coordinated control of power systems with hvdc …...dissertation eth no. 21723 coordinated control...

Dissertation ETH No. 21723

Coordinated Control of Power Systems

with HVDC Links

A dissertation submitted toETH Zurich

for the degree ofDoctor of Sciences

presented by

Alexander Fuchs

Dipl.-Ing., Dresden University of TechnologyM.S., University of Texas at Austin

born 29.04.1982citizen of Germany

accepted on the recommendation of

Prof. Dr. Manfred Morari, examinerProf. Dr. Göran Andersson, co-examiner

Prof. Dr. Ian Hiskens, co-examiner

2014

Coordinated Control of Power Systems

with HVDC Links

Alexander Fuchs

Dissertation ETH Zürich No. 21723

Abstract

Today’s power systems are large networks of electrical energy sources and load compo-nents, connected over long distances mainly through alternating current (AC) trans-mission and distribution grids. This basic structure currently undergoes significantchanges to ensure the stability, availability and sustainability of the future electricalenergy infrastructure. As a result, power system operation and development faces sev-eral challenges, including volatile injections from distributed renewable energy sources,variable energy demands and tighter security margins.

The development of high voltage direct current (HVDC) links provides an alternativesolution for the efficient and flexible transmission of electrical energy, that can supportthe future power system along multiple dimensions. First, during normal operation,HVDC links provide an increased controllability of the AC power system’s operatingpoint. Power system operators can use HVDC links to optimize the AC power flow inthe network in order to avoid congestions and to achieve an economic gain. Secondly,during dynamic situations, HVDC links can be used by a fast automatic grid controllerto support the power system’s transient stability.

This thesis studies the control of HVDC injections in power systems during dynamicscenarios. Coordinated HVDC control has a large potential for the dynamic perfor-mance of power systems, for instance by increasing the damping of power oscillations,but is currently not exploited in a systematic way. The aim is to develop a frameworkfor power system control through HVDC transmission links. Starting with results forclassical AC networks, the thesis presents power system models, operation approachesand network planning methods in the context of dynamically controlled HVDC links.

The modeling of power systems with HVDC links has to incorporate several physi-cal and operational constraints imposed by the HVDC links and the surrounding ACnetwork. A characterization of the resulting constraints on the HVDC injections isparticular important if the HVDC links are to be used for dynamic power system con-trol. Classical capability charts of HVDC links assume a strong AC network connectedto a single HVDC link with diminishing impedances in the AC transmission system.This results in simple active and reactive power bounds on the HVDC injections. Inthis thesis, it is shown that this concept can be generalized to constraint sets that arecharacterized without these simplifying assumptions. The constraint set of admissible

iv Abstract

HVDC injections is found to be convex, but may deviate strongly from the classicalcapability charts.

Most operation schemes for HVDC links use constant power references for the HVDCconverters. The references are typically updated in hourly or 15-minute-intervals ac-cording to global load flow adjustments of the grid operator. One existing control ap-proach for frequency oscillation damping uses local converter measurements and a linearcontroller to manipulate the active HVDC power transmission. This thesis presents ageneral strategy for a coordinated power system control using HVDC links and globalpower system measurements. First, all available models, constraints and objectivesare summarized in a power system control problem. The corresponding optimizationis then repeatedly carried out over a time horizon to determine the HVDC injectionsthat best support the dynamic power system performance. This approach, known asModel Predictive Control (MPC), can react flexibly to disturbances and changes of thepower system operating conditions. It is shown that HVDC links with a global MPCbased grid controller effectively damp power system oscillations, keep the system insynchronism and accompany global power system set point changes.

The effectiveness of HVDC links for power system control depends on the locationof the HVDC links in the surrounding AC network. The primary criterion for theselection of new HVDC locations concerns the mitigation of load flow congestions andthe potential economic gain. In this thesis, it is shown how these aspects can becombined with a novel criterion evaluating the suitability of a given HVDC location fordynamic power system control. A performance measure is introduced to quantify thepower system controllability under constraints and general disturbances. To allow theefficient evaluation, the performance measure is reformulated as a semidefinite program.The resulting placement algorithm ranks HVDC locations according to the performanceimprovement they bring to a given power system.

In conclusion, even power systems with a relatively small share of HVDC transmissioncapacity can have a large benefit from the coordinated control of the HVDC injections.The simulation studies show a significant improvement of the dynamic power systemperformance after disturbances, both in small benchmark systems and a large modelof the European power system.

Zusammenfassung

Die heutigen Elektrizitiätsversorgungssysteme sind grosse Netze von elektrischen Erzeu-gungs- und Lastkomponenten die über lange Entfernungen durch wechselstrombasierteÜbertragungs- und Verteilnetze verbunden sind. Diese Grundstruktur unterliegt der-zeitig starken Änderungen um die Stabilität, Verfügbarkeit und Nachhaltigkeit derzukünftigen Energieversorgung sicherzustellen. Dabei ergeben sich für den Netzbetriebvielseitige Herausforderungen, zum Beispiel durch unstetige Einspeisungen von erneu-erbaren Energiequellen, veränderliche Nachfrageprofile und engere Sicherheitsmargen.

