modeling and control of vsc-hvdc transmissions409908/fulltext02.pdf · 2011. 4. 12. · 3 modeling...

Modeling and Control of

VSC-HVDC Transmissions

HÉCTOR F. LATORRE S.

Doctoral ThesisRoyal Institute of TechnologySchool of Electrical Engineering

Electric Power SystemsStockholm, Sweden, 2011

TRITA-EE 2011:024ISSN 1653-5146ISBN 978-91-7415-924-0

School of Electrical EngineeringElectric Power Systems

Royal Institute of TechnologySE-100 44 Stockholm

Sweden

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan fram-lägges till o�entlig granskning för avläggande av teknologie doktorsexamenonsdagen den 04 maj 2011 kl 09:00 i H1, Teknikringen 33, Kungl TekniskaHögskolan, Stockholm.

c⃝ Héctor F. Latorre S., April 2011

Tryck: Universitetsservice US-AB

Abstract

Presently power systems are being operated under high stress level conditionsunforeseen at the moment they were designed. These operating conditionshave negatively impacted reliability, controllability and security margins.

FACTS devices and HVDC transmissions have emerged as solutions tohelp power systems to increase the stability margins. VSC-HVDC trans-missions are of particular interest since the principal characteristic of thistype of transmission is its ability to independently control active power andreactive power.

This thesis presents various control strategies to improve damping of elec-tromechanical oscillations, and also enhance transient and voltage stabilityby using VSC-HVDC transmissions. These control strategies are based ondi�erent theory frames, namely, modal analysis, nonlinear control (Lyapunovtheory) and model predictive control. In the derivation of the control strate-gies two models of VSC-HVDC transmissions were also derived. They areInjection Model and Simple Model. Simulations done in the HVDC LightOpen Model showed the validity of the derived models of VSC-HVDC trans-missions and the e�ectiveness of the control strategies.

Furthermore the thesis presents an analysis of local and remote infor-mation used as inputs signals in the control strategies. It also describes anapproach to relate modal analysis and the SIME method. This approach al-lowed the application of SIME method with a reduced number of generators,which were selected based on modal analysis.

As a general conclusion it was shown that VSC-HVDC transmissionswith an appropriate input signal and control strategy was an e�ective meansto improve the system stability.

Keywords: Control Lyapunov Functions, Energy Function, Modal Analy-sis, Model Predictive Control, Power Oscillation Damping, SIME, Transient

iii

iv

Stability, Voltage Stability, VSC-HVDC Transmission.

Acknowledgment

First I want to thank Professor Lennart Söder for the invitation to apply tothe open position for this PhD program and for accepting me.

I deeply thank Mehrdad Ghandhari for his excellent supervision, guid-ance and endless patience during the course of the research.

I also want to thank all the members of my reference group for veryinteresting meetings, helpful discussions and meaningful comments duringthe research. The �nancial support of the project from the ELEKTRAprogram is gratefully acknowledged.

I extend my personal thanks to the sta� of Electric Power Systems fora friendly environment to study and perform research. Special thanks toBrigitt Högberg for all her valuable help and assistance.

I want to thank my friends Gloria, Juan Carlos and Rigoberto for theircare and support

Finally, many thanks to my family for their endless love and support.

v

List of Acronyms

AVR Automatic Voltage Regulator

COI Center Of Inertia

CLF Control Lyapunov Function

CSC Current Source Converter

DFIG Doubly Fed Induction Generators

FACTS Flexible Alternate Current Transmission System

HVDC High Voltage Direct Current

IGBT Insulated Gate Bipolar Transistor

LCC Line Commutated Current

MPC Model Predictive Control

PMU Phasor Measurement Units

POD Power Oscillation Damping

PSS Power System Stabilizer

PWM Pulse Width Modulation

RHC Receding Horizon Control

SIME Single Machine Equivalent

SIMPOW Simulation Power Systems

vii

viii

SMIB Single Machine In�nite Bus

SPM Structure Preserving Model

STATCOM Static Compensator

SVC Static Var Compensator

TCSC Thyristor Controlled Series Capacitor

UPFC Uni�ed Power Flow Controller

VSC Voltage Source Converter

WSCC Western Systems Coordinating Council

Contents

Contents ix

1 Introduction 1

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

1.2 Changes in Power Systems . . . . . . . . . . . . . . . . . . . . 2

1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Objective and scope . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Contributions of This Thesis . . . . . . . . . . . . . . . . . . 6

1.6 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Background and Challenges 11

2.1 Power System Modeling . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Synchronous Machines . . . . . . . . . . . . . . . . . . 12

2.1.2 Static Load . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.3 Multi-Machine Power System . . . . . . . . . . . . . . 16

2.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 N−1 Criterion . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Power System Stability . . . . . . . . . . . . . . . . . 19

2.3 FACTS and HVDC Transmission . . . . . . . . . . . . . . . . 21

2.4 VSC-HVDC Transmission in Power Systems . . . . . . . . . . 22

3 Modeling of VSC-HVDC Transmissions 25

3.1 Injection Model . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 HVDC Light Open Model . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Limits Capability Converters . . . . . . . . . . . . . . 29

ix

x CONTENTS

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Control of VSC-HVDC Transmissions 33

4.1 Modal Analysis-based Control Strategy . . . . . . . . . . . . . 334.2 CLF-based Control Strategies . . . . . . . . . . . . . . . . . . 37

4.2.1 Local Information . . . . . . . . . . . . . . . . . . . . 374.2.2 SMIB and Remote Information . . . . . . . . . . . . . 39

4.3 Local and Remote Signals Analysis . . . . . . . . . . . . . . . 444.3.1 Local Signals . . . . . . . . . . . . . . . . . . . . . . . 444.3.2 Remote Signals . . . . . . . . . . . . . . . . . . . . . . 454.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Approach SIME - Modal Analysis . . . . . . . . . . . . . . . . 464.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 Voltage or Reactive Control Mode . . . . . . . . . . . . . . . 484.5.1 Modes of Operation . . . . . . . . . . . . . . . . . . . 484.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Model Predictive Control (MPC)-based Control Strategy 51

5.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . 515.1.1 Model Predictive Control . . . . . . . . . . . . . . . . 51

5.2 Application of MPC to VSC-HVDC Transmissions . . . . . . 535.2.1 General MPC-based design approach . . . . . . . . . 535.2.2 VSC-HVDC transmission model . . . . . . . . . . . . 535.2.3 Application of the MPC-based approach to POD . . . 545.2.4 Extremely Randomized Trees . . . . . . . . . . . . . . 55

5.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 565.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 Conclusions and Future Work 67

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 71

Chapter 1

Introduction

1.1 Historical Background

It can be said that it was in 1882 when the era of transmission of electricenergy began and it was precisely based on dc transmission. The Pearl

Street Station provided New York's �nancial district with electrical energyproduced by the dynamos that Thomas Alva Edison's laboratory had de-veloped. In November - December 1887, Nikola Tesla introduced a systemfor alternating current generators, transformers, motors, wires and lights.The advantages of transmitting ac electrical energy over long distances com-pared to dc transmission were made clear in this very early beginings. Thedominant position of dc transmission was brief and some decades later actransmission overtook the lead [1].

During the decade of 1930s dc transmission began to be considered againas an option of transmitting electrical energy, but this time at high voltagelevels. Important research e�orts on High Voltage Direct Current (HVDC)converter technology were undertaken in this decade. In the decade of 1950sHVDC technology was boosted probably because the technology seemed tobe commercially feasible. In fact in 1950 the �rst commercial order foran HVDC system was given to ASEA by Vattenfall for a 20 MW, 100 kmundersea cable between the Swedish mainland and the island of Gotland.The project was commissioned in March 1954. The converters in this projectwere based in mercury arc-valves [2].

In 1970 HVDC transmission had a revolutionary change when thyristorswere used in a converter for the �rst time. Once again it was in the Swedish

1

2 CHAPTER 1. INTRODUCTION

mainland - Gotland project where this technology was used. ASEA addeda thyristor-based converter group to the dc link. Since then, thyristors-based converters have been the technology used for dc transmission and itis known as Current Source Converter (CSC-HVDC) or also known as LineCommutated Current (LCC-HVDC) or commonly referred to as ClassicalHVDC [2].

A new relevant step in dc transmission happened in the late 1990s whenthe Insulated Gate Bipolar Transistor (IGBT) was introduced within themain building block of the valves [3]. This new component allowed thedevelopment of Voltage Source Converters (VSC) at high voltage levels. The�rst commercial VSC-HVDC transmission project was carried out by ABB.The project was a 50 MW underground cable, interconnecting the southernpart of the island of Gotland with the northern end [4].

Both types of dc transmission CSC-HVDC and VSC-HVDC are maturetechnologies. However, CSC-HVDC transmission is more appropriate forlarge power transfers. CSC-HVDC has reached dc voltage levels of 800 kVand transmission capacities of 6400 MW.

VSCs have had important technology developments in the last 15 years.From the ±80 kV and 50 MW rated values of the system in Gotland, VSC-HVDC transmission has now reached voltage levels of up to ±320 kV andtransmission capacity up to 1200 MW. It is di�cult to compare it with CSC-HVDC transmission considering that the gap-time between both technologiesis roughly 40 years.

Nowadays ac transmission is the dominant type of transmission and itwill be for very long time, but dc transmission has taken again an importantrole in power systems. It might be that someday in the future HVDC takesback the dominant position in the transmission of energy.

1.2 Changes in Power Systems

Electricity markets and environmental rules are two main factors that haveled the implementation of major changes in the way transmission systems aredesigned, constructed and operated. For instance, electricity markets havemade utilities to perform an optimization of the assets in order to remaincompetitive and survive. On the other hand strong environmental rules havemade it more di�cult to obtain licenses for the construction of new electricalprojects, specially overhead transmission lines.

1.2. CHANGES IN POWER SYSTEMS 3

The demand for electric energy is in permanent growth. Although thegrowing trend slows down in periods of worldwide economic crisis, as theone experienced in 2008, the curve of consumption always keeps its positivegradient. Speci�cally for the crisis in 2008, it was expected that most of thenations would begin to return to the rate of growth they had before the crisiswithin the next 12 - 24 months [5].

These driving forces (electricity markets, environmental rules and growthof electrical energy demand) have driven the power systems to operate underthe most stressing conditions ever observed because:

• Patterns of scheduled generation do not exist anymore

• Interconnections of power systems that were initially constructed toimprove reliability levels through scheduled interchange or back upconnections, are now used for trading (even though they were not de-signed for this purpose).

• The introduction of electricity markets have put in stand by or hasdelayed the investments in new projects of expansion [6].

• The power system components are now being operated more closely totheir thermal ratings [7].

Power systems are becoming more stressed by the presence of low fre-quency electromechanical oscillations because the number of interconnectionswith other power systems increases every day. Furthermore, long distancepower trading puts more stress on the existing transmission system. As aresult, low frequency oscillations involving weakly damped interarea modesbecome more pronounced risking system security [8] and lowering transmis-sion capacity. Over the years, many incidents of system outage resultingfrom these oscillations have been reported, being the August 10, 1996 black-out in the WSCC system of United States [9] one of the most well knownexamples.

Transmission lines have become more heavily loaded than ever before,giving rise to a risk of voltage instability [10]. Large disturbances in heavilystressed power systems result in voltage drops and often to voltage insta-bility and collapse. Some examples are the blackout USA and Canada inAugust 14, 2003 [11]; the blackout in Sweden and Denmark in September23, 2003 [12]; and the blackout in south Greece in July 12, 2004 [13].


