfer
functions and transfer matrices,” IEEE Trans. Ind. Electron., vol. 54, no. 4, pp.
2239–2248, August 2007.
[21] P. M. Anderson, B. L. Agrawal, and J. E. V. Ness, Subsynchronous resonance in
power systems. Piscataway, NJ: IEEE Press, 1989.
[22] K. R. Padiyar, Analysis of subsynchronous resonance in power systems.
States of America: Kluwer Academic Publishers, 1999.
[23] E. W. Kimbark, Direct Current Transmission, Vol. I.

New York: Wiley, 1971.

[24] J. Arrillaga, High Voltage Direct Current Transmission, 2nd Edition.
The Institution of Electrical Engineers, 1998.
[25] B. M. Weedy and B. J. Cory, Electric Power Systems, 4th Edition.
England: John Wiley & Sons Ltd., 1998.
186

United

London:

West Sussex,

References
[26] J. Setr´ us and L. Bertling, “Introduction to HVDC technology for reliable electrical
e
power systems,” in The 10th International Conference on Probablistic Methods
Applied to Power Systems, Rincon, Puerto Rico, 2008.
[27] S. Corsi, A. Danelli, and M. Pozzi, “Emergency-stability controls through HVDC
links,” in IEEE Power Engineering Society Summer Meeting, Chicago, USA, 2002.
[28] T. Smed and G. Andersson, “Utilizing HVDC to damp power oscillations,” IEEE
Trans. Power Delivery, vol. 8, no. 2, pp. 620–627, April 1993.
[29] W. Breuer, M. Lemes, and D. Retzmann, “Perspectives of HVDC and FACTS for
system interconnection and grid enhancement,” in Cigr´ Workshop: Brazil-Chinae
India Summit Meeting on HVDC and Hybrid Systems, Rio de Janeiro, Brazil, 2006.
[30] G. Andersson, P. Donalek, R. Farmer, and N. Hatziargyriou, “Causes of the 2003
major grid blackouts in North America and Europe, and recommended means to
improve system dynamic performance,” IEEE Trans. Power Syst., vol. 20, no. 2,
pp. 1922–1928, November 2005.
[31] M. Bahrman and B. Johnson, “The ABCs of HVDC transmission technologies,”
IEEE Power Energy Mag., vol. 5, no. 2, pp. 32–44, March/April 2007.
[32] P. Kundur, Power System Stability and Control.
1994.

New York: McGraw-Hill, inc.,

[33] C. V. Thio, J. B. Davies, and K. L. Kent, “Commutation failures in HVDC transmission systems,” IEEE Trans. Power Delivery, vol. 11, no. 2, pp. 946–957, April
1996.
[34] L. Zhang and M. H. Bollen, “Characteristics of voltage dips in power systems,”
IEEE Trans. Power Delivery, vol. 15, no. 2, pp. 827–832, April 2000.
[35] L. Zhang and L. D¨ fn¨ s, “A novel method to mitigate commutation failures in
o a
HVDC systems,” in PowerCon 2002, Kunming, China, 2002.
[36] “Capacitor commutated converters (CCC) HVDC interconnections: digital modeling and benchmark circuit,” Cigr´ Working Group B4.34-352, Tech. Rep., 2008.
e
[37] N. Flourentzou, V. G. Agelidis, and G. D. Demetriades, “VSC-based HVDC power
transmission systems: An overview,” IEEE Trans. Power Electron., vol. 24, no. 3,
pp. 592–602, March 2009.
[38] A. Lindberg, “PWM and control of two and three level high power voltage source
converters,” Royal Institute of Technology, Stockholm, Sweden, Licentiate Thesis,
1995.
187

References
[39] S. Allebrod, R. Hamerski, and R. Marquardt, “New transformerless, scalable modular multilevel converters for HVDC-transmission,” in IEEE Power Electronics
Specialists Conference - PESC 2008, Rhodes, Greece, 2008.
[40] J. Rodr´guez, J.-S. Lai, and F. Z. Peng, “Multilevel inverters: a survey of topologies,
ı
controls, and applications,” IEEE Trans. Ind. Electron., vol. 49, no. 5, pp. 724–738,
August 2002.
[41] A. Lesnicar and R. Marquardt, “A new modular voltage source inverter topology,”
in 10th European Conference on Power Electronics and Applications-EPE 2003,
Toulouse, France, 2003.
[42] M. Hagiwara and H. Akagi, “Control and experiment of pulsewidth-modulated
modular multilevel converters,” IEEE Trans. Power Electron., vol. 24, no. 7, pp.
1737–1746, July 2009.
[43] M. Glinka and R. Marquardt, “A new AC/DC multilevel converter family,” IEEE
Trans. Power Electron., vol. 52, no. 3, pp. 662–669, June 2005.
[44] “It is time to connect,” http://www.abb.com/hvdc, accessed at February 2010, Tech.
Rep.
[45] L. Zhang, L. Harnefors, and H.-P. Nee, “Power-synchronization control of gridconnected voltage-source converters,” IEEE Trans. Power Syst., vol. 25, no. 2, pp.
809–820, May 2010.
[46] B. T. Ooi and X. Wang, “Voltage angle lock loop control of the boosted type PWM
converter for HVDC application,” IEEE Trans. Power Electron., vol. 5, no. 2, pp.
229–235, April 1990.
[47] G. Joos, L. Mor´ n, and P. Ziogas, “Performance analysis of a PWM inverter VAR
a
compensator,” IEEE Trans. Power Electron., vol. 6, no. 3, pp. 380–391, July 1991.
[48] J. Svensson, “Voltage angle control of a voltage source inverter, application to a
grid-connected wind turbine,” in 6th European Conference on Power Electronics
and Applications, Sevilla, 1995.
[49] ——, “Simulation of power angle controlled voltage source converter using a linear
quadratic method in a wind energy application,” in 5th Workshop on Computers in
Power Electronics, IEEE, 1996.
[50] B. K. Bose, Power Electronics and Variable Frequency Drives.
IEEE Press, 1996.
188