Die Entwicklung der Hochspannungs-Gleichstrom-Übertragung (HGÜ) bietet eine al-ternative Lösung für die effiziente und flexible elektrische Energieübertragung, die daszukünftige Elektrizitiätsversorgungssystem in mehrere Richtungen unterstützen kann.Erstens bietet eine HGÜ-Verbindung im Normalbetrieb dem Netzbetreiber zusätzlicheSteuerbarkeit des Lastflusses im Wechselstromnetz. Die HGÜ-Verbindung kann somitdazu beitragen Netzengpässe zu vermeiden und den wirtschaftlichen Gewinn zu erhö-hen. Weiterhing kann eine HGÜ-Verbindung in dynamischen Situationen durch einenautomatischen Netzregler genutzt werden um die transiente Netzstabilität zu unter-stützen.

Die vorliegende Arbeit untersucht die Regelung von HGÜ-Einspeisungen in Elektri-zitiätsversorgungssystemen während dynamischer Situationen. Die koordinierte HGÜ-Regelung hat ein grosses Potential für das dynamische Verhalten von Energienet-zen, zum Beispiel durch eine verbesserte Dämpfung von Leistungsoszillationen, aberwird gegenwärtig nicht systematisch genutzt. Das Ziel ist die Entwicklung einer HGÜ-basierten Regelungsstrategie für Elektrizitiätsversorgungssysteme. Beginnend mit klas-sischen Methoden für Wechselstromnetze entwickelt diese Arbeit Modelle, Betriebs-strategien und Planungsmethoden für Energienetze mit dynamisch geregelten HGÜ-Verbindungen.

Die Modellierung von Elektrizitiätsversorgungssystemen mit HGÜ-Verbindungen er-fordert die Berücksichtigung mehrerer technischer und betriebsbedingter Beschrän-kungen die sich durch die HGÜ-Verbindungen und das umgebende Wechselstrom-netz ergeben. Eine Charakterisierung der resultierenden Bedingungen an die HGÜ-Einspeisungen ist besonders wichtig wenn die HGÜ-Verbindungen zur dynamischenNetzregelung verwendet werden. Klassische Leistungsdiagramme der HGÜ-Verbindun-

vi Zusammenfassung

gen basieren auf der Annahme eines starken Wechselstromnetzes das mit einem einzel-nen HGÜ-Umrichter verbunden ist. Daraus ergeben sich einfache Schranken der mög-lichen Wirk- und Blindleistungseinspeisungen. In dieser Arbeit wird gezeigt wie diesesKonzept ohne diese vereinfachenden Annahmen verallgemeinert werden kann. Es wirdgezeigt dass die resultierende Menge der möglichen HGÜ-Einspeisungen konvex ist undvon den klassischen Leistungsdiagrammen stark abweichen kann.

Die meisten Betriebsstrategien für HGÜ-Verbindungen verwenden konstante Wirk-und Blindleistungseinspeisungen. Die Referenzwerte werden typischerweise stündlichoder in 15-Minuten-Intervallen mit globalen Lastflussänderungen durch den Netzbetrei-ber angepasst. Ein bekannter Ansatz zur Dämpfung von Frequenzschwingungen ver-wendet lokale Messwerte an den Umrichtern und einen linearen Regler zur Anpassungder HGÜ-Wirkleistungen. Diese Arbeit entwickelt eine allgemeine Strategie zur koordi-nierten Regelung von Elektrizitiätsversorgungssystemen mit HGÜ-Verbindungen undglobalen Messwerten. Zuerst werden alle Modelle, Beschränkungen und Zielstellungenin einem Netzregelungsproblem zusammengefasst. Anschliessend wird die zugehörigeOptimierung wiederholt über einen Zeithorizont durchgeführt um die HGÜ-Einspeisungzu bestimmen die das dynamische Netzverhalten am besten unterstützt. Diese Herange-hensweise, genannt modellprädiktive Regelung, kann flexibel auf Störungen und Ände-rungen des Netzbetriebes reagieren. Es wird gezeigt das HGÜ-Verbindungen mit einemglobalen modellprädiktiven Regler gezielt Leistungsoszillationen dämpfen, die Synchro-nizität des Netzes erhalten und Änderungen des Betriebspunktes begleiten können.

Die Möglichkeiten von HGÜ-Verbindungen zur Netzregelung hängen von ihrem Ortim umgebenden Wechselstromnetz ab. Das Hauptkriterium zur Auswahl von neuenHGÜ-Verbindungen bewertet die Beseitigung von Lastflussengpässen und den mögli-chen wirtschaftlichen Gewinn. Diese Arbeit zeigt wie diese Aspekte durch ein neuesKriterium ergänzt werden können, das den Nutzen der HGÜ-Verbindung für die Netz-regelung quantifiziert. Ein Leistungsmass bestimmt die Netzregelbarkeit unter Berück-sichtigung von Beschränkungen und allgemeinen Störungen. Zur effizienten Bewert-barkeit wird das Leistungsmass als semidefinites Optimierungsproblem formuliert. Derresultierende Algorithmus ordnet die Platzierungen der HGÜ-Verbindungen nach ihremNutzen für die Netzregelung.