The conclusion that can be drawn from these blackouts is that powersystems have de�nitely become more stressed. In addition, reliability, con-trollability and security margins have been negatively a�ected due to thesestringent operating conditions. Power systems are more vulnerable and op-erate with more risk of not ful�lling the N − 1 criterion, as the case of theblackout in Europe during November 4th, 2006 [14]. The root of this blackoutwas the disconnection of a transmission line under heavy loading conditions.That disconnection led to a cascading e�ect, more transmission lines weretripped and �nally causing the blackout.

On the other hand, FACTS devices and HVDC transmissions haveemerged as important solutions to help power systems to increase stabilitymargins [15, 16]. Some of these power electronics-based components have themain function of controlling reactive power (SVC and STATCOM) and someothers to control active power (as TCSC and CSC-HVDC transmission). Allthese devices are also capable of damping electromechanical oscillations. Asubstantial amount of research has been devoted to this topic.

VSC-HVDC transmission is a type of power electronics-based systemrelatively young compared to the devices mentioned above. VSCs have thebig advantage that they have two degrees of freedom, i.e., they are able tocontrol both active and reactive power, independent of each other. Thisfeature gives a high potential for remarkably enhancing transient stability,providing voltage support, increasing damping and controlling the power�ow in the system in a larger scale compared to other components.

The development of IGBT with higher rated capabilities has allowedVSC-HVDC transmission to be included as a very feasible solution amongtransmission system owners for grid expansion. The south west link projectin Sweden is one example [17].

1.3 Motivation

Transmission systems are being operated closer to their stability limits whichmay lead to reduced damping of electromechanical oscillations and as a con-sequence to decreased system stability margins. For this reason, when powersystems are to be expanded, alternatives that allow the increase of powertransfer and stability margins are the �rst options considered.

VSC-HVDC transmissions with appropriate input signals and controlstrategies can signi�cantly contribute to the e�cient utilization of transmis-

1.4. OBJECTIVE AND SCOPE 5

sion networks (under normal conditions or post-fault conditions) thanks totheir fast response capabilities, and their ability to independently controlactive and reactive output power from their converters. By utilizing VSC-HVDC transmission in a power system, there are great possibilities for:

• Enhancing transient stability.

• Improving damping of low frequency electromechanical power oscilla-tions.

• Enhancing voltage stability.

• Decreasing undesirable loop (or parallel) �ows.

Therefore, VSC-HVDC transmissions in addition to increase the grid's trans-mission capacity, they also contribute in keeping transmission systems (undernormal conditions or post-disturbed conditions) stable.

1.4 Objective and scope

The main aim of this project is to analyze the impact of a VSC-HVDCtransmission on a power system, when it is embedded in the transmissionnetwork.

More speci�c objectives de�ning the scope of this research are:

• To derive control strategies for damping electromechanical oscillationsusing linear control, nonlinear control and model predictive control.

• To analyze VSC-HVDC transmissions' electromechanical oscillationsdamping capabilities by controlling either active power or reactivepower or both simultaneously.

• To use the SIngle Machine Equivalent (SIME) method within controlstrategies and to �nd a possible relation between the information ob-tained from SIME and modal analysis.

• To analyze the use of local and remote information within controlstrategies.


• To analyze possible interactions between the control system of VSC-HVDC transmissions and other controllers of the power systems underthe di�erent modes of operation.

• To formulate a control strategy to improve post-fault conditions bycontrolling the power �ow in the system.

1.5 Contributions of This Thesis

The main contributions are:

1. Modeling

1.1. Injection Model. An Injection Model of a VSC-HVDC trans-mission was derived and evaluated. In several numerical examplesthe results of the Injection Model were compared with results ofa more detailed model (an HVDC Light Open Model providedby ABB). The dynamic behavior of the power system was verysimilar when using either model. This contribution is presentedin Paper II.

1.2. Simple Model. This model is a simpli�cation of the InjectionModel. In the Simple Model either the active or reactive powerare directly modulated by a control strategy. This model is usefulin the derivation or testing of control strategies. This contributionis presented in Paper I.

2. Control

2.1. An energy function for a VSC-HVDC transmission. En-ergy functions for most systems using power electronic devicesare available in the literature. However for the speci�c case ofVSC-HVDC transmission there are no prior references in the lit-erature. In this project an energy function for a power systemwith an embedded VSC-HVDC transmission has been derived.This contribution is presented in Paper I.

2.2. A Control Lyapunov Function (CLF)-based control strat-

egy for a VSC-HVDC transmission using local informa-

tion. A CLF for enhancing transient stability was derived. The

1.5. CONTRIBUTIONS OF THIS THESIS 7

control strategy also showed to be very e�ective in damping elec-tromechanical oscillations. As the case of energy functions, CLF-based control strategies are known for most of power electronics-based systems. However by the time the control strategy wasderived, no prior references in the literature were found. Thecontrol strategies resulted in a function that uses the frequenciesof the buses the converters were connected to. We assume thesefrequencies to be local information. This contribution is presentedin Paper II.

2.3. A CLF-based control strategy for a VSC-HVDC trans-

mission using remote information. A second CLF-based con-trol strategy for improving transient and small signal stability wasderived. The control strategy was derived for a Single MachineIn�nite Bus (SMIB), but it can be applied to any power systemsize. This second control strategy resulted in a function of thespeed and angle of generators. The control strategy simultane-ously modulates both the active and reactive part of the VSCs.This contribution is presented in Paper V.

2.4. An approach to relate information between the SIME

method and modal analysis. The SIME method relies oninformation from all generators in power systems, meanwhile withthe approach proposed in this thesis, only some generators areconsidered. This contribution is presented in Paper IV.

2.5. Power �ow control. VSC-HVDC transmissions connected insynchronous system are very e�ective means to control the power�ow in transmission networks, not only because of the control ofthe power itself, but also because it simultaneously increases thestability margin. This contribution is presented in Paper II andPaper IV.

2.6. Wind park and VSC-HVDC transmission coordination

for Power Oscillation Damping (POD). A coordination be-tween wind park and a distant VSC-HVDC transmission for PODis suggested. The coordination shows how the damping in thesystem improves with the coordination. This contribution is pre-sented in Paper V.

2.7. A Model Predictive Control (MPC)-based control strat-

egy for a VSC-HVDC transmission. This control strategy


helps to improve power system stability. Even though the controlstrategy is computational expensive, it shows to be an interestingtool to analyze the power systems in o�ine mode. This contribu-tion is presented in Chapter V

1.6 Thesis outline

The remaining chapters of this thesis are organized as follows:

Chapter 2 gives a brief description of power system modeling and chal-lenges.

Chapter 3 describes the VSC-HVDC transmission models used in this the-sis.

Chapter 4 summarizes the derived control strategies. It also presents acontrollability and observability analysis of a VSC-HVDC in the Nordic32A power system. Furthermore, it shows a comparison when remotesignals and local signals are used as input of a POD controller. Thechapter ends with a summary of voltage stability analysis, and showsan analysis of the interaction between the VSC-HVDC transmission,operating under either voltage control or reactive power control, andAutomatic Voltage Regulators (AVRs) in the neighborhood.

Chapter 5 describes the application of Model Predictive Control in VSC-HVDC transmission for improving damping.

Chapter 6 presents the conclusions of the project and suggests some ideasfor future work.

1.7 List of Publications

The following papers were published during the course of this project.

• H. Latorre, M. Ghandhari and L. Söder, �Control of a VSC-HVdc Op-erating in Parallel with AC Transmission Lines�, IEEE PES Trans-mission and Distribution, Conference and Exposition, Venezuela,Aug. 2006. Paper I in this thesis.

1.7. LIST OF PUBLICATIONS 9

• H. Latorre, M. Ghandhari and L. Söder, �Application of Control Lya-punov Functions to Voltage Source Converters-based High Voltage Di-rect Current for Improving Transient Stability�, Power Tech Conf.,Switzerland, Jul. 2007.

• H. Latorre, M. Ghandhari and L. Söder, �Multichoice Control Strategyfor VSC-HVdc�, 2007 iREP Symposium, USA, Aug. 2007.

• H. Latorre, M. Ghandhari and L. Söder, �Active and Reactive PowerControl of a VSC-HVdc�, Electric Power Systems Research, vol. 78,no. 10, pp. 1756-1763, Oct. 2008. Paper II in this thesis.

• H. Latorre, M. Ghandhari and L. Söder, �Use of Local and RemoteInformation in POD Control of a VSC-HVdc�, Power Tech Conf., Ro-mania, Jul. 2009. Paper III in this thesis.

• H. Latorre and M. Ghandhari, �Improvement of Voltage Stability byUsing VSC-HVdc�, IEEE Transmission and Distribution, Conferenceand Exposition, South Korea, Oct. 2009

• H. Latorre and M. Ghandhari, �Improvement of Power System Stabilityby Using VSC-HVdc�, Electrical Power & Energy Systems, vol. 33,no.2, pp. 332-339, Feb. 2011. Paper IV in this thesis.

• K. Elkington, H. Latorre and M. Ghandhari, �Operation of DoublyFed Induction Generators in Power Systems with VSC-HVDC Trans-mission�, in International Conference on AC and DC Power Transmis-sion, England, Oct. 2010. Paper V in this thesis.

Chapter 2

Background and Challenges

This chapter gives a brief description of how the main parts of power systemsare modeled. It also describes the main challenges that power systems arefacing.

2.1 Power System Modeling

The basic structure of an electrical power system can be divided into threemain parts: Generation, Transmission and Distribution. The di�erent partsof the power system operate at di�erent voltage levels. Typically generationand distribution are classi�ed within the medium voltage range (between1 kV and 100 kV). Transmission is classi�ed in high voltage (between 100 kVand 300 kV) or extra-high voltage (between 300 kV and above).

Generation is produced by converting mechanical energy appearing onthe shaft of turbines into electrical energy. This conversion is almost univer-sally done by the use of synchronous generators. The synchronous generatorsfeed their electrical power into the transmission system via a step-up trans-former.

A signi�cant advantage of electrical energy is that it can be generatednear to a primary energy resource and then be transmitted over long dis-tances to load centers. The energy lost in a transmission system is propor-tional to the square of the current. For this reason transmission lines operateat high or extra-high voltage. Transmission networks connect power stationsinto one system, transmit and distribute power to load centers.

11

12 CHAPTER 2. BACKGROUND AND CHALLENGES

Demand for electrical power is never constant and changes continuouslythroughout the day. The changes in demand of individual consumers maybe fast and frequent. However, the higher the voltage level from wherethe consumers are connected, the smaller and smoother the changes are.Consequently, the total power demand at the transmission level changes ina more or less predictable way.

2.1.1 Synchronous Machines

A synchronous generator consists basically of a stator, on which the three-phase armature winding is normally wound, and a rotor (or �eld), on whichthe dc �eld winding is wound. The rotor can also be equipped with addi-tional short-circuited damper windings to reduce mechanical oscillations inthe rotor. The stator is represented by three magnetic axes, a, b and c,each corresponding to one of the phase windings. On the other hand, therotor is represented by two axis, namely the direct axis (d -axis), which is themagnetic axis of the �eld winding, and the quadrature axis (q-axis), whichis the axis of symmetry between two poles.

In the analysis of power system dynamics, synchronous machines aretypically represented by three models, namely: two-axis model, �ux-decaymodel (or one-axis model) and classical model.