Piscataway, NJ:

References
[51] L. Harnefors and H.-P. Nee, “Model-based current control of ac machines using
the internal model control method,” IEEE Trans. Ind. Applicat., vol. 34, no. 1, pp.
133–141, January/February 1998.
[52] L. Harnefors, “On analysis, control and estimation of variable-speed drives,” Ph.D.
dissertation, Royal Institute of Technology, Stockholm, Sweden, 1997.
[53] H. Xie, “On power-system benefits, main-circuit design, and control of STATCOMS with energy storage,” Ph.D. dissertation, Royal Institute of Technology,
Stockholm, Sweden, 2009.
[54] J. W. Choi and S. K. Sul, “Fast current controller in three-phase AC/DC boost converter using d-q axis crosscoupling,” IEEE Trans. Power Electron., vol. 13, no. 1,
pp. 179–185, January 1998.
[55] M. P. Kazmierkowski and L. Malesani, “Current control techniques for three-phase
voltage-source PWM converters: A survey,” IEEE Trans. Ind. Electron., vol. 45,
no. 5, pp. 691–703, October 1998.
[56] T. Kawabata, T. Miyashita, and Y. Yamamoto, “Dead beat control of three phase
PWM inverter,” IEEE Trans. Power Electron., vol. 5, no. 1, pp. 21–28, January
1990.
[57] L. Malesani, P. Mattavelli, and S. Buso, “Robust deat-beat current control for PWM
rectifiers and active filters,” IEEE Trans. Ind. Applicat., vol. 35, no. 3, pp. 613–620,
May/June 1999.
[58] C. B. Jacobina, M. B. R. Corrˆ a, R. F. Pinheiro, E. R. C. Silva, and A. M. N.
e
Lima, “Modeling and control of unbalanced three-phase systems containing PWM
converters,” IEEE Trans. Ind. Applicat., vol. 37, no. 6, pp. 1807–1815, November/December 2001.
[59] H.-S. Song and K. Nam, “Dual current control scheme for PWM converter under
unbalanced input voltage conditions,” IEEE Trans. Ind. Electron., vol. 46, no. 5,
pp. 953–959, October 1999.
[60] M. Bongiorno, “On control of grid-connected voltage source converters: mitigation of voltage dips and subsynchronous resonances,” Ph.D. dissertation, Chalmers
University of Technology, Gothenburg, Sweden, 2007.
[61] Y. Jiang, “Active and reactive power control for transmission systems with voltage
source converters,” Ph.D. dissertation, Royal Institute of Technology, Stockholm,
Sweden, 1997.
189

References
[62] L. Zhang, H.-P. Nee, and L. Harnefors, “Analysis of stability limitations of a VSCHVDC link using power-synchronization control,” Submitted to IEEE Trans. on
Power Syst.
[63] 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. on Power Syst.
[64] P. M. Anderson and A. A. Fouad, Power System Control and Stability, 2nd Edition.
New York: IEEE Press, 2003.
[65] T. V. Cutsem and C. Vournas, Voltage Stability of Electric Power Systems.
Springer, inc., 1998.
[66] C. Taylor, Power System Voltage Stability. New York: McGraw-Hill, inc., 1994.
[67] I. J. T. F. on Stability Terms and Definitions, “Definition and classification of power
system stability,” IEEE Trans. Power Syst., vol. 19, no. 2, pp. 1387–1401, May
2004.
[68] F. P. Demello and C. Concordia, “Concept of synchronous machine stability as
affected by excitation control,” IEEE Trans. Power App. Syst., vol. PAS-88, no. 4,
pp. 189–202, April 1969.
[69] M. Klein, C. J. Rogers, and P. Kundur, “A fundamental study of inter-area oscillations in power systems,” IEEE Trans. Power Syst., vol. 6, no. 3, pp. 914–921,
August 1991.
[70] E. V. Larsen and D. A. Swann, “Applying power system stabilizers Part I: general
concepts,” IEEE Trans. Power App. Syst., no. 6, pp. 3017–3024, June 1981.
[71] M. Noroozian and G. Andersson, “Damping of inter-area and local modes by use
of controllable components,” IEEE Trans. Power Delivery, vol. 10, no. 4, pp. 2007–
2012, October 1995.
[72] D. J. Hill, “Nonlinear dynamic load models with recovery for voltage stability studies,” IEEE Trans. Power Syst., vol. 8, no. 1, pp. 166–176, February 1993.
[73] T. V. Cutsem, “Voltage instability: phenomena, countermeasures, and analysis
methods,” Proceedings of the IEEE, vol. 88, no. 2, pp. 208–227, February 2000.
[74] B. Gao, G. K. Morison, and P. Kundur, “Voltage stability evaluation using modal
analysis,” IEEE Trans. Power Syst., vol. 7, no. 4, pp. 1529–1542, November 1992.
[75] M. K. Pal, “Discussion of voltage stability evaluation using modal analysis,” IEEE
Trans. Power Syst., vol. 7, no. 4, pp. 1529–1542, November 1992.
190