Zusammenfassend können auch Elektrizitiätsversorgungssysteme mit einem kleinenAnteil an HGÜ-Verbindungen einen grossen Vorteil aus der koordinierten Regelungder HGÜ-Einspeisungen ziehen. Die Simulationsuntersuchungen zeigen eine signifikan-te Verbesserung des dynamischen Verhaltens nach Netzstörungen, sowohl mit kleinenPrüfsystemen als auch mit einem grossen Modell des Europäischen Netzes.

Dedicated to my family.

Acknowledgements

The completion of the PhD and the writing of this thesis was a great experience thatinvolved the help and support from many people.

My greatest gratitude and appreciation go to my advisor Prof. Manfred Morari whohas provided support throughout all of my PhD studies. His invaluable advice, theconstructive discussions and the encouragement to pursue individual project directionscreated a unique academic working environment. It was a great privilege to work withyou at the Automatic Control Laboratory!

I am equally indebted to Prof. Göran Andersson and Prof. Ian Hiskens who agreedto be my co-examiners, but also shared the broad expertise of themselves and theirresearch groups.

Especially in the beginning of the PhD, my work was greatly enhanced by two people,Sébastien Mariéthoz and Mats Larsson. In many long discussions they helped patientlyto get started with the research at the intersection of automatic control and powersystems. I also thank Walther Sattinger and Turhan Demiray for the data exchangeand programming support to create realistic power system simulations.

The HVDC project formed the core of this research and was greatly supported, bothfinancially and intellectually, by Swisselectric Research and ABB Corporate Research,Switzerland. I thank Markus, the fellow PhD student on this project, for the long-lasting and fruitful collaboration.

The work would have been much less fun without all the inspiring people broughttogether at the Automatic Control Laboratory. Daniel, Colin, Miroslav, Melanie, Hel-fried, Philipp, George, Joe, Martin, Claudia, Thomas, Maryam, Roy, Bart, Stefan,Robin, Peyman, Robert and everybody else - it was great to work, learn and spendtime with all of you. A special thanks goes also to Alice, Martine, Alain and Markusfor their help with the administration at ETH and beyond.

Finally, all my love goes to Kerstin, Liam, Andrin, Jannik and Emilia, for everything.

Alexander FuchsZürich, 2014

Contents

Abstract iii

Zusammenfassung v

Acknowledgements ix

1 Introduction 1

1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Overview and contributions . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Models and constraints of power systems with HVDC links . . . 3

1.2.2 Power systems control through HVDC links . . . . . . . . . . . 4

1.2.3 Placement of HVDC links in power systems . . . . . . . . . . . 5

1.2.4 Outlook to distributed grid control . . . . . . . . . . . . . . . . 6

2 Models and constraints of power systems with HVDC links 9

2.1 Power system modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Power system network equations . . . . . . . . . . . . . . . . . . 9

2.1.2 Power system dynamic equations . . . . . . . . . . . . . . . . . 12

2.1.2.1 General dynamics . . . . . . . . . . . . . . . . . . . . . 12

2.1.2.2 Simple power system dynamics . . . . . . . . . . . . . 12

2.2 HVDC constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Origin of HVDC constraints . . . . . . . . . . . . . . . . . . . . 14

2.2.1.1 Equality constraints . . . . . . . . . . . . . . . . . . . 15

2.2.1.2 Inequality constraints . . . . . . . . . . . . . . . . . . 15

2.2.1.3 Constraint parameterization . . . . . . . . . . . . . . . 16

2.2.2 HVDC terminal at a constant voltage bus . . . . . . . . . . . . 17

2.2.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2.2 Constraint derivation . . . . . . . . . . . . . . . . . . . 17

2.2.3 HVDC terminal in an AC network . . . . . . . . . . . . . . . . 20

2.2.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.3.2 Constraint derivation . . . . . . . . . . . . . . . . . . . 23

xii Contents

2.2.4 HVDC links in an AC network . . . . . . . . . . . . . . . . . . . 24

2.2.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.4.2 Constraint derivation with variable DC voltage . . . . 26

2.2.4.3 Constraint derivation with coupled AC power injections 28

2.2.4.4 Constraint derivation with coupled DC powers . . . . . 32

2.2.5 Constraint derivation from local power system parameters . . . 352.2.5.1 HVDC terminal at a constant voltage bus . . . . . . . 35

2.2.5.2 HVDC terminal in an AC network . . . . . . . . . . . 35

2.2.5.3 HVDC links in an AC network . . . . . . . . . . . . . 37

3 Power system control through HVDC links 41

3.1 Control system overview . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Grid controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.1 Power system performance objective . . . . . . . . . . . . . . . 43

3.2.2 Measurement and estimator system . . . . . . . . . . . . . . . . 45

3.2.3 Local damping control . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.4 Coordinated Model Predictive Control . . . . . . . . . . . . . . 473.2.4.1 Linear power system prediction model . . . . . . . . . 47