For transient stability analysis, synchronous machines are normally sim-pli�ed by making the following assumptions [18]:

1. The stator transients are neglected.

2. All the stator resistances are neglected.

3. During the transient state, the rotor speed is near synchronous speed.

4. The rotor transient saliency is neglected, i.e., x′q = x′d (in the two-axismodel) and x′q = xq = x′d (in the one-axis model)

Let us consider a generator connected to a power system as shown inFigure 2.1. In the two-axis model it is assumed that the q-axis is equippedwith a short-circuited damper winding. With the assumptions mentionedabove, the dynamics of generator k using this model are represented by the

2.1. POWER SYSTEM MODELING 13

Uk k

Power

System

Genk

Figure 2.1: Generator k connected to a power system

following di�erential equations:

δk = ωk (2.1)

ωk =1

M(Pmk

− Pek −Dkωk) (2.2)

E′qk

=1

T ′dok

(Efk −

xdkx′dk

E′qk

+xdk − x′dk

x′dkUk cos(δk − θk)

)(2.3)

E′dk

=1

T ′qok

(−xqkx′qk

E′dk

−xqk − x′qk

x′qkUk sin(δk − θk)

)(2.4)

and Pek is given by

Pek =E′

qkUk sin(δk − θk) + E′

dkUk cos(δk − θk)

x′dk(2.5)

where,

ωk is the di�erence between the rotor speed and the synchronous speed, i.e.,ωgk − ωs

δk is the rotor angle, δk = ωkt+ δ0k

E′qk

is the q-axis transient emf

E′dk

is the d -axis transient emf

Pmkis the mechanical power

Pek is the electrical power

Dk is the damping constant


Mk is the inertia constant

Efk is an emf proportional to the �eld voltage

xdk is the d -axis synchronous reactance

x′dk is the d -axis transient reactance

xqk is the q-axis synchronous reactance

x′qk is the q-axis transient reactance

T ′dok

is the d -axis transient open-circuit time constant

T ′qok

is the q-axis transient open-circuit time constant

Uk is the magnitude of the voltage of the generator terminal bus where thesynchronous generator is connected

θk is the phase angle of Uk

The one-axis model does not consider dynamics due to the damper-winding in the q-axis (di�erential equation (2.4)). In this way, the dynamicsof the Flux-Decay model are described by (2.1) to (2.3), where Pek is givenby:

Pek =E′

qkUk sin(δk − θk)

x′dk(2.6)

since x′dk = xqk = x′qk (assumption 4).

The classical model is the simplest of all the synchronous machine models.It is also called the constant voltage behind the transient reactance x′d model.In this model the �ux produced by the �eld winding and the current through�eld winding are assumed to be constant. This assumption results in asecond-order system described by (2.1) and (2.2), where Pek is given by (2.6)and E′

qkis constant.

2.1.2 Static Load

A typical power system may consist of several hundreds buses at the trans-mission level, but there could be a hundred thousand buses at the distri-bution levels. For this reason, when power systems are analyzed only the


transmission level is considered, and distribution networks are replaced byequivalent loads, referred to as composite loads.

The demand of the composite load normally depends on the bus voltageand the frequency. The functions describing the dependency of the activeand reactive load demand on the voltage and frequency P (U, f) and Q(U, f)are known as the static load characteristics. Characteristics P (U) and Q(U)taken at constant frequency are voltage characteristics while characteristicsP (f) and Q(f) taken at constant voltage are frequency characteristics. Char-acteristics of composite loads depend on characteristics of their individualcomponents.

The determination of a simple and valid composite load models is not aneasy task, and it is still subject of intensive research. Composite loads maybe represented by static or dynamic models or a combination of both. Thefollowing models are the most popular ones used in power systems analy-sis [19].

Constant Power, Current or Impedance

The simplest load models assume one of the following conditions:

• a constant power demand (P )

• a constant current demand (I)

• a constant impedance (Z)

A constant power model is voltage invariant and allows loads with a sti�voltage characteristics. This model is often used in load �ow calculations,but it is generally unsatisfactory for transient stability analysis, in the pres-ence of large voltage variations. In the constant current model the loaddemand changes linearly with the voltage and is a good approximation forthe real power demand of a mix of resistive and motor devices. The con-stant impedance model assumes that the load power changes proportionallyto the squared of the voltage and represents some lighting loads well, but itis not suited for sti� loads. To obtain more general voltage characteristics,the bene�ts of each of these models can be combined by using the so-calledpolynomial or ZIP model consisting of the sum of the constant impedance


(Z), constant current (I), and constant power (P ) terms:

P = Po

[a1

(U

Uo

)2

+ a2

(U

Uo

)+ a3

]

Q = Qo

[a4

(U

Uo

)2

+ a4

(U

Uo

)+ a6

] (2.7)

where, Uo, Po and Qo are normally taken as the values at initial operatingconditions. The parameters of this polynomial model are coe�cients (a1 toa6) and the power factor of the load.

Exponential Load Model

In this model the power is related to the voltage by:

P = Po

(U

Uo

)mp

Q = Qo

(U

Uo

)mq (2.8)

where, mp and mq are the model parameters. Note that by setting theparameters to 0, 1 or 2 the load can be represented by constant power,constant current or constant impedance, respectively.

Frequency-Dependent Load Model

Frequency dependence is usually represented by multiplying either a poly-nomial or exponential load model by a factor [1+af (f − fo)], where f is theactual frequency, fo is the rated frequency and af is the model frequencysensitivity parameter.

2.1.3 Multi-Machine Power System

Power systems consist of several transmission lines which interconnect gen-erator stations with load centers. Since electrical energy cannot be stored inlarge quantities, there is always a balance between the total power generatedand the sum of the total power consumed and losses.

Let us consider a multi-machine power system with n generators and atransmission network with N buses with all values expressed in p.u. Fur-thermore, consider the following assumptions:


1. Generators are described by one-axis model

2. Constant mechanical power at each generator, Pmk

3. Loads are represented by the static model

4. The transmission network is lossless.

Considering the �rst assumption the generator can be modeled as shownin Figure 2.2, where the reactance of the k-th step-up transformer is includedin x′dk . The generator dynamics generator are given by:

δk = ωk (2.9)

ωk =1

M(Pmk

− Pek −Dkωk) (2.10)

The voltage at the internal bus is given by the magnitude E′qk

and thephase angle δk.

'

kqE '

kdj x

kgI kU k

θk

δ

Figure 2.2: Synchronous generator one-axis dynamic circuit

Let Ybus, of order (N×N), be the admittance matrix of the transmissionnetwork. Since the transmission network is lossless, the kl-th element of theadmittance matrix is de�ned by Ybuskl = j Bkl.

The complex power supplied by the generator k, injected into its terminalBus k is de�ned by:

Pgk = Re(UkI∗gk)

=E′

qkUk sin(δk − θk)

x′dkQgk = Im(UkI

∗gk)

=E′

qkUk cos(θk − δk)− U2

k

x′dk

(2.11)

Concerning the transmission network, the complex power injected intoBus k is given by:


for k = 1, . . . , n

Pk =

N∑l=1

BklUkUl sin(θk − θl) +E′

qkUk sin(θk − δk)

x′dk

Qk = −N∑l=1

BklUkUl cos(θk − θl) +U2k − E′

qkUk cos(θk − δk)

x′dk

(2.12)

and for k = (n+ 1), . . . , N

Pk =N∑l=1

BklUkUl sin(θk − θl)

Qk = −N∑l=1

BklUkUl cos(θk − θl)

(2.13)

Let PLk+ j QLk

be the complex power of the load at Bus k. Then, fork = 1, . . . , N , the power �ow equations (2.12) and (2.13) can be written as:

0 = Pk + PLk

0 = Qk +QLk

(2.14)

Because generators are assumed to be one-axis modeled, the vectors xand y are:

x = [δ1 . . . δn, ω1 . . . ωn, E′q1 . . . E

′qn]

T

y = [θ1 . . . θN , U1 . . . UN ]T

then (2.1) to (2.3) and (2.14) can be rewritten as:

x = f(x, y)

0 = g(x, y)(2.15)

The multi-machine power system is therefore described by a set of dif-ferential equations related to the generators and a set of algebraic equationsrelated to the transmission network (balance of power). This dynamic modelgiven by the set of di�erential-algebraic equations is known as Structure Pre-serving Model (SPM).

2.2. CHALLENGES 19

2.2 Challenges

2.2.1 N−1 Criterion

Power systems are frequently subjected to various types of disturbanceswhich may be small, in the form of load changes and control actions, orlarge in the form of a short circuit on a transmission line or the loss of alarge generator. The system must, however, be able to adjust to changingconditions and operate satisfactorily despite these disturbances. Hence, theoperation of a power system is usually subject to security and reliabilitystandards required by the system operator. A main principle underlyingthese standards is the so called N − 1 criterion.

De�nition 1 The N-1 criterion states that:

• The power system must be operated at all times such that after anunplanned loss of an important generator or transmission facility itwill remain in a secure state, (i.e. to avoid blackouts and wide scaleconsumer disconnections).

• Furthermore, when a such loss occurs the system must be returned toa new N-1 secure state within a speci�ed time (normally within 15-20minutes) to withstand a possible new loss.

2.2.2 Power System Stability

Power system stability is of fundamental importance for system security.Power system stability is classi�ed into di�erent categories as follows [20].

1) Rotor angle stability refers to the ability of the synchronous ma-chines of an interconnected power system to remain in synchronismafter being subjected to a disturbance. After the disturbance, instabil-ity may take the form of increasing angular swings of some generatorsleading to their loss of synchronism with other generators. Loss of syn-chronism can occur between one machine and the rest of the system,or between groups of machines, with synchronism maintained withineach group after separating from each other. Rotor angle stability maybe classi�ed as follows:


a) Small-signal stability, which is concerned with the ability of thepower system to maintain synchronism under small disturbances.(It mostly deals with damping of electromechanical oscillatorymodes, which are typically in the frequency range of 0.1 to 2 Hz).

b) Transient stability, which is concerned with the ability of thepower system to maintain synchronism when subjected to a largedisturbance, such as a short-circuit on a transmission line. Tran-sient stability depends on the initial operating conditions of thesystem as well as the type, severity and location of the distur-bance. Transient stability is also referred to as �rst swing stabil-ity.

2) Voltage stability refers to the ability of a power system to maintainsteady voltages at all buses in the system after being subjected toa disturbance from a given initial operating condition. In this caseinstability may take the form of a progressive fall or rise of voltages atsome buses.

When a power system enters to a new operating state after a disturbance,in this new state, all variables are still within an acceptable range due to the�rst condition in De�nition 1. However, the system may be weakened, andsome transmission lines may become more loaded than was planned. As a re-sult, weakly damped interarea modes may become more pronounced, riskingsystem security. Low frequency electromechanical oscillations are inherent inlarge interconnected power systems. These oscillations, commonly referredto as "interarea oscillations", are associated with groups of synchronous gen-erators in one geographical region swinging with respect to other groups ina di�erent region interconnected through tie-lines. Adequate damping ofthese oscillations is a pre-requisite for the secure operation of the system.Over the years, many system outages, product of poorly damped oscillationshave been reported, one example being the August 10, 1996 blackout in theWSCC system of United States of America [9].

Furthermore, large disturbances in heavily loaded transmission systems(i.e. when the system is in a new operating state after experiencing a largedisturbance, and the second condition of De�nition 1 has not been satis-�ed yet) may lead to a serious voltage instability problem. Voltage collapseusually occurs in the aftermath of a large disturbance in a heavily stressedpower system. This results in increased reactive power losses and voltage

2.3. FACTS AND HVDC TRANSMISSION 21

drops. Voltage drops lead to the initial load reduction bringing load restora-tion control mechanisms into action. It is the dynamics of these controlsthat have often lead to voltage instability and collapse [21]. The blackoutsin the USA and Canada in August 14, 2003 [11]; in Sweden and Denmarkin September 23, 2003 [12]; and in south Greece in July 12, 2004 [13] arerepresentative examples of voltage instability incidents.