References
[76] D. Lee, “Voltage and power stability of HVDC systems,” Ph.D. dissertation, Royal
Institute of Technology, Stockholm, Sweden, 1997.
[77] M. Bahrman, E. V. Larsen, R. J. Piwko, and H. S. Patel, “Experience with HVDCturbine-generator torsional interaction at Square Butte,” IEEE Trans. Power App.
Syst., vol. PAS-99, pp. 966–975, May/June 1980.
[78] I. S. R. W. G. of the System Dynamic Performance Subcommitte Power System
Engineering Committe, “Terms, definitions and symbols for subsynchronous oscillations,” IEEE Trans. Power App. Syst., vol. PAS-104, no. 6, pp. 1326–1334, June
1985.
[79] I. S. R. W. G. of the System Dynamic Performance Subcommitte, “Reader’s guide
to subsynchronous resonance,” IEEE Trans. Power Syst., vol. 7, no. 1, pp. 1326–
1334, February 1992.
[80] L. Harnefors, “Analysis of subsynchronous torsional interaction with power electronic converters,” IEEE Trans. Power Syst., vol. 22, no. 1, pp. 305–313, February
2007.
[81] I. M. Canay, “A novel approach to the torsional interaction and electrical damping
of the subsynchronous machine. Part I: Theory,” IEEE Trans. Power App. Syst.,
vol. PAS-101, no. 10, pp. 3630–3638, October 1982.
[82] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and
Design, 2nd Edition. West Sussex, England: John Wiley & Sons Ltd., 2005.
[83] A. J. Laub and B. C. Moore, “Calculation of transmission zeros using QZ techniques,” Automatica, vol. 14, no. 6, pp. 557–566, November 1978.
[84] C. B. Moler and G. W. Stewart, “An algorithm for generalized matrix eigenvalue
problems,” SIAM Journal on Numerical Analysis, vol. 10, no. 2, pp. 241–256, April
1973.
[85] R. Ottersten, “On control of back-to-back converters and sensorless induction machine drives,” Ph.D. dissertation, Chalmers University of Technology, Gothenburg,
Sweden, 2003.
[86] L. Zhang and H.-P. Nee, “Multivariable feedback design of VSC-HVDC connected
to weak ac systems,” in PowerTech 2009, Bucharest, Romania, 2009.
[87] J. D. Ainsworth, A. Gavrilovic, and H. L. Thanawala, “Static and synchronous
compensations for HVDC transmission converters connected to weak ac systems,”
in Cigr´ General Session, France, 1980.
e
191

References
[88] M. Morari and E. Zafiriou, Robust Process Control.
sey: Prentice-Hall, Inc., 1989.
[89] B. W. Bequette, Process Control.
2003.

Englewood Cliffs, New Jer-

Upper Saddle River, NJ: Prentice-Hall, Inc.,

[90] H. Kwakernaak, “Robust control and H∞ -optimization–tutorial paper,” Automatica, vol. 29, no. 2, pp. 255–273, 1993.
[91] K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control. Englewood Cliffs,
New Jersey: Prentice Hall, 1996.
[92] J. C. Doyel, “Analysis of feedback systems with structured uncertainties,” IEE Proceedings, vol. 129, no. 6, pp. 242–250, November 1982.
[93] A. Packard and J. C. Doyel, “The complex structured singular value,” Automatica,
vol. 29, no. 1, pp. 71–109, 1993.
[94] L. Zhang, L. Harnefors, and H.-P. Nee, “Modeling and control of VSC-HVDC links
connected to island systems,” in IEEE Power and Energy Society General Meeting,
Minneapolis, USA, 2010.
[95] C. V. Thio and J. B. Davies, “New synchronous compensators for the nelson river
HVDC system-planning requirements and specifications,” IEEE Trans. Power Delivery, vol. 6, no. 2, pp. 922–928, April 1991.
[96] G. Zhang, Z. Xu, and H. Liu, “Supply passive networks with VSC-HVDC,” in
IEEE Power Engineering Society Summer Meeting, Columbia, Canada, 2001.
[97] C. K. Sao and P. W. Lehn, “Intentional islanded operation of converter fed microgrids,” in IEEE Power Engineering Society General Meeting, Minneapolis, Minnesota, USA, 2006.
[98] C. Du, E. Agneholm, and G. Olsson, “Comparison of different frequency controllers for a VSC-HVDC supplied system,” IEEE Trans. Power Delivery, vol. 23,
no. 4, pp. 2224–2232, October 2008.
[99] ——, “VSC-HVDC system for industrial plants with onsite generators,” IEEE
Trans. Power Delivery, vol. 24, no. 3, pp. 1359–1366, July 2009.
[100] M. Parniani and M. R. Iravani, “Computer analysis of small-signal stability of
power systems including network dynamics,” IEE Proc.-Gener. Transm. Distrib.,
vol. 142, no. 6, pp. 613–617, 1995.
192

References
[101] D.-J. Kim, Y.-H. Moon, and H.-K. Nam, “SSR small-signal stability analysis program of power systems and its application to IEEE benchmark systems,” in PowerTech 2007, Lausanne, 2007.
[102] Y. H¨ fner, H. Duch´ n, K. Lind´ n, and M. Hyttinen, “Improvement of subsyna
e
e
chronous torsional damping using VSC-HVDC,” in PowerCon 2002, Kunming,
China, 2002.
[103] N. Prabhu and K. R. Padiyar, “Investigation of subsynchronous resonance with
VSC-based HVDC transmission systems,” IEEE Trans. Power Delivery, vol. 24,
no. 1, pp. 433–440, January 2009.
[104] “HVDC system control for damping of subsynchronous oscillations,” EPRI Research Project 1425-1, Tech. Rep., 1982.
[105] R. J. Piwko and E. V. Larsen, “HVDC system control for damping of subsynchronous oscillations,” IEEE Trans. Power App. Syst., vol. PAS-101, no. 7, pp.
2203–2211, July 1982.

193

References

194

Appendices

195

Appendix A
Fundamentals of the Phasor and the
Space-Vector Theory
This appendix gives a brief review of the two different approaches for ac-circuit analysis,
i.e., the phasor approach and the space-vector approach in Section A.1 and Section A.2.
In the equations, phasors are denoted by capital letters with overline, while space vectors
are denoted by bold letters. In order to avoid confusions with scaling, per unit values are
assumed in the equations for both of the theory. In Section A.3, the implementations of αβ
and dq transformations for controller designs are described.