3.2.4.2 MPC problem formulation . . . . . . . . . . . . . . . . 49

3.3 Simulation of power system benchmark systems . . . . . . . . . . . . . 53

3.3.1 Overview of the simulation examples . . . . . . . . . . . . . . . 53

3.3.2 Single machine infinite bus system . . . . . . . . . . . . . . . . . 54

3.3.2.1 Power system model . . . . . . . . . . . . . . . . . . . 54

3.3.2.2 HVDC model . . . . . . . . . . . . . . . . . . . . . . . 583.3.2.3 Simulation of an AC line fault . . . . . . . . . . . . . . 58

3.3.3 Two area power system . . . . . . . . . . . . . . . . . . . . . . . 64


3.3.3.2 HVDC model . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.3.3 Simulation of an AC line loss . . . . . . . . . . . . . . 65

3.3.3.4 Sensitivity to state constraints . . . . . . . . . . . . . 65

3.3.3.5 Sensitivity to measurement delay . . . . . . . . . . . . 713.3.4 European power system . . . . . . . . . . . . . . . . . . . . . . 77


3.3.4.2 HVDC model . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.4.3 Implementation of the network simulation . . . . . . . 80

3.3.4.4 Loss of a large power plant . . . . . . . . . . . . . . . 80

3.3.4.5 Loss of a large load . . . . . . . . . . . . . . . . . . . . 80

3.3.4.6 Loss of an AC line . . . . . . . . . . . . . . . . . . . . 833.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Contents xiii

4 Placement of HVDC links in power systems 91

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.1.1 Planning of power systems with HVDC links . . . . . . . . . . . 914.1.2 General performance measures for actuator selection . . . . . . 924.1.3 Power system performance measures for HVDC selection . . . . 92

4.2 Performance evaluation of actuators . . . . . . . . . . . . . . . . . . . . 934.2.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . 934.2.2 Problem formulation using LMIs . . . . . . . . . . . . . . . . . 944.2.3 Worst case performance formulation . . . . . . . . . . . . . . . . 944.2.4 Handling of equality constraints . . . . . . . . . . . . . . . . . . 954.2.5 Handling of input inequality constraints . . . . . . . . . . . . . 964.2.6 LMI formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3 Optimal placement of HVDC links . . . . . . . . . . . . . . . . . . . . 984.3.1 HVDC placement as multi-objective optimization . . . . . . . . 984.3.2 Power grid control using HVDC . . . . . . . . . . . . . . . . . . 994.3.3 Performance assessment of HVDC locations . . . . . . . . . . . 1004.3.4 Placement algorithm for a single HVDC link . . . . . . . . . . . 1014.3.5 Placement algorithm for multiple HVDC link . . . . . . . . . . 102

4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.4.1 Illustration of the LMI based performance evaluation . . . . . . 1044.4.2 Placement of HVDC links in the two area power system . . . . 1044.4.3 Placement of HVDC links in the European power system . . . . 119

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5 Outlook 127

5.1 Problems with centralized grid control . . . . . . . . . . . . . . . . . . 1275.2 Towards distributed grid control . . . . . . . . . . . . . . . . . . . . . . 129

A Appendix 133

A.1 Norm dependent scaling of discs . . . . . . . . . . . . . . . . . . . . . . 133A.1.1 Proof of the number of boundary elements . . . . . . . . . . . . 133A.1.2 Proof of the shape of the boundary elements . . . . . . . . . . . 136A.1.3 Proof of the convexity . . . . . . . . . . . . . . . . . . . . . . . 143

Bibliography 145

Publications 153

Curriculum vitae 155

1 Introduction

1.1 Background and motivation

The electrical power grid is one of the most important engineering achievements of the20th century. The conversion of primary energy to electrical energy allows, through theelectrical grid infrastructure, to connect power sources with electrical load devices overlong distances. The availability of electrical energy is essential for virtually all aspectsof daily life, including the supply of water, heating, food, medical care, transport andeconomic infrastructure. Stability and reliability of the electrical power grid is thereforecritical for the well-being of human society.

The power grid in today’s form has its origins more than a century ago. Duringthe 1890s the rivalry between two approaches to the transmission of electrical energy,alternating current (AC) and direct current (DC), became known as the War of the

Currents [Fai12]. Three phase AC transmission prevailed and has ever since dominatedthe development of the electrical power grid throughout the world until today. A deci-sive factor was the convertibility of AC power to high voltage levels using transformers.This greatly reduced the transmission losses compared to DC power, which could atthat time not be directly converted to high voltage levels [Ree12].

The development of point-to-point high voltage direct current (HVDC) connectionscontinued, following the basic structure of a pair of AC-DC-converters located at twobuses in a surrounding AC grid with a DC link in between. The first commercial HVDClink was installed in 1954 between the Swedish coast and the island of Gotland [Ada12],with converter switches based on mercury arc valves. HVDC gained increased interestfrom the 1970s onwards, when semiconductor technology became available for the de-velopment of new efficient converter systems. Today’s HVDC links can be grouped intotwo categories, according to the converter technology used. Line-commutated converterHVDC use thyristors to perform converter switches, providing one degree of freedomto select the injections at the AC terminals. Voltage-source converter HVDC (VSC-HVDC) use insulated-gate bipolar transistors to perform converter switches, providingtwo degrees of freedom to select the injections at the AC terminals [Pad90].