2.3 FACTS and HVDC Transmission

Transmission system owners, and in general all di�erent stakeholders, havehad to carry out an optimization of their assets in order to achieve a com-petitive position and survive in the new deregulated electricity market. Thissituation has submitted the transmission network to high stress conditionsthat increase the probability of blackouts [22]. On the other hand, customersdemand quality electricity supply from their respective supplier, includingconstant voltage, a minimum of interruptions, if any, and constant frequency.

The objective of transmission system expansion is to increase the networkcapacity and e�ectively improve �exibility, reliability and security. Amongmany possible solutions to achieve this objective, FACTS devices and HVDCsystems play an important role. These type of devices/systems have shownto be capable in stabilizing transmission systems, resulting in higher transfercapability [15, 16, 23].

An interesting question is how power electronics-based devices (FACTSand HVDC transmissions) can contribute to keep system stability when an-other contingency occurs before ful�lling the second condition in De�nition 1.

Most FACTS devices and thyristor-based HVDC systems only have onedegree of freedom; i.e., they are able to control either active or reactive power.However, Uni�ed Power Flow Controllers (UPFCs) and VSC-HVDC trans-mission systems are alternatives that have two degrees of freedom. They cancontrol active and reactive power, independent of each other. This featuregives UPFCs and VSC-HVDC transmissions a potential to enhance transientstability, provide voltage support, increase damping, and control the power�ow in a much larger extension compared to other devices/systems.

Even though UPFCs and VSC-HVDC transmissions have similar char-acteristics, from the cost point of view, VSC-HVDC transmissions, have theadvantage of using conventional power transformers. UPFCs require theinstallation of power transformers in series with transmission lines, which


demand much higher levels of insulation, more complex design and a moreelaborate construction and installation.

2.4 VSC-HVDC Transmission in Power Systems

VSC-HVDC transmissions with appropriate input signals and control strate-gies have shown to be e�ective means for stabilizing transmission systemsresulting in higher transfer capabilities and reduced outage risk. Di�erentstudies have shown the capability of VSC-HVDC transmissions for enhanc-ing power system stability [24, 25], providing voltage support [26, 27] and itsapplication in the connection of wind farms [28, 29], just to mention someapplications. This solution enables power grid owners to increase the ex-isting transmission network capacity while improving the operating marginsnecessary for grid stability. As a result, a more reliable power delivery servicecan be provided.

VSC-HVDC transmission is becoming a serious candidate when there isa need for expanding the transmission network thanks to:

• fast power �ow control, and the decrease of undesirable loop �ows,

• stability bene�ts (enhancement to transient and voltage stability, im-provement to electromechanical oscillations damping),

• development of turn o� capable valves with higher rating values,

• possibility to lay transmission cables either overhead, underground orunderwater [30, 31],

• advantages for the connection of o�shore wind farms to the transmis-sion grid [32, 33].

All these features and the advantages over others FACTS devices andthyristor-based HVDC systems are a good indicator that VSC-HVDC trans-missions will play a very important and active role in power systems in thenear future. Therefore, it is of high interest to explore, study, analyze andderive di�erent control strategies and possible ways to control this type ofsystems.

Currently, about twelve VSC-HVDC transmissions operate in the world,with a total capacity close to 2500 MW. To date, six more projects will

2.4. VSC-HVDC TRANSMISSION IN POWER SYSTEMS 23

be constructed in the coming years, which will mean a total capacity of7000 MW by the year 2015.

Chapter 3

Modeling of VSC-HVDCTransmissions

This chapter presents three di�erent models for VSC-HVDC transmissions,namely Injection Model, Simple Model and HVDC Light Open Model. TheInjection Model and Simple Model are used for the derivation of CLF-basedcontrol strategies. The HVDC Light Open Model is used for testing controlstrategies and analyzing the impact of VSC-HVDC transmissions on powersystems. The Injection Model and Simple Model have been presented in PaperI and II.

3.1 Injection Model

The Injection Model is intended for power �ow and electromechanical dy-namics analyses. For this reason it is su�cient to consider the voltage andcurrent phasors in the ac system. The Injection Model can be consideredas an element, which provides adequate interaction with other elements forenhancing the dynamic performance and stability of the system as a whole.Harmonics and dc transient components are neglected, because they nor-mally have a second order e�ect on the active and reactive powers. Voltagesand currents are represented by phasors in the ac network, which vary withtime during transients. The Injection Model is valid for symmetrical condi-tions, i.e., for positive sequence voltages and currents. The model is used inthe derivation and test of control strategies.

25

26 CHAPTER 3. MODELING OF VSC-HVDC TRANSMISSIONS

Figure 3.1 shows a basic structure of a VSC-HVDC link connected inparallel with an ac transmission line.

id

RdCd Cd

VSC VSC

j xt i j xt j

Ps iQs i

Filter Filter

Ps jQs j

Ui Ujj xL

U ci U cj

Figure 3.1: Basic Structure of a VSC-HVDC transmission.

From the connection nodes Bus i and Bus j, a VSC-HVDC transmissioncan be seen as a synchronous machine without inertia where the productionor consumption of active power is independent of the production or consump-tion of reactive power. This interpretation leads to modeling a VSC-HVDCtransmission as two controllable voltage sources in series with a reactance,which represents the impedance of the power transformer. This modeling isshown in Figure 3.2.

j xL

j xt i j xt j

Uj θjUi θi

Ii IjU ci U cj

Ps j,Qs jPs i,Qs i

Figure 3.2: Modeling of a VSC-HVDC transmission.

In the �gure Uci = Uciejγi and Ucj = Ucje

jγj . Psi and Qsi can be indepen-dently controlled by γi and Uci. Likewise Psj and Qsj can be independentlycontrolled by γj and Ucj

It is assumed that dc voltage control keeps the dc voltage magnitudeclose to the rated voltage. Therefore the losses of the converters are assumed

3.2. SIMPLE MODEL 27

constant, regardless of the current through the converters. The losses areconsequently represented as a constant active load. The losses of the dccables are neglected. The relation between Bus i and Bus j is given by theactive power: Psi = −Psj . Figure 3.3 shows the Injection Model.

j xLUj θjUi θi

Psj + j QsjPsi + j Qsi Plosses

Figure 3.3: Injection Model of a VSC-HVDC transmission.

Expressions for Psi, Qsi, Psj and Qsj can be found in Paper II.

3.2 Simple Model

The Simple Model is a variation of the Injection Model where the active andreactive power are controlled directly, i.e., γi, Uci, γj and Ucj do not controlthe active nor the reactive power injected into the respective converter. Themain assumption in the Simple Model is that the control of the VSC has avery fast PI-regulator driving the active and reactive power injected into theVSC. The Simple Model is then described by (5) in Paper II, where ∆Psi,∆Qsi and ∆Qsj are inputs for control strategies or voltage support.

3.3 HVDC Light Open Model

This model is available in the software used for the simulations in this thesis:SIMulation of POWer systems (SIMPOW) [34], which is based on the HVDCLight Model developed by ABB.

The objective of the model is to provide the correct interaction between acand dc systems. It interacts with ac networks through the injected currents.The node voltages of the networks are determined by the Kircho�'s currentlaw. It de�nes an equation for each node, from which the node voltagescan be solved, given the injected currents as function of the node voltages.


j xL

Harmonic

Filter

Transformer

Idc

UdcConverter

Reactor

UacIac

UPCCIPCC

U1

Figure 3.4: Single line diagram of one end of a VSC-HVDC transmission

Hence, the modeling objective is to provide equations of the HVDC Light,which give the phasor of the ac current injected to the ac network as functionof the phasor of the voltage of the ac node. The model is written for transientconditions. The model for steady state conditions is obtained by neglectingall time derivatives.

Reference values for the di�erent controls in the ABB HVDC Light OpenModel Version 1.1.6, such as active power, reactive power and ac voltagereferences can be set by the user and, if necessary, modi�ed during thesimulation. However, the dc voltage reference cannot be controlled by theuser and is set by the model to match the nominal dc voltage of the dc nodes.

Figure 3.4 shows the single line diagram of one end of a VSC-HVDCtransmission. The �gure shows a PWM converter, a series reactor, xL, anac �lter and a transformer.

In the �gure, UPCC and IPCC are the injected voltage and the currentinto the Point of Common Coupling (PCC).

The active power through the converter reactor, xL, is equal to the dcpower injected into the dc nodes. The losses of the PWM converter aremodeled inside and are divided into no-load losses and load losses.

The PWM converter controls the internal ac voltage bus U1, so thatthe real and imaginary part of the current IPCC on the primary side of theconverter transformer correspond to current orders from internal controllers.

For a converter in active power control mode, an internal active powerPI-regulator controls the real part of the current IPCC so that the activepower is equal to the active power reference.

For a converter in dc voltage control mode, an internal dc voltage PI-regulator controls the real part of the current IPCC so that the set value for

3.3. HVDC LIGHT OPEN MODEL 29

Converter

reactor

Harmonic

Filter

Transformer

reactive

power

control

ac

voltage

control

phase

current

limit

converter

voltage

limit

active

power

control

inner

current

limit

dc

voltage

control

Uacref

Qref

Qc

Uac

Udc

Pref

Pc

Udcref

Uac-c

Uac meas

Qmeas

Udc meas

Pmes

Figure 3.5: Overview control of HVDC Light Open Model

the dc node voltage, e.g. 1.0 p.u., is maintained.

For a converter in either reactive power control mode or ac voltage controlmode, an internal PI-regulator controls the imaginary part of the currentIPCC so that either the reactive power or ac voltage is equal to the referencevalue.

Figure 3.5 shows a block diagram of the control of the HVDC Light OpenModel.

In the �gure,∆Pc,∆Qc and∆Uac−c are user inputs and they are intendedfor supplementary control. In this thesis the input ∆Uac−c is not used.

3.3.1 Limits Capability Converters

The limits restricting the operation of the converters are considered in theHVDC Light Open Model. Figure 3.6 shows the restricted operating area ofa P-Q diagram due to the limitation on the dc cable and dc voltage [25].


Max IMax VdcMax Pdc

Q

P

Vac=0,9 pu

Vac=1,0 pu

Vac=1,1 pu

Figure 3.6: P - Q diagram for dc voltage range

There are three main factors that limit the capability of the convertersseen from a power system stability perspective.

1. Maximum current. Maximum current that can �ow through thevalves. This current when multiplied by the ac voltage yields the max-imum MVA circle in the power plane. If the ac voltage decreases, sodoes the MVA capability.

2. Maximum dc voltage. Reactive power is mainly dependent on thevoltage di�erence between the ac voltage the VSC can generate fromthe dc voltage and the ac network. If the ac voltage is high, the dif-ference between the maximum dc voltage and ac voltage is low. Thereactive power capability is then moderate, but increases with decreas-ing ac voltage.

3. Ampacity cables. Maximum dc current that can �ow through thecables.

3.4 Results

It was important to know if both the Injection Model and the Simple Modelprovided an acceptable representation of the VSC-HVDC transmission. In

3.4. RESULTS 31

order to determine the validity of both models, di�erent fault cases were sim-ulated and compared with the HVDC Light Open Model. Simulation resultsshowed that the representation of the VSC-HVDC transmission through theInjection Model and Simple Model was very acceptable and appropriate forderivation and testing of control strategies [35].