A.1 Fundamentals of the phasor theory
The phasor theory is the major mathematical tool for power systems. It has been used
widely in power-flow programs, short-circuit calculations, and stability analysis. In this
section, the fundamentals of the theory are briefly reviewed .
With the simplified three-phase ac system shown Fig. 4.5, the impedance Z of the
ac line is defined as
Z = R + jX

(A.1)

where X = ω1 L is defined as the reactance of the ac line, with ω1 = 2πf1 , where f1 is
the fundamental frequency of the ac system.
In the phasor theory, Ohm’s law still holds for three-phase ac systems if they are
¯
symmetrical, i.e., the current phasor I can be calculated as
¯
¯
¯
¯
¯ V −E = V −E .
I=
Z
R + jX

(A.2)

The phasor relations in (A.2) can also be conveniently visualized by the phasor diagram
as shown in Fig. A.1.
197

Chapter A. Fundamentals of the Phasor and the Space-Vector Theory
Imaginary

V

jωL I
I
RI

θ
E

Real

Fig. A.1 Phasor diagram of the electrical circuit in Fig. 4.5.

The complex power is defined as
¯
¯¯
S = P + jQ = V I ∗
V cos θ + jV sin θ − E
= (V cos θ + jV sin θ)
jX
2
EV
V − EV cos θ
=
sin θ + j
X
X



(A.3)

where the resistance R is neglected in (A.3) for simplicity. From (A.3), the active and
reactive powers P and Q can be identified as
EV
sin θ
X
V 2 − EV cos θ
Q=
.
X
P =

(A.4)
(A.5)

A.2 Fundamentals of the space-vector theory
In contrast to the phasor approach, where only fundamental frequency effects are treated,
the space-vector approach deals with instantaneous values in the electrical circuit. The
basic idea of the space-vector theory comes from the fact that one complex vector y can
be used to represent three-phase variables ya , yb , yc , if any zero-sequence components are
disregarded. As shown in Fig. A.2(a), the complex vector y is defined as
ys =

2


ya + yb ej 3 + yc ej 3
3

(A.6)

where superscript s denotes the stationary αβ reference frame.
At any instant, the rotating vector ys has two corresponding variables yα and yβ on
the αβ (i.e., real and imaginary) axes. To transform the rotating vector to a non-rotating
198

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

β
q

b





y

d

y
a

α

ω1t

yd

yq

α




c

(a)

(b)

Fig. A.2 Reference frame transformations. (a) From three-phase variables to a rotating vector in
the stationary αβ frame. (b) From the stationary αβ frame to the synchronous dq frame.

one, a rotating dq reference frame is introduced, as shown in Fig. A.2(b). The dq frame
rotates in the same direction as ys with a speed that equals the angular frequency ω1 of
the ac system. Consequently, the vector y in the dq frame is related to ys in the stationary
frame by
ys = ejω1 t y.
(A.7)
In the following, the simple ac system in Fig. 4.5 is analyzed by the space-vector approach. The equations for the dynamics of the ac system in Fig. 4.5 can be described by
Kirchhoff’s voltage law in the stationary frame as
L

dis
= vs − Es − Ris
dt

(A.8)

where vs , Es , and is denote the space vectors of the voltage of node V, the voltage of node
E, and the current of the ac line, respectively. To transform the electrical variables from
the stationary frame to the dq frame, the space αβ vectors are written as
Es = Eejω1 t , is = iejω1 t , vs = vejω1 t .

(A.9)

Substituting (A.9) into (A.8) yields the dynamic equation in the dq frame
L

di
= v − E − Ri − jω1 Li.
dt

(A.10)

By comparing (A.10) to (A.2), it can be observed that there is a relationship between
phasors and space vectors. If the same reference node is chosen, e.g., node E, the d and q
components of the space vectors correspond to the real and imaginary parts of the phasors
di
in the steady state ( dt = 0).
For the simple ac system in Fig. 4.5, ac capacitors are not included. However, it can
easily be shown that the same statement holds even with ac capacitors included.
199

Chapter A. Fundamentals of the Phasor and the Space-Vector Theory
Assuming per unit scaling [20], the instantaneous active power P and reactive
power Q are defined in the dq frame as
P = Re {vi∗ } = vd id + vq iq


Q = Im {vi } = vq id − vd iq .

(A.11)
(A.12)

In the steady state, the active and reactive powers defined in (A.11) and (A.12) in the dq
frame equal those defined in (A.4) and (A.5) by the phasor approach.

A.3 Implementation of αβ and dq transformations
In the control block diagram of vector current control (Fig. 3.5) and power-synchronization
control (Fig. 3.10), αβ and dq transformations are applied for controller implementations.
In this section, the mathematical expressions for those blocks are described.

A.3.1 abc-αβ transformation
Following (A.6), the vector ys in the stationary frame is expressed as
ys = yα + jyβ =



2
ya + yb ej 3 + yc ej 3 .
3

(A.13)

Since the zero-sequence quantities in the three-phase variables are neglected, the following relation is established
ya + yb + yc = 0.

(A.14)

Based on (A.13) and (A.14), the αβ components can be derived from the three-phase
variables
yα = ya
1
yβ = √ (yb − yc ).
3

(A.15)

Based on (A.15) and (A.14), the transformation from the αβ components to the threephase variables can also be derived
ya = yα


1
3
yb = − yα +

2
2

1
3
yc = − yα −
yβ .
2
2

200

(A.16)

A.3. Implementation of αβ and dq transformations

A.3.2 αβ-dq transformation
According to (A.7), the vector y = yd + jyq in the synchronous dq frame is related to
ys = yα + jyβ in the stationary αβ frame by
y = e−jωt ys

(A.17)

where ωt is the synchronization variable (from PSL or PLL) of the VSC. Eq. (A.17) can
be written in component form as
yd = yα cos ωt + yβ sin ωt
yq = −yα sin ωt + yβ cos ωt.