The comparison of AC and DC technology as transmission solution with a given

2 1 Introduction

power rating includes economic and operational aspects [Ada12].

From an economic perspective, HVDC connections require a larger investment forthe converter systems at the end points of the DC link. However, since the power istransmitted as DC current, no reactive power losses occur in the transmission link. Incomparison, AC connections have a smaller investment for the end point connection,but require additional reactive power compensation and cable material, increasing thecost per kilometer. The economic break-even between AC and DC connections occursat roughly 100 to 200 kilometer, depending on the power rating, the voltage level,the HVDC technology, the location of the connection and the general energy pricelevel [Laz05]. As a result, HVDC connections were often selected as solution for longtransmission distances [SCSM06], the connection of offshore windfarms [GB13] , islands[AHL+99], or coastal links across seas [NJV+94].

From an operational perspective, HVDC links have the ability to connect otherwisedisconnected asynchronous AC grids. This increases the flexibility to support either ACgrid using bi-directional external infeeds from the HVDC link [Ils94]. An operationaldifficulty is the behavior of HVDC links during faults. Unlike AC currents, DC currentshave no natural zero crossing and must be forced to zero during the opening of aconnection [Fra11]. Further criteria for the selection of AC or DC technology concernsocial and environmental aspects, such as the requirement to put new transmissionlinks underground. The ongoing development towards a hybrid power grid with bothAC and DC components aims at exploiting the advantages of both technologies in orderto maximize the efficiency, reliability and flexibility of the future energy infrastructure[VHG10,CKA13b].

A feature of HVDC technology that is little exploited in today’s grid operation is itspotential as dynamic control device to improve the power system performance duringtransients. A VSC-HVDC link can independently control the active and reactive powerinjected into the surrounding AC grid. This increased flexibility is already used bythe network planner quasi-statically to optimize power flows between generation andload centers of the AC network [HG08, RS08]. However, the response time of HVDCconverters to changes in the injection references is sufficiently fast to also supportthe network during critical system transients due to faults, the loss of power systemcomponents, inter-area oscillations, or variable injections of renewable energy sources.

Dynamic grid controllability becomes more important as the power system operatescloser to its security margins and is particularly valuable during stress scenarios. It istypically ensured by control actions at the power grid’s synchronous generators throughdroop controller, automatic voltage regulators (AVR) and power system stabilizers(PSS) [Kun93]. In addition, flexible AC transmission system (FACTS) devices provideadjustable reactive power compensation at selected buses in the AC grid [KSS07]. The

1.2 Overview and contributions 3

potential dynamic grid controllability from HVDC connections relies on temporaryadjustments of the converter injections into the AC grid. HVDC can provide bothlocal and global dynamic power system support since they directly effect the powerflow between remote areas [LGS08,PAMM11,FMLM11,MNNDSL13].

This thesis focuses on HVDC links as power system control devices in the presence oftransient phenomena in the surrounding power network. Specifically, it adresses threekey aspects of the dynamic operation of HVDC links:

• What are the constraints for the dynamic operation of HVDC links in AC net-

works?

• How should HVDC links be operated to best support the power system during

transients?

• Where should HVDC links be located in a given AC network for best dynamical

support?

The context and contributions related to each of these questions are summarized in thefollowing section. Throughout the thesis, unless stated otherwise, the term „HVDC“refers to the voltage source converter type (VSC-HVDC).

1.2 Overview and contributions

1.2.1 Models and constraints of power systems with HVDC links

Chapter 2 introduces the models for the simulation and control of power systems withHVDC links. It includes material from the publications [FMLM11] and [FIDM14].

The context of the modeling are dynamical phenomena in AC power grids, such aspower oscillations or electromechanical transients. A particular focus is on the dynamicoperation of HVDC links in order to support the system’s security and stability.

Section 2.1 presents a modeling framework for dynamic power system phenomena.The time scale ranges from about 100 milliseconds to a few minutes and includestypical oscillations associated with networks of synchronous generators [KRK91]. Thepresented modeling is largely based on classical power system modeling approaches[MBB08,Kun93,And13], modified to incorporate the effect of HVDC injections.

Section 2.2 studies the constraint formulation for HVDC injections during staticand dynamic scenarios in the surrounding AC grid. The origin of the constraintsare the physical bounds on the current and voltage changes at the HVDC converters.Classical HVDC bounds for grid operation are given by the PQ-chart, a symmetricbound in the complex plane spanned by active and reactive power HVDC converter

4 1 Introduction

injections [HEMN02, Kir09, BK11]. The PQ-chart assumes the AC terminal voltageto be constant, which is only true for static scenarios in very large power grids andneglects the impact of HVDC injection changes. Furthermore, no coupling between theconverters through the AC or DC system is accounted for. It is shown how to deriveimproved analytical HVDC bounds without these simplifying assumptions, in order tocapture the constraints during dynamic situations in realistic grids more accurately.The operating range of HVDC links are shown to form a convex set that can differsignificantly from the classical PQ-chart.

1.2.2 Power systems control through HVDC links

Chapter 3 presents a control framework for the stabilization of electromechanical tran-sients through the help of HVDC links. It includes material from the publications[FMLM10], [FMLM11] and [FIDM14].