Chapter 4

Control of VSC-HVDCTransmissions

This chapter summarizes the derivation of three di�erent control strategiesused in this thesis. One control strategy is based on modal analysis. The othertwo are based on nonlinear control theory, more speci�cally on Lyapunovtheory. The control strategies have been used in Papers I to V. The chapteralso presents an analysis of the use of local and remote signals as an inputto the control strategies. This analysis was presented in Paper III. Thischapter study also presents an approach to relate the SIME method and modalanalysis. This has been presented in Paper IV. A description of the impact ofthe two modes of operation (reactive power control and voltage control mode)are also included in this chapter. This description was presented in PaperIV.

4.1 Modal Analysis-based Control Strategy

Let us consider the SPM in Section 2.1.3, but with an input (control action)vector u. The set of di�erential-algebraic equations (2.15) becomes:

x = f(x, y, u)

0 = g(x, y, u)(4.1)

Moreover, let us consider that the output of the system, Y, is given by:

Y = h(x, y, u) (4.2)

33

34 CHAPTER 4. CONTROL OF VSC-HVDC TRANSMISSIONS

Let x0 and u0 be the equilibrium point of the system. Also let the stateand input vectors be x = x0 and u = u0 in steady state. The linearized formof (4.1) and (4.2) around the equilibrium point is:

∆x = A∆x(t) +B∆u

∆Y = C∆x+D∆u(4.3)

where:

∆x is the state vector (dimension nx × 1)

∆u is the input vector (dimension r × 1)

∆Y is the output vector (dimension m× 1)

A is the state matrix (dimension nx × nx)

B is the input matrix (dimension nx × r)

C is the output matrix (dimension m× nx)

D is the (feedforward) matrix which de�nes the proportion of input whichappears directly in the output (dimension m× r)

The eigenvalues of matrix A are given by

det(A− λI) = 0 (4.4)

The nx solutions of λ = λ1, . . . , λnx are the eigenvalues of A, which can bereal or complex. The complex eigenvalues are of particular interest in powersystems because they correspond to oscillatory modes. The real part of theeigenvalues determine the damping, and the imaginary part the frequencyof an oscillation. A negative part indicates a damped oscillation whereas apositive real part undamped oscillations. For a complex pair of eigenvalues:

λ = σ ± jωp (4.5)

where the frequency of oscillation is given by:

f =ωp

2π(4.6)

4.1. MODAL ANALYSIS-BASED CONTROL STRATEGY 35

and the damping ratio is given by

ζ =−σ√σ2 + ω2

p

(4.7)

Controllability and observability of an oscillatory mode are measures thathelp to determine the best possible location for a controllable device and thebest input signal for POD. Let us denote Φ as the modal matrix of righteigenvectors and Ψ as the modal matrix of left eigenvectors. The controlla-bility and observability matrices are de�ned as:

B′ = ΨB

C ′ = CΦ(4.8)

A mode is uncontrollable if the corresponding row of the matrix B′ iszero. A mode is unobservable if the corresponding column of the matrixC ′ is zero. If a mode is either uncontrollable or unobservable, a feedbackbetween the output and input does not have e�ect on the mode.

Controllability of an oscillatory mode is dependent on the location of theelectrical equipment. A higher controllability of a critical mode results in asmall compensation requirement for a given modal damping.

Observability of oscillatory modes depends on the feedback signal se-lected.

Residue is a measure that depends on both controllability and observ-ability. A higher residue magnitude implies higher sensitivity of the corre-sponding eigenvalue to controller gain [36]. It is then desirable to have highcontrollability and high observability of a mode of oscillation to achieve highdamping.

If we assume that:

• Y is not a direct function of u (i.e., D = 0)

• m = r = 1, i.e., single input single output

• eigenvalues are distinct, i.e. λi = λj

it can be shown that the transfer function of the system (4.3) can be ex-pressed in partial fractions as [37]:

G(s) =

nx∑i=1

Ri

s− λi(4.9)


where Ri is the residue of G(s) at λi, and it is de�ned as:

Ri = CΦiΨiB (4.10)

By having a feedback transfer function of the form H(s, k) = kH(s),where k is a constant gain between output and input, it can be shown thatfor small values of gain (with s = λi) [38]:

∆λi = RiH(λi, k)

∆λi = kRiH(λi)(4.11)

Equation (4.11) implies that the position of a selected eigenvalue (ormode) can be changed by tuning the parameters of the transfer functionH(s = λi) using the residue Ri.

In this thesis the regulator H(s) consisted of three parts: a wash-out�lter, a phase compensating �lter and a gain. The regulator is shown inFigure 4.1.

1+s T1

1+s T2

s Tw

1+s Tw

PODKp

Input

Wash-out

Filter

Lead-lag

FilterGain

Figure 4.1: Damping Regulator

The wash-out �lter serves as a high-pass �lter, with the time constant Tw

high enough to allow signals associated with oscillations to pass unchanged.The value of Tw is not critical and may be in the range of 1 to 20 seconds[39].

The phase compensation block provides the appropriate phase-lead orphase-lag characteristic. For practical reasons, the argument of the phasecompensator is limited to a maximum of 60◦ [39]. This is the reason why inpractice two or more �rst-order blocks may be used to achieve the desiredcompensation.

The stabilizer gain determines the amount of damping introduced bythe regulator. Ideally, the gain should be set at a value corresponding tomaximum damping; however, it is often limited by other constraints.

The residue of the oscillatory mode of interest, which gives the directionof the eigenvalue departure for a small change of the system parameters, is

4.2. CLF-BASED CONTROL STRATEGIES 37

given by equation (4.10). In order to move the eigenvalue λi to a desiredlocation and considering only the real part, the argument of H(s = λi), ϕ,must be ϕ = 180◦ − arg(Ri); see Figure 4.2.

Desired*

Actual*

Im

Re

arg(Ri)

Ri

Figure 4.2: Direction of the eigenvalue departure

Hence the tuning parameters for the regulator is performed with theresidue technique [38].

4.2 CLF-based Control Strategies

4.2.1 Local Information

In order to derive a CLF-based control strategy for a VSC-HVDC transmis-sion it is necessary to �rst �nd a Lyapunov function, i.e., a function thatful�lls the conditions described in Section 3.1 of Paper II. The Lyapunovfunction can be found through two main steps. The �rst step is to �nd aLyapunov function for an uncontrolled power system without a VSC-HVDCtransmission. This Lyapunov function is denoted as Vuncontrolled. The secondstep is to �nd a Lyapunov function for an uncontrolled power systems withan uncontrolled VSC-HVDC transmission. This second Lyapunov functionis denoted as VHV dc

uncontrolled.

There is no established formulation for determining Lyapunov functions.However, since Lyapunov functions are closely related to the energy of thesystem, functions of energy are Lyapunov function candidates.

In the development of energy functions for multi-machine power systems,and also in order to clearly distinguish between the forces that accelerate thewhole system and those that tend to separate the system into di�erent parts,it is convenient to transform the system (2.15) into the so called Center OfInertia (COI) reference frame. The position of the COI is de�ned by:


δCOI =1

MT

n∑k=1

Mkδk (4.12)

ωCOI =1

MT

n∑k=1

Mkωk (4.13)

where,

MT =n∑

k=1

Mk

Furthermore (assuming zero inherent damping),

ωCOI =1

MT

n∑k=1

(Pmk− Pek) =

PCOI

MT(4.14)

Next the state variables δk and ωk are transformed to the COI variablesas:

δk = δk − δCOI

ωk = ωk − ωCOI

(4.15)

These COI variables are constrained by:

n∑k=1

Mkδk =0 (4.16)

n∑k=1

Mkωk =0 (4.17)

The load buses are also transformed to the COI reference frame by

θk = θk − δCOI (4.18)

By taking the derivative in time of (4.15) gives:

˙δk = δk − δCOI = ωk − ωCOI = ωk (4.19)

˙ωk = ωk − ωCOI =1

Mk

(Pmk

− Pek −Mk

MTPCOI

)(4.20)


Thus the system (2.15) in the COI reference frame is expressed as

˙δk = ωk (4.21)

˙ωk =1

Mk

(Pmk

− Pek −Mk

MTPCOI

)(4.22)

E′qk

=1

T ′dok

(Efk −

xdkx′dk

E′qk

+xdk − x′dk

x′dkUk cos(δk − θk)

)(4.23)

and Pek given by

Pek =E′

qkUk

x′dksin(δk − θk) (4.24)

For power systems in the COI reference frame, with n-generators de-scribed by the one-axis model and n + N -load buses, the energy functiongiven by (14) in Section 3.1.1 in Paper II is available in the literature [40, 41].This energy function is the Lyapunov function for the uncontrolled powersystem without VSC-HVDC transmission.

When an uncontrolled VSC-HVDC transmission is included in the uncon-trolled power system, the energy function is given by (21) in Section 3.1.1 inPaper II. By using this energy function as a Lyapunov function candidate forthe controlled system, i.e., a power system with a controllable VSC-HVDCtransmission, the following control strategy is obtained:

∆Psi = kfij (4.25)

where k is a positive gain and fij is the frequency di�erence between thebuses where the VSCs are connected.

4.2.2 SMIB and Remote Information

Let us consider the Single Machine In�nite Bus (SMIB) shown in Figure 4.3.

By using the Injection Model, the SMIB system becomes as shown inFigure 4.4 where Psi and Qsi are given by:


1 0

jxLjX’d

Ui=UiE’q

HVDC

Un=

Psi + j Qsi

Figure 4.3: SMIB power system

1 0

jxLjX’d

E’q Ui=Ui Un=

j xt i

Uci i

Psi ,Qsi

Figure 4.4: Injection Model VSC-HVDC transmission in a SMIB system

Psi =Ui(sin(θi)u1 − cos(θi)u2)

xti

Qsi =U2i − Ui(cos(θi)u1 + sin(θi)u2)

xti

(4.26)

If the generator of the SMIB system is represented by the classical modelwith no damping, the dynamics of the system is described by (4.27) :

δ = ω

ω =1

M(Pm − Pe)

(4.27)

where M is the inertia of the generator, Pm is the mechanical power and Pe

is the electrical power.


Kirchho�'s current law at node i gives:

E′q − Ui

jx′d=

Ui − Un

jxL+

Ui − Uci

jxti(4.28)

By solving (4.28) for Ui the following is obtained:

Ui =

(E′

q

x′d+

Un

xL+

Uci

xti

)(1

x′d+

1

xL+

1

xti

)−1

(4.29)

The generator electrical power is given by:

Pe = Re(E′q I

∗g ) (4.30)

The generator current is de�ned by:

Ig =E′

q − Ui

jx′d(4.31)

By inserting (4.29) in (4.31) and substituting (4.31) in (4.30), the electricalpower is given by:

Pe =E′

qUn

xL + x′d +x′dxL

xti

sin δ +E′

quci1

xti + x′d +x′dxti

xL

sin δ

−E′

quci2

xti + x′d +x′dxti

xL

cos δ

(4.32)

The dynamics of the system are now described by the following expres-sion:

x = f0(x) + f1(x)uci1 + f2(x)uci2 (4.33)

which is an a�ne system de�ned by (11) in Section 3.1 in Paper II. In (4.33):

f0(x) =

ω

1M (Pm − Pemax sin δ)

f1(x) =

0

−Cu sin δ

f2(x) =

0

Cu cos δ


where

Pemax =E′

qUn

xL + x′d +x′dxL

xti

Cu =E′

q

M(xti + x′d +

x′dxti

xL

)For the SMIB system without VSC-HVDC transmission the dynamic of

the system arex = f0(x)

whose energy function is given by [42],

V(x) = 1

2Mω2 − Pmδ − Pemax cos δ + Co (4.34)

where Co is a constant such that V(x) = 0 at x = x0.By having a VSC-HVDC transmission in the system (see(4.33)) and ap-

plying the CLF concept described in Section 3.1 in Paper II, the followingstabilizing control laws are obtained

uci1 = ku1ω sin δ (4.35)

uci2 =− ku2ω cos δ (4.36)

Single Machine Equivalent (SIME)

The SIngle Machine Equivalent (SIME) method is a hybrid direct-temporaltransient stability method, which transforms the trajectories of a multi-machine power system into the trajectory of a single machine equivalentsystem of the form:

δSIME =ωSIME (4.37)

ωSIME =1

M(PmSIME − PeSIME ) (4.38)

whose parameters (which are derived from multi-machine power system) aretime-varying. Basically, the SIME method deals with the post-fault con�gu-ration of a power system subjected to a disturbance which presumably drivesit to instability. Under such condition, the SIME method uses a time-domain


program in order to identify the mode of separation of its machines into twogroups, namely critical and non-critical machines which are successively re-placed by a two-machine equivalent. Then, this two-machine equivalent isreplaced by a single machine equivalent system. By de�nition, the criticalmachines are the machines responsible for the loss of synchronism [18].