(A.18)

Similarly, the vector ys = yα + jyβ in the stationary αβ frame is related to y = yd + jyq
in the synchronous dq frame by
ys = ejωt y
(A.19)
which can also be written in component form as
yα = yd cos ωt − yq sin ωt

yβ = yd sin ωt + yq cos ωt.

(A.20)

201

Chapter A. Fundamentals of the Phasor and the Space-Vector Theory

202

Appendix B
Jacobian Transfer Matrix
In this appendix, some mathematical derivations and proofs related to Jacobian transfer
matrix are given. In Section B.1, the transfer functions of the ac Jacobian transfer matrix
for VSCs connected to an impedance source neglecting the ac capacitor at the filter bus
are derived. In Section B.2, a mathematical proof of the instability of the dc Jacobian
transfer matrix for a π-link dc circuit is given.

B.1 Derivation of the transfer functions in Table 4.1
B.1.1 Transfer function JPθ (s)
In a synchronous reference frame, by neglecting the ac capacitor Cf , the dynamic equation
of the main circuit in Fig. 4.4 can be written in the dq frame as
di
L = v − E − Ri − jω1 Li
(B.1)
dt
where R = Rn + Rc , L = Ln + Lc , and i = ic = in . Eq. (B.1) can also be written in
component form as
did
= V cos θv − E0 − Rid + ω1 Liq
L
dt
diq
L
= V sin θv − Riq − ω1 Lid .
(B.2)
dt
Let the voltage magnitude of the VSC be kept constant, i.e., V = V0 . If the operating
points in (B.2) are denoted with subscript 0 and the deviations around the operating points
are denoted with the prefix ∆, (B.2) can be linearized using
θv = θv0 + ∆θv , id = id0 + ∆id , iq = iq0 + ∆iq

(B.3)

where cos θv and sin θv can be linearized as
sin (θv0 + ∆θv ) ≈ sin θv0 + cos θv0 ∆θv

cos (θv0 + ∆θv ) ≈ cos θv0 − sin θv0 ∆θv .

(B.4)
203

Chapter B. Jacobian Transfer Matrix
Substituting (B.3) and (B.4) into (B.2), and keeping only the deviation parts yields the
linearized form of (B.2)
d∆id
= −V0 sin θv0 ∆θv − R∆id + ω1 L∆iq
dt
d∆iq
L
= V0 cos θv0 ∆θv − R∆iq − ω1 L∆id .
dt

L

(B.5)

Moreover, by applying the Laplace transform to (B.5), the transfer functions of ∆id vs.
∆θv , and ∆iq vs. ∆θv are obtained
ω1 L cos θv0 − (sL + R) sin θv0
∆θv
(sL + R)2 + (ω1 L)2
ω1 L sin θv0 + (sL + R) cos θv0
∆iq = V0
∆θv .
(sL + R)2 + (ω1 L)2

∆id = V0

(B.6)

According to the definition in (A.11), the instantaneous active power P from the filter bus
to the ac system is
P = Re {uf i∗ } .
(B.7)
Linearizing (B.7) yields the expression of the active-power deviation as
∆P = Re {i∗ ∆uf + uf 0 ∆i∗ }
0

(B.8)

or in component form
id0

∆P =

iq0

T

∆uf d
∆uf q

+

uf d0

T

uf q0

∆id
∆iq

.

(B.9)

The current vector i0 = id0 + jiq0 at the operation point can be derived by
Uf 0 ejθu0 − E0
i0 =
Rn + jω1 Ln

(B.10)

which yields
Uf 0 sin θu0
E0 − Uf 0 cos θu0
, iq0 =
.
(B.11)
ω1 Ln
ω1 Ln
The resistance Rn in (B.10) is neglected in (B.11) to simplify the expression. The voltage
vector of the filter bus at the operating point, uf 0 = uf d0 + juf q0 , can be expressed as
id0 =

uf d0 = Uf 0 cos θu0 , uf q0 = Uf 0 sin θu0 .

(B.12)

By linearization of the expression
Ln
204

di
= uf − E − Rn i − jω1 Ln i
dt

(B.13)

B.1. Derivation of the transfer functions in Table 4.1
and subdivision into d and q components, the voltage-deviation components ∆uf d and
∆uf q can be written as
∆uf d = sLg ∆id − ω1 Lg ∆iq

∆uf q = sLg ∆iq + ω1 Lg ∆id .

(B.14)

The resistance Rn is also neglected in (B.14) to simplify the expression. By substituting
(B.6), (B.11), (B.12), and (B.14) into (B.9), the dynamic relation between ∆P and ∆θv
is found to be
a0 s2 + a1 s + a2
∆θv
(B.15)
∆P =
(sL + R)2 + (ω1 L)2
JPθ (s)

where
L
(k3 − k1 )
ω1
R
a1 =
(k3 − k1 ) − Lk2
ω1
a2 = ω1 Lk3 − Rk4 .
a0 =

(B.16)

The k parameters used in (B.16) are defined as
k1 = V0 Uf 0 cos (θv0 − θu0 )
k2 = V0 Uf 0 sin (θv0 − θu0 )
k3 = E0 V0 cos θv0

k4 = E0 V0 sin θv0 .

(B.17)

B.1.2 Transfer function JQθ (s)
According to the definition in (A.12), the instantaneous reactive power Q from the filter
bus to the ac system is
Q = Im {uf i∗ } .
(B.18)
Linearizing (B.18) yields the expression of the reactive-power deviation. Thus,
∆Q = Im {i∗ ∆uf + uf 0 ∆i∗ }
0

(B.19)

or in component form
∆Q =

id0
−iq0

T

∆uf q
∆uf d

+

uf d0
uf q0

T

−∆iq
∆id

.