The basic structure of the HVDC grid controller manipulating the active and reactivepower at the converter terminals is introduced in Section 3.1. The HVDC links are tra-ditionally controlled based on local measurements at the converter terminals [DLL03].With the growing infrastructure for wide area measurement systems (WAMS), a super-visory control layer can process measurements of the entire power system in a coordi-nated way. This supervisory layer allows the static optimization of the power system’sload flow [HG08] but can also be used to enhance the system during transients.

Section 3.2 presents an automatic grid control scheme to manipulate the AC powerinjections at the HVDC terminals. A suitable power system performance objective isselected, capturing the desired power system behavior [Kun93] while also leading toa tractable control problem formulation [Mac01]. Following a discussion of availabledata acquisition techniques, two grid control schemes, based on local and global powersystem measurements, are introduced.

The first control approach is a state of the art HVDC damping controller using lo-

cal measurements at the converter terminals [Eri08, SA93]. Related control schemesaddress the damping of inter-area oscillations using other active power system compo-nents, including flexible AC transmission systems (FACTS) and power system stabiliz-ers (PSS) [Sad06,AESMF96,Kun93].

The second grid control approach for HVDC links developed in Section 3.2 is a global

centralized model predictive control (MPC) scheme. Current grid controller schemesusing HVDC and global power system measurements include rule based adjustmentsof the power injections [LGS08] as well as a weighted HVDC coordination based onlinear quadratic gaussian (LQG) control [PAMM11]. The general MPC approach hasthe advantage to explicitly account for the principal system dynamics and constraints


of the actuator components [BBM13]. The power system dynamics, constraints andobjectives introduced in Section 2.1, Section 2.2 and Section 3.2.1 are used to formulatean MPC power system control problem with the HVDC injections as manipulatedvariables. The resulting MPC based grid controller can in principle also coordinate theHVDC links with FACTS, PSS and other active power system components. With arestricted communication structure, it can also be developed as a distributed controlscheme [MNNDSL13].

In Section 3.3, the local HVDC damping controller and the MPC based grid con-troller are demonstrated with three benchmark power systems. A simple single machineinfinite bus system [Kun93] with an HVDC link allows a closed form solution of thenetwork equations to illustrate the power system dynamics and the resulting MPCformulation. A classical four generator power system [Kun93] with an added HVDClink demonstrates the damping capabilities of inter-area oscillations in a general powersystem with an arbitrary number of generators. It also demonstrates the robustness totime delays due to the computation or communication of the control and measurementsignals. A detailed dynamical model of the European power grid [Haa06] with multi-ple exising or planned HVDC links and thousands of states demonstrates the dampingcapabilities of the grid controller in a complex power system.

1.2.3 Placement of HVDC links in power systems

Chapter 4 studies the planning problem for the selection of the location of new HVDClinks. It includes material from the publications [FM13a] and [FM13b].

The main factor for the planning and construction of new HVDC links are economicconsiderations during static scenarios [CKA13a, Laz05]. This chapter approaches theplanning problem from the perspective of dynamic power system performance. Thegoal is to select HVDC locations that provide the best performence for a grid controllermanipulating the HVDC injections as presented in Chapter 3.

The stability and dynamic controllability of the power grid is an important aspect forthe planning of power systems regarding extensions of the AC grid and active controldevices such as PSS systems. [KSS07,Vou95,CE05]. Dynamic controllability can alsoserve as partial decision criteria when investigating candidate locations for new HVDCconnections in a power grid with no or already existing HVDC links.

Section 4.1 discusses general and power system specific performance measures asin [Kun93, KSLP93] regarding the closed loop performance evaluation of power gridswith HVDC links operated by dynamic grid controllers. The approaches are comparedregarding their suitability for a range of operating conditions, power flows and powersystem disturbances [VBF91,DEK75].

6 1 Introduction

Section 4.2 formulates a performance measure to evaluate and compare constrainedactuators for general disturbances. In addition to the aspects discussed in Section4.1, the performance measure is also required to provide comparable result based on atractable computation, since it has to be evaluated for a potentially large number ofactuators and combinations thereof. A time domain integral performance measure isintroduced, evaluating the system’s trajectories for a range of disturbances and actu-ator limitations modeled as ellipsoidal constraints. The performance measure is thanreformulated a semidefinite program (SDP) using classical results of linear matrix in-equalities (LMIs) [BEFB94].

The SDP based performance measure is related to other controllability metrics, likethe trace of the controllability gramian [ZDG+96]. While such metrics do not accountexplicitly for constraints on the system inputs and the disturbances, they admit efficientalgorithms to select combinations of actuators that maximize the controllability [SL14,DG06].

Section 4.3 introduces an algorithm for the selection of HVDC locations, applyingthe dynamic performance measure introduced in Section 4.2 to systems representingpower grids with HVDC links. The dynamic performance measure is put into contextwith other aspects of power system planning as a multi-objective optimization problem[EG02]. For the placement of multiple HVDC an exhaustive and a recursive placementalgorithm is proposed, where the latter uses a heuristic to place individual links for thesake of complexity reduction.