The parameters in (4.37) are:

MC =∑i∈C

Mi (4.39)

MNC =∑j∈NC

Mj (4.40)

The angle and the speed of the SIME are expressed by:

ωSIME = ωC − ωNC (4.41)

δSIME = δC − δNC (4.42)

with

δC =1

MC

∑i∈C

Miδi δNC =1

MNC

∑j∈NC

Mjδj (4.43)

ωC =1

MC

∑i∈C

Miωi ωNC =1

MNC

∑j∈NC

Mjωj (4.44)

where

ωC : equivalent speed of the critical machines

ωNC : equivalent speed of the noncritical machines

δC : equivalent rotor angle of the critical machines

δNC : equivalent rotor angle of the noncritical machines

MC : total inertia constant of the critical machines

MNC : total inertia constant of the noncritical machines


By using the SIME method, the CLF-based control strategy (4.35) canthen be applied to a multi-machine power system. The state variables δ andω are replaced by δSIME and ωSIME [43]. The control strategy becomes:

uci1 = ku1ωSIME sin δSIME (4.45)

uci2 =− ku2ωSIME cos δSIME (4.46)

4.3 Local and Remote Signals Analysis

This analysis consisted of using di�erent input signals in the derived controlstrategies presented earlier. The test power system was the Nordic 32A [44].Three cases were analyzed, two for local signals and one for remote signals.

1. Local signals

1.1. Modal analysis-based control strategy

1.2. CLF-based control strategy

2. Remote signals

2.1. CLF-based control strategy

The CLF-based control strategy described in Section 4.2.1 was also usedin the case of remote signal, but the input signal was a function of measure-ments from remote places.

4.3.1 Local Signals

Modal analysis was performed to �nd the most suitable location where toconnect the VSC-HVDC transmission, and also the most appropriate localinput signal for POD between current and power. This resulted in the highestpossible residue that the VSC-HVDC transmission could have (see Table IIIin Paper III ).

For the �rst case the local signal selected was the current. Power signalsare normally chosen since they contain information of the voltage and thecurrent. Nevertheless, in our case the current showed to have larger observ-ability (see table II in Paper III ). This was a similar result obtained from

4.3. LOCAL AND REMOTE SIGNALS ANALYSIS 45

POD using Power System Stabilizers (PSSs) and PMUs [45]. The signal wasused in the control strategy described in Section 4.1.

For the second case the local signal was the frequency di�erence betweenbuses where the VSC-HVDC transmission was connected. This current sig-nal was used in the control strategy (4.25).

4.3.2 Remote Signals

Modal analysis was also performed in this case to �nd the most participatinggenerator of each group in the interarea mode. The two selected generatorswere the sources for remote information. The signal measured from eachgenerators was the speed and the rotor angle. As mentioned above, thecontrol strategy used in this case was based on (4.25), but additionally anadaptive gain was included. The adaptive gain was a function of the rotorangles of the chosen generators and calculated as: | sin(δ1 − δ2)|. Then thecontrol strategy became:

∆Psi = k(ω1 − ω2)(| sin(δ1 − δ2)|)

4.3.3 Results

Simulations showed that using local current as input signal gave as goodresult as the remote signal. It is important to highlight that this is the bestresult that can be expected when using the local signal since the VSC-HVDCtransmission was purposely located to give the highest possible residue. Fur-thermore, current magnitudes have a uniform content throughout an entiretransmission line (modeled wit π model), as opposed to the angles of othervariables and frequencies [45�47].

For the second local signal the results were similar to those obtained withthe current magnitude.

As the VSC-HVDC transmission changed location (and of course its con-trollability) the information contained in the local signals also changed (andtherefore their observability). Now local signals did not have as good impactas the remote signals. This was even clearer in the case of frequency di�er-ence signal. The variation of the frequency in one end of the VSC-HVDCtransmission had essentially the same variation of the frequency at the otherend. Information about the interarea mode was practically lost and reduceddamping was provided.


Remote signals of course remained unaltered when the VSC-HVDC trans-mission changed its location. This con�rms that local signals may lack ob-servability of signi�cant interarea modes depending on their location [48].

4.4 Approach SIME - Modal Analysis

The SIME method needs information from all machines in the power system.Although PMUs might seem like an attractive option, they currently do notmeasure machine speeds and angles, and so these quantities would need tobe estimated. Moreover the cost-bene�t relation is highly unfeasible withtoday's technology.

One possible way to circumvent the lack of access to all generator quan-tities measurements in a power system and still use the SIME method is toselect the most relevant generators. Relevant generators can have a di�erentmeaning depending on the purpose for obtaining information. In this thesisrelevant generators mean generators that participate in a speci�c mode ofoscillation and can provide information for increasing damping.

The most relevant generators in an oscillatory mode can be identi�edby carrying out modal analysis. In modal analysis the relative mode-shapecan be plotted to observe if there is a coherent group of machines oscillatingagainst another group of machines. Furthermore, the participation matrixcan give some idea of what generators should be selected from each coherentgroup. Once the generators are selected, the two groups of machines can beconformed and the SIME method applied.

This approach has the main advantage that information from all genera-tors is not needed and measurements are required only at the most relevantgenerators. From the implementation point of view this approach wouldrequire a signi�cant lower number of PMUs to be installed.

Measurements from the selected generators are enough (in our case) toobtain a very good input signal for the POD control strategy without sacri�c-ing too much damping compared to the signal obtained when measurementsfrom all generators is collected.

However there is a price to be paid when using this approach. By usingmodal analysis it is not possible to know what the critical and noncriticalmachines are. This is very important because some control strategies mightresult negative regulator gain, hence providing negative damping. Neverthe-

4.4. APPROACH SIME - MODAL ANALYSIS 47

less, preliminary analysis or experience in the behavior of the system cangive ideas of what group can be determined as critical or noncritical.

Another aspect important to mention is that modal analysis is performedon a speci�c operating point. The critical and noncritical machines identi�-cation depends on the initial equilibrium point, the type of fault and location.For some fault cases generators can be classi�ed as critical machines, but forsome others, the same generators might be identi�ed as noncritical machines.

4.4.1 Results

This approach was tested in the Nordic 32A power system, which is shown inFigure 4.5. The power system has a total of 21 generators. Modal analysisshowed that there is an interarea mode between the north part and central-southwest part with a frequency of 0.5 Hz. The left eigenvalues showed that9 generators were in one coherent group (north part), and the remaining 12were in the other group.

It was assumed that generators in the north part were the critical ma-chines and the other group were the noncritical machines. The selection ofgenerators from which the information was going to be measured was carriedout by using the participation matrix and con�rmed with simulations. Mostof generators from the north part had high participation in the interareamode. 7 out of 9 generators were selected for the reduced group of criticalmachines. On the other hand, the number of generators in the central andsouth west part was lower. 5 out of 12 generators were chosen for the reducedgroup noncritical machines.

The measurements taken from each generator was the speed. The controlstrategy used was the speed di�erence.

Simulations were done considering information from all machines and alsowith the reduced number of generators. The main conclusion was that thedamping achieved with the reduced number of generators was practically thesame as the damping provided with all generators. However it is importantto note that in this simulation the approach worked very well, but for sim-ulations with di�erent initial conditions, the machine groups might change.It is common to �nd that once some generators were classi�ed as critical ornoncritical, they remained as critical or noncritical for many di�erent faultsand fault locations, but this cannot be assumed as a rule.


4071

4072

4011

4012

4022 4021

4031 4032

4042

4041

4046

4043 4047

4044

4045

4051

4061

4062

4063

1011

1012

1013

1014

10221021

20312032

1044

1043

1041 1045

1042

50% 50%

37,5%

40%

40%

Nordic 32A

Cigré System

External

South

West

Central

North

400 kV

220 kV

130 kV

Figure 4.5: Nordic 32A test power system

4.5 Voltage or Reactive Control Mode

4.5.1 Modes of Operation

A VSC-HVDC transmission has the capability to control either the reactivepower injected into the VSC or the ac voltage at the node where the VSC is

4.5. VOLTAGE OR REACTIVE CONTROL MODE 49

connected (or nearby nodes).

Furthermore, a supplementary control for POD can be used in bothmodes of operation. This supplementary control can be activated to worksimultaneously with the supplementary control used in the active power con-trol or be activated when there is no dc power transfer. A VSC-HVDC trans-mission operating with Pdc = 0 is similar to having two STATCOMs in thepower system.

4.5.2 Results

Simple Model

Only the ac voltage control mode was used in the Injection Model. Fig-ure 4.6 shows the bock diagram where a POD regulator is included on bothconverters (i and j denote the connection nodes of the VSCs).

The use of the Injection Model in simulations showed that higher powersystem stability could be achieved if the POD signal was used in the activepower mode control and ac voltage control mode was set in both converters.This can be observed in the results presented in Paper I

+-

Uio

Ui

input1+S T2i

1+S T1ikqi

PI ctrl

+

+Qsi

S Tw

1+S Tw

+-

Ujo

Uj

1+S T2j

1+S T1jkqj

PI ctrl

+

+Qsj

S Tw

1+S Twinput

Figure 4.6: ac voltage control mode in the Injection Model

The bene�ts of having a supplementary control in the voltage controlmode were seen when the power through the dc-link was suddenly inter-rupted. It was assumed that the converters were still available after theinterruption. This case was described in Paper I.


When the POD signals were used in both VSCs, the results showed thatsome coordination had to be done in the converters since negative dampingcould appear as observed in the simulations.

HVDC Light Open Model

A deeper analysis was done by using the HVDC Light Open Model, since allthe limits that a�ected the VSCs are included in the model. In the InjectionModel it was assumed that the VSC-HVDC transmission had available thecomplete MVA circle for the modulation of P and Q.

When the VSC-HVDC transmission is operating in reactive power controlmode, interactions with nearby controllers, mainly with AVR in generators,could be seen. This is mainly the case when an part of the system has lostreactive power support (as in the case of generator outage). This case ispresented in Paper IV.

In the ac-voltage control mode there was no interaction with closebygenerators, since the VSCs could generate more reactive power. Howeverthe interaction can become again a problem if the VSCs reach any limits.

One possible strategy to avoid possible negative interactions with gener-ators and also to increase the stability margin is to set a change in the dcpower �ow. By increasing the active power injected into the VSC, the acpower will decrease and so will the current. This decrease results in lowerreactive power losses and a better voltage pro�le. This power �ow strategywas used in Paper II and IV

Chapter 5

Model Predictive Control(MPC)-based Control Strategy

This chapter describes the application of MPC for POD. Parts of this chapterfollow the discussion in [49].