(B.20)

205

Chapter B. Jacobian Transfer Matrix
By substituting (B.6), (B.11), (B.12), and (B.14) into (B.20), the dynamic relation between ∆Q and ∆θv is found to be
∆Q =

a0 s2 + a1 s + a2
∆θv
(sL + R)2 + (ω1 L)2

(B.21)

JQθ (s)

where
L
(k4 − k2 )
ω1
R
a1 =
(k4 − k2 ) − Lk1
ω1
a2 = ω1 L (k4 − 2k2 ) − R (k3 − 2k1 ) .

a0 =

(B.22)

B.1.3 Transfer function JUf θ (s)
The voltage magnitude at the filter bus can be expressed as
Uf =

u2 d + u2 q
f
f

(B.23)

which can be linearized by using
Uf = Uf 0 + ∆Uf , uf d = uf d0 + ∆uf d , uf q = uf q0 + ∆uf q .

(B.24)

Substituting (B.24) into (B.23) and only keeping the deviation parts yields the linearized
form of (B.23) as
uf d0
uf q0
∆uf d +
∆uf q .
(B.25)
∆Uf =
Uf 0
Uf 0
By substituting (B.6) into (B.14), and further substituting (B.14) and (B.12) into (B.25),
the dynamic relation between ∆Uf and ∆θv is found to be
a0 s2 + a1 s + a2
∆Uf =
∆θv
(sL + R)2 + (ω1 L)2

(B.26)

JUf θ (s)

where
a0 = −LLn k2 /Uf 0

a1 = −RLn k2 /Uf 0

2
a2 = −ω1 LLn k2 /Uf 0 − RLn k1 /Uf 0 .

(B.27)

If the resistance R in (B.27) is neglected, JUf θ (s) can be simplified as
JUf θ (s) = −
206

Ln
V0 sin (θv0 − θu0 ) .
L

(B.28)

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

B.1.4 Transfer function JPV (s)
To derive the transfer functions related to the voltage-magnitude deviation ∆V at the
VSC, let θv be kept constant, i.e., θv = θv0 . Eq. (B.2) can be linearized using
V = V0 + ∆V, id = id0 + ∆id , iq = iq0 + ∆iq .

(B.29)

Substituting (B.29) into (B.2), and keeping only the deviation parts yields the linearized
form of (B.2) as
d∆id
= ∆V cos θv0 − R∆id + ω1 L∆iq
dt
d∆iq
L
= ∆V sin θv0 − R∆iq − ω1 L∆id .
dt

L

(B.30)

Moreover, by applying the Laplace transform to (B.30), the following transfer functions
are obtained for ∆id and ∆iq vs. ∆V
ω1 L sin θv0 + (sL + R) cos θv0
∆V
(sL + R)2 + (ω1 L)2
−ω1 L cos θv0 + (sL + R) sin θv0
∆V.
∆iq =
(sL + R)2 + (ω1 L)2

∆id =

(B.31)

By substituting (B.11), (B.12), (B.14) and (B.31) into (B.9), the dynamic relation between
∆P and ∆V /V0 is found to be
∆P =

a0 s2 + a1 s + a2
∆V
2
2 ·
(sL + R) + (ω1 L) V0

(B.32)

JPV (s)

where
L
(k4 − k2 )
ω1
R
a1 =
(k4 − k2 ) + Lk1
ω1
a2 = ω1 Lk4 + Rk3 .
a0 =

(B.33)

B.1.5 Transfer function JQV (s)
By substituting (B.11), (B.12), (B.14) and (B.31) into (B.20), the dynamic relation between ∆Q and ∆V /V0 is found to be
∆Q =

a0 s2 + a1 s + a2
∆V
2
2 ·
(sL + R) + (ω1 L) V0

(B.34)

JQV (s)

207

Chapter B. Jacobian Transfer Matrix
where
L
(k1 − k3 )
ω1
R
a1 =
(k1 − k3 ) − Lk2
ω1
a2 = ω1 L (2k1 − k3 ) + R (k4 − 2k2 ) .
a0 =

(B.35)

B.1.6 Transfer function JUf V (s)
By substituting (B.31) into (B.14), and further substituting (B.14) and (B.12) into (B.25),
the dynamic relation between ∆Uf and ∆V /V0 is found to be
∆Uf =

a0 s2 + a1 s + a2
∆V
2
2 ·
(sL + R) + (ω1 L) V0

(B.36)

JUf V (s)

where
a0 = LLn k1 /Uf 0
a1 = RLn k1 /Uf 0
2
a2 = ω1 LLn k1 /Uf 0 − RLn k2 /Uf 0 .

(B.37)

If the resistance R in (B.37) is neglected, JUf V (s) can be simplified as
JUf V (s) =

Ln
V0 cos (θv0 − θu0 ) .
L

(B.38)

B.2 Proof of the instability of the dc Jacobian transfer
matrix
In the following, a mathematical proof of the instability of the dc Jacobian transfer matrix
Gdc (s) for operating points that Pdc10 = 0 and Pdc20 = 0 is given. Gdc (s) is a linear
model of the dc π-link in Fig. 4.25(b).
The poles of Gdc (s) are the eigenvalues of the A matrix in (4.78), which has the
characteristic equation
det(λI − A) = λ3 + a1 λ2 + a2 λ2 + a3 = 0
208

(B.39)

B.2. Proof of the instability of the dc Jacobian transfer matrix
with the coefficients
Rdc
Ldc
Pdc10 Pdc20
Pdc10 Rdc
Pdc20 Rdc
1
1
a2 = 2
+ 2
+ 2
+
+
udc10 u2 Cdc1 Cdc2 udc10 Ldc Cdc1 udc20 Ldc Cdc2 Ldc Cdc1 Ldc Cdc2
dc20
Pdc10 u2 + Pdc20 u2 + Pdc10 Pdc20 Rdc
dc20
dc10
a3 =
.
(B.40)
2
Udc10 u2 Ldc Cdc1 Cdc2
dc20
a1 =

Pdc10
2
udc10 Cdc1

+

Pdc20
2
udc20 Cdc2

+

To prove the instability of Gdc (s), it is sufficient that any one of the three coefficients in
(B.40) is negative. The following is a proof of
a3 < 0.