Section 4.4 applies the algorithms from Section 4.3 to the selection of HVDC locationsin the four generator test system and the European transmission system used for thedemonstration of automatic grid control schemes in Section 3.3. It is shown how thedamping performance is successively improved through the addition of HVDC links.Additionally it is shown that the dynamic performance improvement is not dependenton the length of the HVDC link, although a connection of remote areas might be ofadvantage from an economic or a power flow perspective [CKA13a].

The approach presented in this chapter can also be used to evaluate the effectivenessof FACTS, PSS and other power system control devices or the combinations thereof.

1.2.4 Outlook to distributed grid control

Chapter 5 concludes the thesis with an outlook for future developments in the contextof power system control through HVDC links.

The centralized coordinated control approach has strong requirements for the com-munication and computation infrastructure. Section 5.1 discusses the developments


needed to meet each of these requirements, and therefore achieve a fully coordinatedHVDC control. As a possible compromise, Section 5.2 outlines a decomposition struc-ture for a distributed control approach, with weaker requirements for the data andinformation exchange.

2 Models and constraints of power

systems with HVDC links

This chapter introduces the basic power system model for the study of electromechan-ical transients. The dynamic AC power system model uses mostly standard nota-tion [MBB08,Kun93], adjusted to incorporate HVDC links with variable injections. Adynamic capability chart is derived to characterize the set of admissible HVDC injec-tions.

The models and constraints are used in later chapters for simulation purposes, butalso serve to derive simplified models for the computation of power system controllers.

2.1 Power system modeling

This section summarizes the dynamic power system equations and introduces the nota-tion used for the control problem formulation with the HVDC injections as manipulatedvariable.

2.1.1 Power system network equations

The network equations are a set of algebraic equations that determines the power flowin an AC network. They are required for steady state analysis and optimization of thepower network, for instance for optimal power flow. They are also part of the dynamicpower system model.

Consider a power network as depicted in Fig. 2.1, comprised of nodes connected byAC transmission lines and HVDC links. Each of the nodes in the network representsan AC bus, with an AC voltage and AC current injections from generators, loads andHVDC terminals. The buses are grouped into three disjoint categories:

1. buses connected to an AC generator (green in Fig. 2.1)

2. buses connected to an HVDC terminal (red in Fig. 2.1)

3. buses connected to no AC generator or HVDC terminal (blue in Fig. 2.1)

10 2 Models and constraints of power systems with HVDC links

Figure 2.1: Meshed AC power network with generator nodes (green), HVDC nodes

(red) and pure load nodes (blue), connected by AC transmission lines

(black) and HVDC links (orange).

Generator buses are connected to the rest of the network through a transformer, mod-eled as a separate branch. Loads, modeled as constant impedances, can be connectedto all non generator nodes.

All AC quantities are assumed to be balanced between the three phases and modeledas complex phasors in a dq0-frame, rotating at the frequency of the AC system. Givena network with ng generator buses (category 1), nh HVDC buses (category 2) and nlpure load buses (category 3), denote the vector of complex voltages at these buses attime t ∈ R by

{Vg(t), Vh(t), Vl(t)} ∈ Cng × C

nh × Cnl . (2.1)

The i’th element of Vg(t) is denoted by the phasor Vg,i(t) ∈ C and represents the voltageat the i’th generator bus. Similar definitions are used for the complex currents Ig(t)and Ih(t), injected from the generators and HVDC terminals into the AC system. Tosimplify notation, the time dependency is omitted subsequently.

The currents and voltages are linked by the ng + nh + nl Kirchhoff equations of thenetwork,

IgIh0

=

Y1 Y2 Y3

Y T2 Y4 Y5

Y T3 Y T

5 Y6

VgVhVl

, (2.2)

with Yi being constant complex matrices of appropriate dimension, computed from the

2.1 Power system modeling 11

impedance model of the transformers, the AC transmission lines and the loads. Notethat by definition, there are no currents injected at the load buses. Furthermore, thepowers injected at the generator and HVDC buses are given by

Sg,i = Vg,i · I∗g,i i = 1, ..., ng (2.3)

Sh,i = Vh,i · I∗h,i i = 1, ..., nh , (2.4)

forming the complex vectors

Sg = [Sg,1, ..., Sg,ng ]T = diag(I∗

g ) ·Vg (2.5)

Sh = [Sh,1, ..., Sh,nh]T = diag(I∗

h) ·Vh . (2.6)

In total, (2.2) and (2.5)-(2.6) contain

nequations = 2 ·ng + 2 ·nh + nl (2.7)

equations innvariables = 3 ·ng + 3 ·nh + nl (2.8)

variables, leavingnfree = nvariables − nequations = ng + nh (2.9)

free variables to define the entire network flow. Typically, ng variables are given by thevoltage vector Vg, which depends on the dynamic generator states. For HVDC basedpower system control, the remaining nh variables can be defined as the injections atthe HVDC terminals.