5.1 Theoretical background

5.1.1 Model Predictive Control

Model Predictive Control (MPC) or Receding Horizon Control (RHC) is aform of control in which the current control action is obtained by solvingon-line, at each sampling instant, a �nite horizon open loop optimal controlproblem, using the current state of the power system as the initial state; theoptimization yields an optimal control sequence and the �rst control in thissequence is applied to the power system. Examples where model predictivecontrol may be advantageously employed include unconstrained nonlinearsystems and time varying systems [50].

MPC is not a new method of control design. It essentially solves standardoptimal control problems. One main di�erence respect to other controllertechniques is that it solves the optimal control problem online for the currentstate of the system, rather than determining a feedback policy online (thatprovides the optimal control for all states). The online solution is obtainedby solving an open loop optimal control problem in which the initial state isthe current state of the system being controlled. It requires the open loop

51

52CHAPTER 5. MODEL PREDICTIVE CONTROL (MPC)-BASED

CONTROL STRATEGY

optimal control problem to be solvable in a reasonable time (compared withthe system's dynamics) and the use of a �nite horizon.

The open loop optimal control problem

The system to be controlled is usually described, or approximated, by anordinary di�erential equation but, since the control is normally piecewiseconstant, di�erence equations are used

x(n+ 1) =f(x(n), u(n)) (5.1)

y(n) =h(x(n)) (5.2)

where f(·) is implicitly de�ned by the originating di�erential equation thathas an equilibrium point at the origin f(0, 0) = 0. x is the vector of statevariables, y the output of the system and u denotes a control action. x andu must satisfy x(n) ∈ X and u(n) ∈ U . Usually U is a compact subset of Rm

and X a closed subset of Rn, each set containing the origin in its interior.

For the event (x, n) (i.e., for state x at instant n) the cost function isde�ned by:

C(x, n,u) =

N∑i=1

c(x(n+ i), u(n+ i− 1)) (5.3)

where u = u(n), u(n+ 1), . . . , u(n+N − 1). The terminal time increaseswith time n and is often referred to as receding horizon or time horizon, N .

At event (x, n) the optimal control problem of minimizing C(x, n,u)subject to the control, state and terminal constraints is solved yielding theoptimizing control sequence

u0(x, n) = {u0(n, x(n+ 1)), u0(n+ 1, x(n+ 2)), . . . ,

u0(n+N − 1), x(n+N)}(5.4)

and the cost function

C0 = C(x, n,u0(x, n)) (5.5)

The �rst control u0(n, x(n+1)) in the optimal sequence u0(x, n) is appliedto the system (at instant n).

5.2. APPLICATION OF MPC TO VSC-HVDC TRANSMISSIONS 53

5.2 Application of MPC to VSC-HVDC

Transmissions

5.2.1 General MPC-based design approach

Model predictive control is a decision-making technique that is commonlyapplied to time-variant �nite-time control problems. These problems areusually characterized by a discrete-time process, for which the dynamicsf : X × U × {0, 1, ..., N − 1} → X correspond to the generic equation:

x(n+ 1) = f(x(n), u(n), n) (5.6)

where N is the control horizon, X the space of system states, U the set of Ppossible control actions, and x(n) ∈ X and u(n) ∈ U the system state andcontrol action at time instant n, respectively.

The MPC strategy selects a control action to apply at instant n by iden-tifying a sequence of H successive control actions that maximize a cost func-tion C : X × UH 7→ R given the current state x(n). Then, it applies the�rst control action of this sequence and, at the next time step, it reproducesthe same process. An optimal control problem is solved at every discreteinstant. By recomputing an open loop sequence at every instant n, it is pos-sible mitigate the sub-optimality problems related, among others, to the factthat (5.6) may not represent perfectly the dynamics of the real system andthat the initial state may not be known exactly. To compute the optimalcontrol sequence, an exhaustive search over all the PH possible sequencesof actions could be computed. However, even for relatively small values ofP and H, such a procedure is generally computationally expensive. In anattempt to limit the computational burden associated with the identi�cationof optimal sequence actions, the A-star (also denoted as A*) algorithm canbe used [51].

5.2.2 VSC-HVDC transmission model

For the sake of simplicity in the manipulation of the information of statesvariables, control actions and computation of open loop sequences, the In-jection Model described in Section 3.1 is used.


CONTROL STRATEGY

5.2.3 Application of the MPC-based approach to POD

The vector x(n) gathers values of the state variables at instant n, the controlvariable u(n) corresponds to the modulation of the control variable of theVSC (i.e., ∆uci1 or ∆uci2) at instant n. In order to restrict the set ofpossible values for ∆uci(n) and reduce computation time, the search spaceU is discretized and restricted to P values. Although the real-time estimationof both x(n) and a function f that represents well the power system dynamicsis di�cult, it is assumed that both the current state and the system dynamicsare perfectly known.

The cost function C corresponds to the sum of instantaneous costsc(x(n + 1)), c(x(n + 2)), ..., c(x(n + H)), with c(x) re�ecting the degree ofinstability of the system in state x:

C(x(n), u(n), . . . , u(n+H − 1)) =H∑i=1

c(x(n+ i)) (5.7)

The instantaneous cost function c : X 7→ R is de�ned by

c(x(n)) =

D(x)−Dmin if ∃ i, j ∈ 1, . . . , NG such that

∥δi(n)− δj(n)∥ < δmax

cpen otherwise

(5.8)

where NG represents the number of generators, D(x) is a transient stabilityindex, Dmin its minimum value, and cpen a large constant value that stronglypenalizes system states outside the region of stability of the system. Inparticular, this value should be chosen large enough to ensure that if thereexists a sequence of actions that maintain the system inside its stabilitydomain, then a sequence of action that leads to instability is necessarilysuboptimal. Di�erent transient stability indices can be used. Three possibletransient stability indices, available in the literature can be: DP , acceleratingpower; DC , coherency index; andDE energy index. These indices are de�nedas follows

1) The accelerating power index

DP (x(n)) =

NG∑i=1

(ωi(n)− ωCOI(n))(θi(n)− θCOI(n)) (5.9)

5.2. APPLICATION OF MPC TO VSC-HVDC TRANSMISSIONS 55

where ωi(n) and θi(n) are the rotor angle and speed of generator i atinstant n, respectively. ωCOI(n) and θCOI(n) represent the rotor angleand rotor speed of the center of inertia of the systems at instant n.

2) The coherency index

DC(x(n)) =

NG∑i=1

Gi(x(n))(ωi(n)− ωCOI(n)) (5.10)

with

Gi(x(n)) = Pmi(n)− Pi(n)−Mi

MTPCOI(n) (5.11)

where Pm(n) is the mechanical power received by generator i at instantn, Pi(n) is the electrical power output at instant n, and

3) The energy index

DE(x(n)) =

NG∑i=1

(ωi(n)− ωCOI(n))2 (5.12)

5.2.4 Extremely Randomized Trees

The estimation of the function f that represents the power system dynamicsand the states variables estimation is a di�cult task because several factorshave to be considered. Type, location and duration of disturbances and thepost-fault topology are some examples of the factors that a�ect estimation.

One method that can be used in the task of �nding the function f andthe prediction of the new states variables based on current states variablesand applied action is �Extremely Randomized Trees� [52].

Extremely randomized trees is a method for supervised classi�cation andregression problems. The method essentially consists of randomizing stronglyboth attributes (input variables) and the choice of cut-points while splittinga tree node. In the extreme case, it builds totally randomized trees whosestructures are independent of the output values of the learning sample. Thestrength of the randomization can be tuned to problem speci�cs by the ap-propriate choice of a parameter.


CONTROL STRATEGY

The Extra-Trees algorithm builds an ensemble of unpruned decision orregression trees according to the classical top-down procedure. The maindi�erences with other tree-based ensemble methods are that it splits nodesby choosing cut-points fully at random and that it uses the whole learningsample to grow the trees.

The output of the method is a target variable that de�nes the supervisedproblem. When the output of the method is categorical it is said to bea classi�cation problem. When the output is numerical it is said to be aregression problem. The latter is the case in MPC-based control strategies.

Extremely randomized trees are then used for two purposes:

1) To �nd the function f that models the power system dynamics.

2) To estimate the next state of the system x(n+1), when for the currentstate x(n) when an action u(n) is applied.

The function f is computed using a data set or learning set. This learningset is obtained by running several disturbances in the power system andapplying random actions. The generation of the learning set is as follows:at the �rst simulation time step, the states variables, x(n), are stored anda random action, u(n), is applied. In the second time step the new statesvariables are the result of the applied action. This new states variables arex(n + 1). This three variables conform the �rst sample of the learning set{x(n), u(n), x(n+1)}. A new random action is applied. In the third step thenew states variables, x(n+2), will be the result of applying the random actionu(n + 1) to the state x(n + 1) and the second sample of the learning set isstored {x(n+1), u(n+1), x(n+2)}. The process is repeated until the desiredsimulation time is completed. As the number of samples grows, the learningset becomes richer and the function f more accurate. High accuracy of fcan be achieved if the simulated disturbance perturbs the states variables.

Trees are generated once the learning set is obtained. The estimation ofthe next state of the power system for the current state and applied actioncan then be computed.

5.3 Numerical Example

For testing purposes a very simple power system is considered. It is a radialsystem with one generator, a transmission corridor with a VSC-HVDC trans-mission and parallel ac transmission lines, and an in�nite bus. Figure 5.1

5.3. NUMERICAL EXAMPLE 57

shows the power system. The VSC-HVDC is represented by the InjectionModel.

VSC-HVDC

1 23 45

L1-1

L1-2

L2-1

L2-2

Figure 5.1: Test Power System

A cost function is calculated by using the energy index. Because there isonly one generator, the energy index becomes:

DE(x(n)) = (ω(n))2 (5.13)

and the cost function is de�ned by:

c(x(n)) =

{D(x)−Dmin if ∥δ(n)∥ < δmax

cpen otherwise(5.14)

with δmax = 180◦ and cpen = 100A three phase short circuit at Bus 5 is applied. The short circuit is

cleared after 120 ms. There is no disconnection of ac transmission lines.The MPC-based control strategy is compared with another nonlinear

control strategy in order to have a better understanding of its performance.The second control strategy is the CLF-based control strategy described inSection 4.2.2.

∆uci1 = k1ω sin(δ) (5.15)

∆uci2 = −k2ω cos(δ) (5.16)

where i is the converter connected to Bus 3.

Performance without dc power control nor ac voltage control

The �rst objective is to verify that the MPC-based control strategy workscorrectly. For this reason the VSC-HVDC transmission does not have con-trol on the dc power, the reactive power or the ac voltage. The control of


CONTROL STRATEGY

the VSC-HVDC transmission is exclusively for damping. Figure 5.2 and Fig-ure 5.3 show the control diagrams. POD is only applied in converter i (Bus3).

1ci ou

1∆

ciu 1

1seT s+ 1ci

u

+

Figure 5.2: Control of the real partuci1

2∆

ciu 1

1seT s+ 2ci

u

+

2ci ou

Figure 5.3: Control of the imagi-nary part uci2

In the MPC two di�erent possible actions are de�ned, i.e., P = 2. Thepossible actions are ∆ucimax and ∆ucimin , which in this case are +0.05 p.u.and -0.05 p.u. respectively. These possible actions are applied to ∆uci1 and∆uci2. The number of successive actions is also set to two, i.e., H = 2.