(B.41)

It is obvious that the denominator of a3 in (B.40) is positive. Thus, (B.41) is identical to
Pdc10 u2 + Pdc20 u2 + Pdc10 Pdc20 Rdc < 0.
dc20
dc10

(B.42)

From the main circuit of the π-link model in Fig. 4.25(b), the following equality is established
Pdc10
Pdc20
= idc0 = −
.
(B.43)
udc10
udc20
For VSC-HVDC applications, it is apparent that udc10 and udc20 have the same polarity.
Consequently, it follows from (B.43) that
Pdc10
<0
Pdc20

(B.44)

Pdc10 Pdc20 < 0.

(B.45)

or
Based on the inequality in (B.45), by dividing Pd10 Pd20 at both sides, (B.42) can be rewritten as
u2
u2
dc20
+ dc10 + Rdc > 0.
(B.46)
Pdc20 Pdc10
Substituting
Pdc10 = udc10 idc0
Pdc20 = −udc20 idc0

udc20 = udc10 − Rdc idc0

(B.47)

2Rdc > 0

(B.48)

into (B.46), yields
which apparently holds. In other words, for other operating points when Pdc10 = 0 and
Pdc20 = 0, Gdc (s) is unstable as long as the dc-transmission line is not lossless.
209

Chapter B. Jacobian Transfer Matrix

210

Appendix C
Technical Data of the Test System
This appendix gives the technical data of the VSC-HVDC system, the synchronous generator, and the induction motor that are used in the thesis. A 50 Hz ac system is assumed.

C.1 The VSC-HVDC link

Table C.1 Technical data of the VSC-HVDC system. AC: per unit based on 350 MVA and 195 kV.
DC: per unit based on 350 MW and 150 kV.

Rated power PdcN

350 MW

Rated ac voltage (Line-to-line) UacN

195 kV

Nominal ac system frequency f1

50 Hz

Phase-reactor reactance ω1 Lc

0.2 p.u.

Phase-reactor resistance Rc

0.01 p.u.

Converter topology

two-level

PWM switching frequency fsw

1650 Hz

Maximum valve current Imax

1.08 p.u.

Rated direct voltage UdcN

±150 kV

DC capacitance Cdc

0.015 p.u.

Converter transformer rating

380 MVA

Converter transformer ratio

195 kV/400 kV

Transformer leakage inductance Ltv

12%
211

Chapter C. Technical Data of the Test System

C.2 The synchronous generator

Table C.2 Technical data of the synchronous generator. Per unit based on 150 MVA and 20 kV.

Rated power MVAg

150 MVA

Rated terminal voltage (Line-to-line) UacN

20 kV

Nominal ac system frequency f1

50 Hz

Stator resistance Ra

0.0045 p.u.

Stator leakage inductance Ll

0.13 p.u.

Synchronous inductance Ld , Lq

1.79 p.u., 1.71 p.u.





Transient inductance Ld , Lq

0.169 p.u., 0.228 p.u.

′′

′′

Subtransient inductance Ld , Lq


0.135 p.u., 0.2 p.u.



Transient time constant Td0 , Tq0
′′

4.3 sec, 0.85 sec
′′

Subtransient time constant Td0 , Tq0

0.032 sec, 0.05 sec

Inertia constant H

6.175 sec

Damping coefficient KD

0.005

Step-up transformer rating

150 MVA

Step-up transformer ratio

20 kV/400 kV

Transformer leakage inductance Ltg

14%

The parameters of the equivalent circuit can be calculated by the technical data given in
Table C.2:
1. The mutual inductances Lad and Laq are
Lad = Ld − Ll = 1.66 p.u.
Laq = Lq − Ll = 1.58 p.u.

(C.1)

2. The leakage inductance of the field circuit Lf d is


Lf d

212

Lad (Ld − Ll )
=
= 0.04 p.u.

Lad − (Ld − Ll )

(C.2)

C.3. The induction motor
3. The leakage inductances of the damping circuits L1d , L1q and L2q are
′′

L1d
L1q

Lad Lf d (Ld − Ll )
=
= 0.0057 p.u.
′′
Lad Lf d − (Lad + Lf d )(Ld − Ll )

Laq (Lq − Ll )
=
= 0.1045 p.u.
Laq − (L′q − Ll )
′′

L2q

Laq L1q (Lq − Ll )
=
= 0.245 p.u.
Laq L1q − (Laq + L1q )(L′′ − Ll )
q

(C.3)

4. The resistance of the field circuit Rf d is
Rf d =

Lad + Lf d
= 0.0013 p.u.

Td0 ω1

(C.4)

5. The resistances of the damping circuits R1d , R1q and R2q are
Lad Lf d
1
L1d +
Lad + Lf d
Td0 ω1
Laq + L1q
=
= 0.0063 p.u.

Tq0 ω1
1
Laq L1q
= ′′
L2q +
Laq + L1q
Tq0 ω1

R1d =
R1q
R2q

′′

= 0.0044 p.u.

= 0.0218 p.u.

(C.5)

C.3 The induction motor
Table C.3 Technical data of the induction motor. Per unit based on 30 MVA and 12 kV.

Rated power MVAm

30 MVA

Rated terminal voltage (line-to-line) UacN

12 kV

Stator resistance Rs

0.0034 p.u.

Stator leakage inductance Lsλ

0.09 p.u.

Rotor resistance Rr

0.007 p.u.

Rotor leakage inductance Lrλ

0.08 p.u.

Magnetizing inductance Lm

1.9 p.u.