0 0.5 1 1.5 2 2.5 325

30

35

40

45

50

Time ( s )

δ(

deg

)

No controlMPCCLF

Figure 5.4: POD by controlling the real part, uci1

For both control strategies (MPC-based and CLF-based) two cases aresimulated. The �rst case uses the input of the control of the real part of


the voltage inserted by the VSC (Figure 5.2). The simulation results can beobserved in Figure 5.4.

The second case uses the input of the control of the imaginary part ofthe inserted voltage (Figure 5.3). Figure 5.5 shows the result.

0 0.5 1 1.5 2 2.5 325

30

35

40

45

50

Time ( s )

δ(

deg

)

No controlMPCCLF

Figure 5.5: POD by controlling the imaginary part, uci2

Four main observations are worth to mention.

1) MPC-based control strategy works and the power system is damped.

2) There is a ripple remaining in the system and the power system doesnot return to the initial equilibrium point.

3) During the transient (�rst swing) both control strategies give aboutthe same result. Although after 1 s, CLF has a better performancebecause the system reaches an equilibrium point (no ripple as in theMPC strategy).

4) Both control strategies have a better performance when the POD signalis connected to the control of the imaginary part of the voltage insertedby the VSC, i.e., uci2.


CONTROL STRATEGY

The ripple observed when using the MPC-based control strategy is dueto bang-bang control. The possible actions that can be applied by the MPCstrategy are either ∆ucimax or ∆ucimin . Figure 5.6 shows the output of thecontrol in Figure 5.3.

0 0.5 1 1.5 2 2.5 30.34

0.36

0.38

0.4

0.42

0.44

0.46

Time ( s )

u2

(p.u

.)

MPCCLF

Figure 5.6: Controllable variable uci2

It can be seen that after 1 s, when the large oscillations are damped, thecontrol begins to chatter. This behavior can be explained as follows: Thecost function uses the energy index (5.13), which is a function of the speed.The oscillations of the speed are dramatically damped after 1,2 s. At thistime the MPC strategy selects to apply ∆ucimin . This change makes thespeed of the generator to increase a little, enough to make the MPC strategyto choose ∆ucimax in the next step. The gap-time between steps is 30 ms.In 30 ms the VSC has not reached ∆ucimin and already receives the order ofchanging to ∆ucimax . The VSC inserts ∆ucimax for three consecutive stepswhen the generator speed decreases again. The MPC strategy selects onceagain the action ∆ucimin and the loop starts over. Since ∆uci2 has a smallpositive mean value, uci2 does not return to the pre-fault operating point.

One extra simulation was done for the MPC-based control strategy. Inthis case the objective was to observe the impact of having more possible


actions, P , and more successive applied actions, H. The following threecases were simulated:

1) P = 2, {∆ucimax , ∆ucimin}. Two successive actions, H = 2. This isthe case presented above.

2) P = 5, {∆ucimax , ∆ucimax/2, 0,∆ucimin/2 , ∆ucimin}. Two successiveactions, H = 2.

3) P = 3, {∆ucimax , 0 , ∆ucimin}. Three successive actions, H = 3.

Simulation results in Figure 5.7 show that there is no di�erence in the�rst swing. All cases behaved similarly. A di�erence becomes apparent oncelarge oscillations are damped. Although all cases still show ripple, when thenumber of possible actions is greater than two, P > 2, the ripple is reduced.There is no relevant di�erence between the second case and the third case.

0 0.5 1 1.5 2 2.5 332

34

36

38

40

42

44

46

48

50

Time ( s )

δ(

deg

)

P=2 H=2P=5 H=2P=3 H=3

Figure 5.7: Damping comparison for di�erent combinations of P and H

Figure 5.8 shows the output of the controller in Figure 5.3. From this�gure it is possible to see that less ripple is obtained for cases with P > 2because the order given by the MPC is now changing between 0 and ∆ucimax .


CONTROL STRATEGY

For the case P = 2 the order given is changing between ∆ucimax and ∆ucimax

as described above.

0 0.5 1 1.5 2 2.5 30.34

0.36

0.38

0.4

0.42

0.44

0.46

Time ( s )

uci2

(p.u

.)

P=2 H=2P=5 H=2P=3 H=3


Performance under dc power control and ac voltage control

Control of the ac voltage at both ends of the VSC-HVDC transmission andof the active power are now included in the Injection Model. Figures 5.9and 5.10 show the control block diagrams. The input of the POD signalis available for both controls uci1 and uci2 after the PI controller for acvoltage and active power, respectively. However, as observed from the resultswithout PI-control, better results are obtained when the POD signal is usedin the control of the imaginary part of the voltage inserted by the VSC.Therefore, only ∆uci2 is used.

The power system shown in Figure 5.1 is used again. A short circuit atBus 5 is applied. The short circuit is cleared after 120 ms by disconnectingthe ac lines L1-1 and L2-1. Simulation results are shown in Figure 5.11.

As in previous simulations, both control strategies perform similarly forthe �rst swings. After the transient (t > 1, 2 s) the MPC-based control


1ciuUK

s

+-refU

iU

1

1 seT s+ixpu

+

1∆ ciu

Figure 5.9: Control ac Voltage

2ciu

+-refP

siP

+

iypuPK

s

2∆ ciu

1

1 seT s+

Figure 5.10: Control dc Power

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 536

38

40

42

44

46

48

50

52

Time ( s )

δ(

deg

)

No PODMPCCLF

Figure 5.11: Comparison damping


CONTROL STRATEGY

strategy shows the same pattern as the CLF strategy, but again there isripple.

In this case, small oscillations can be seen in the MPC strategy. Fig-ure 5.12 illustrates the controllable variable uci2, which increases due to theactive power control of the VSC-HVDC transmission. However no ripple isobserved for the MPC-based control strategy. The reason is that the strategyis always selecting the action ∆ucimin after 1.2 s.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

Time ( s )

uci2

(p.u

.)

MPCCLF


The MPC-based control strategy has shown that works very well forthe transient period (�rst seconds after the fault) and results showed sameperformance when compared with the other control strategy. However theripple produced by the chattering e�ect of the MPC strategy is clearly adrawback. One can conclude that the chattering behavior is directly relatedto the de�ned set of possible actions, but as it was shown in Figure 5.7,di�erent sets of P did not help in avoiding the ripple.

As a countermeasure a small change could be introduced in the MPC-based control strategy. Cost function calculations de�ne the control actionthat should be applied, therefore a condition in the cost function can helpto reduce or eliminate ripple. A possible condition is that the control action

5.4. COMMENTS 65

selected by the cost function can only be applied if the energy index, (5.13),is greater than a certain value, otherwise no action is applied. When thecondition is not ful�lled, the electromechanical oscillations must be rathersmall and the inherent damping of the power system will help them to vanish.

5.4 Comments

This application of MPC to VSC-HVDC transmission to provide dampingin a power system is a collaborative work performed with the System andModeling Research Unit of the University of Liège.

This application will be studied in a larger power system. The study willinclude also a comparative analysis of all three transient stability indices. Apaper with the outcome of this investigation will be submitted to a journal.

Chapter 6

Conclusions and Future Work

This �nal chapter summarizes the main conclusions obtained in this thesis.It also suggests further research that can be pursued as a continuation of thisresearch.

6.1 Conclusions

This project has focused on four main topics:

1) derivation of control strategies,

2) analysis of the impact of local and global measurements in controlstrategies,

3) interaction of VSC-HVDC transmission controllers with other nearbycontrollers, and

4) improvement of power �ow conditions.

The following conclusions can be drawn for each main topic, besides one �nalconclusion about VSC-HVDC transmission.

1. Control Strategies

• Four control strategies were derived. Three di�erent theoreticalwork frames were used in the derivation, namely nonlinear control(Control Lyapunov Function (CLF)), modal analysis and modelpredictive control (which is also nonlinear control). CLF was

67

68 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

used in the derivation of two control strategies. CLF-based andMPC-based strategies showed that they signi�cantly enhancedtransient stability in power systems. All control strategies showedtheir e�ectiveness in the increase of damping of electromechanicaloscillations. Although the nonlinear-based control strategies weremainly aimed to enhance transient stability, they also showedtheir capability to damp electromechanical oscillations.

• With the exception of the CLF-based control strategy derivedfrom SMIB, the control strategies resulted in controlling only oneof the two controllable variables of VSCs. These control strategieswere used as input to either modulate active power or reactivepower. For both cases the control strategies allowed VSC-HVDCtransmissions to increase damping.

2. Local and remote information

• Remote signals usually contain more information than local sig-nals and therefore become more appropriate for the control strate-gies. However in some cases the use of local signals can result inequal or even better results than the use of remote signals. Thiswas seen when the VSC-HVDC transmission was connected indi�erent locations in a power system. This is directly related tothe residue (controllability and observability) obtained from theapplication of modal analysis.

• The selection of the most relevant generators for SIME methodthrough modal analysis showed to be a reliable alternative whenit is not possible to have access to the measurements from allgenerators in a power system.

3. Controllers interaction

• When the VSC-HVDC transmissions operates in reactive powercontrol mode, some interactions with AVR controllers in genera-tors nearby the VSCs can be seen. This was the result of losingreactive power in the area (a generator trip). It was shown that,in these cases, the system becomes weak (or it is already a weakarea under undisturbed conditions), the VSC-HVDC transmission

6.2. FUTURE WORK 69

should operate in voltage control mode or increase the dc power�ow.

4. Power �ow conditions

• VSC-HVDC transmissions can greatly improve post-fault condi-tions in the power system by controlling power �ows. Controlstrategies based on voltage magnitudes measured at connectionnodes of VSCs or reactive power conditions at the nearby busesresulted in an improvement of the post fault stability margins.

5. VSC-HVDC - general conclusion

• It was shown that VSC-HVDC transmissions with the appropriatecontrol strategy are able to improve stability margins in power sys-tems. It has been shown that to increase power transfer capacityin the transmission grid by installing either a new ac transmis-sion line or a VSC-HVDC transmission, the latter signi�cantlyenhances transient and voltage stability, improves the dampingof electromechanical oscillations and contributes towards the e�-cient utilization of the transmission system.

6.2 Future Work

The following is a list of ideas that can be considered for future work

• Power system stability improvement can be achieved when controlstrategies are used for modulation of active power and reactive powerin VSCs. Future work could determine an adaptive controller that�nds the optimum relation between how much active power and howmuch reactive power should be modulated in the event of a distur-bance. This optimum relation would translate into an improvement ofstability margins.

• An initial approach to �nd a relation between SIME method and modalanalysis was carried out. However this relation was found for a speci�coperating point only (in the case of modal analysis) and for some spe-ci�c faults (in the case of the SIME method). A deeper analysis and amore general relation between could be developed.

70 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

• An approach to coordinate wind farms and POD from the VSC-HVDCtransmission was presented. The proposed coordination basically con-sidered the wind power production and the gain of the control strategyfor active power. A larger scale coordination of the controllers can bepursued. Coordination wind power production, POD control of windpower, POD control and voltage control of the VSC-HVDC transmis-sion can be included in a more general approach.

• MPC showed to be e�ective for damping of electromechanical oscilla-tions. Future work in this area can focus on how to avoid the smallripple that resulted after having damped the �rst swings occurringafter a disturbance.

• MPC assumes that the control has access to all state variables in thepower system. An interesting improvement in the MPC-based controlstrategy is to make an analysis on how to reduce the number of statesvariables without sacri�cing damping e�ectiveness.

• It would be interesting to perform an analysis of the e�ectiveness of thederived control strategies in a real time digital simulator (for instanceOPAL-RT simulator available at KTH). Moreover a further analysisof the issues found in the reactive power control mode in weak areascould also be done in the simulator.

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