Inertia constant J = 2H

3.7267 sec

Damping coefficient KD

0.005

Step-up transformer rating

30 MVA

Step-up transformer ratio

12 kV/400 kV

Transformer leakage inductance Ltm

8%
213

Chapter C. Technical Data of the Test System

214

Appendix D
List of Symbols and Abbreviations
Symbols
A

system matrix

AS

desired attenuation for the sensitivity function

AT

desired attenuation for the complementary sensitivity function

a

coefficient

B

input matrix

C

output matrix

Cdc

capacitance of the single dc capacitor dc-link representation

Cdc1 , Cdc2

dc capacitances of the π-link representation

Cf

capacitance of the ac capacitor connected to the filter-bus of the
VSC-HVDC system

Cn

capacitance of a series capacitor

D

feedthrough matrix

De

damping torque

e

control error signal

E

voltage vector of a constant-frequency stiff voltage source

ef d

field voltage of the synchronous generator

eg

terminal voltage vector of the synchronous generator

em

terminal voltage vector of the induction motor

ev

terminal voltage vector of the VSC-HVDC converter

f1

nominal frequency of the ac system

fsw

PWM switching frequency

FPLL

PLL controller
215

Chapter D. List of Symbols and Abbreviations
Fn

notch filter for the direct-voltage controller

F

low-pass filter transfer matrix

H

inertia constant of the synchronous generator or the induction motor

Hac

inertia of the ac system

Hdc

relative rotational inertia of a dc link

HHP

high-pass filter

HLP

low-pass filter

I

identity matrix

i

current vector

−1

j
J, J

Jacobian transfer matrix and transfer function

J

process model of J

Jp

non-invertible part of J

Jn

invertible part of J

K, K

controller transfer matrix and transfer function

kc

load compensation ratio of the VSC

Ke

synchronizing torque

ku

gain of alternating-voltage control of the VSC

kv

gain of high-pass current control of the VSC

kg

load compensation ratio of the synchronous generator

kp

gain of the power-synchronization loop of the VSC

Kr

prefilter of direct-voltage control

L

inductance

Lc

phase-reactor inductance of the VSC

Ldc

dc inductance of the π-link representation

P

active power

PacN

ac power base value

PdcN

dc power base value

p

pole

Q

reactive power

R

resistance

Rdc

dc resistance of the π-link representation

r

reference signal

216

S

sensitivity function

s

Laplace operator

Td

time delay

Tdc

time constant of the dc capacitor

Tp

period time

Te

electrical torque

Tm

mechanical torque

T

complementary sensitivity function

u

input vector

UacN

alternating-voltage base value

UdcN

direct-voltage base value

uf

filter-bus voltage vector

v

VSC voltage vector

W1

sensitivity weighting function

W3

complementary sensitivity weighting function

X

reactance

y

output vector

z

zero

ZRHP

RHP zero

α

bandwidth

δ

load angle of the synchronous generator

γ

parameter of the anti-windup scheme

L

Laplace transform

µ

structured singular value

ω1

fundamental angular frequency

ωr

rotor angular frequency

ωn

resonance frequency of Fn

ωS

crossover frequency of W1

ωT

crossover frequency of W3

φ

power factor angle

ψ

flux

ρc

a state variable to represent HHP (s) in state-space form

σ

singular value

217

Chapter D. List of Symbols and Abbreviations
θ

phase angle

θu

load angle of the VSC at the filter bus

θv

load angle of the VSC at the VSC bus

τ

time constant of a low-pass filter

ξ

parameter of Fn

Subscripts
a,b,c

phase quantities

ac

alternating current quantity

c

VSC quantity

d

d component in the dq frame

dc

direct current quantity

f

filter-bus values of the VSC

g

synchronous generator quantity

I

imaginary component in the R − I frame

i

integral control parameter

l

load quantity

m

induction motor quantity

max

maximum value

min

minimum value

n

ac-system quantity

N

base value

nom

nominal value

p

proportional control parameter

pcc

PCC quantity

PLL

PLL quantity

R

real component in the R − I frame

ref

reference value

s

stator quantity

t

transformer quantity

q

q component in the dq frame

v

VSC quantity

0

operating point value

218



infinity norm

Superscripts
c

quantity in the converter dq frame

g

synchronous generator quantity

m

induction motor quantity

P

control parameter related to active-power control

PLL

control parameter related to PLL

Q

control parameter related to reactive-power control

ref

reference value

s

quantity in the synchronous αβ-frame

U

control parameter related to alternating-voltage control

Abbreviations
ac

alternating current

APC

active-power control

AVC

alternating-voltage control

CCC

capacitor-commutated converter

conj

conjugate

CSC

current-source converter

dc

direct current

det

determinant

emf

electromotive force

FACTS

flexible ac transmission systems

GTO

gate turn off

HP

high pass

HVDC

high-voltage direct-current

IGBT

insulated-gate bipolar transistor

Im

imaginary

IM

induction motor

IMC

internal model control

LCC

line-commutated converter

LHP

left-half plane

219

Chapter D. List of Symbols and Abbreviations
LP

low pass

max

maximum

min

minimum

MIMO

multi-input multi-output

MPC

maximum power curve

M2C

modular-multilevel converter

nom

nominal

NPC

neutral-point clamped

OMIB

one-machine infinite-bus

PCC

point-of-common-coupling

PI

proportional-integral

PLL

phase-locked loop

PSL

power-synchronization loop

PSS

power-system stabilizer

p.u.

per unit

PWM

pulse-width modulation

Re

real

RHP

right-half plane

RLC

resistive-inductive-capacitive

RPC

reactive-power control

SC

synchronous condenser

SCR

short-circuit ratio

SG

synchronous generator

SISO

single-input single-output

SM

synchronous machine

SSDC

subsynchronous damping controller

SSR

subsynchronous resonance

SSTI

subsynchronous torsional interaction

STATCOM

static synchronous compensator

TOV

transient over voltage

UIF

unit interaction factor

VSC

voltage-source converter

220

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