If the HVDC injections are specified through the currents Ih, all AC quantities canbe directly expressed in terms of the free variables, using

IgVhVl

=

Mg1 Mg2

Mh1 Mh2

Ml1 Ml2

(

VgIh

)

, (2.10)

with the constant complex matrices

Mg1 = Y1 − (Y2 Y3)

(

Y4 Y5

Y T5 Y6

)−1 (Y T

2

Y T3

)

, (2.11)

Mg2 = (Y2 Y3)

(

Y4 Y5

Y T5 Y6

)−1 (I

0

)

. (2.12)

and similar expressions for Mh1,Mh2,Ml1 and Ml2. The transformation from (2.2) to(2.10) can be done for all practical power systems, forming a connected graph [DB10].


If the HVDC injections are specified through the powers Sh, the AC quantities areimplicitly expressed with Ih as solution of the system of equations

Sh = diag(I∗h) · (Mh1Vg +Mh2Ih) . (2.13)

Mathematically, there can be multiple or no solutions to (2.13). Closed form solutionsare only possible for power networks with a single HVDC terminal. For systems withtwo HVDC terminals (i.e. a single HVDC link), all solutions can be obtained bydetermining the roots of a sixth order univariate polynomial. Practically, only one ofthese solutions corresponds to physical values within reasonable bounds. For systemswith multiple HVDC links, one has to resort to numerical approaches to determinespecific local solutions.

2.1.2 Power system dynamic equations

2.1.2.1 General dynamics

From a control perspective, the general dynamic power system model is given by asystem of differential algebraic equations

x(t) = f(x(t), z(t), u(t)) (2.14)

0 = g(x(t), z(t), u(t)) (2.15)

y(t) = h(x(t), z(t), u(t)) (2.16)

with{x(t), z(t), u(t), y(t)} ∈ R

nx ×Rnz × R

nu ×Rny (2.17)

denoting the dynamic states, the algebraic states, the power system inputs and theoutputs of interest. The functions f , g and h contain nx, nz and ny elements, respec-tively.

The dynamic equations in f include the generator dynamics, relevant dynamics of theHVDC links and all local control schemes like the primary frequency control, automaticvoltage regulators or power system stabilizers. The algebraic equations in g include thenetwork equations, AC and DC power constraints and saturation functions of the localcontrollers.

2.1.2.2 Simple power system dynamics

A simple example of (2.14)-(2.16) for a basic model of electromechanical oscillations inan AC network with HVDC links can be formulated using the so called swing equations,

2Hiθi = Pm,i − Pe,i −Diθi i = 1, ..., ng . (2.18)

2.2 HVDC constraints 13

For a network of ng generators, the dynamic state x has then 2 ·ng elements, formedby the generator angles θi and angular speeds ωi = θi. The inertia Hi, the dampingcoefficient Di and the mechanical generator powers Pm,i are assumed to be constant.The electrical power drawn from the generator is given by

Pei = ℜ(Sg,i) = ℜ(Vg,iI∗g,i) (2.19)

where the complex generator voltage

Vg,i = |Vg,i|ej· θi (2.20)

has a constant magnitude |Vg,i|. The generator currents Ig are given by the reducednetwork equation

Ig = Mg,1Vg +Mg,2Ih (2.21)

as in (2.10).

If the HVDC injections are specified through the currents Ih, the equations (2.18)-(2.21) can be reduced to an ordinary differential equation. The input u is then formedby the real and imaginary parts of Ih and no algebraic state z is required to describethe system.

If the HVDC injections are specified through the complex powers at the terminals,Sh, an additional constraint (2.13) is required to describe the system as an algebraicdifferential equation. The input u is then formed by the real and imaginary parts ofSh and the algebraic state z is formed by the real and imaginary parts of Ih.

The power system output of interest, y, must be measured or estimated and is usedto formulate the control objective. For many examples in this thesis, these are the nggenerator frequencies ωi.

Local controllers manipulating the generator’s Pm,i or |Vg,i| are not included in thismodel. Also, the HVDC is modeled as an ideal source, that instantaneously injects therequired current or power. The model can be extended to a more detailed descriptionof the generators or the HVDC link dynamics [MBB08]. All methods described in thisthesis do not change as long as the power system model is still of the form (2.14)-(2.16).

2.2 HVDC constraints

This section summarizes the constraints on HVDC injections into AC networks. Inpractise, HVDC converters have security mechanisms preventing violations of physi-cal and operational constraints. However, for the automatic grid controller, a prioriknowledge of the HVDC constraints is important, to compute inputs with a predictableimpact on the power system.


Vh1jX

Ih1 Vc1 VDC1

RDC

IDC VDC2 Vc2

jX

Ih2 Vh2

Figure 2.2: Steady state equivalent circuit model for HVDC constraint computation

During operation, the HVDC constraints change over time, since they depend on theAC network’s dynamic state. The resulting state dependent set of admissible HVDCinjections is derived in three steps:

1. Injection from an HVDC terminal at a constant voltage bus. The AC networktopology and the coupling between HVDC terminals is neglected.

2. Injection from an HVDC terminal at a bus connected to an AC network. TheAC network topology is accounted for, the coupling with other HVDC terminalsis neglected.

3. Injection from HVDC links in a meshed AC network. The AC network topologyand the coupling between HVDC terminals is accounted for.

The first step covers the classical capability chart for a single HVDC terminal [ABB12].The following steps consider power systems of increasing complexity and reduce thenumber of simplifying assumptions.

2.2.1 Origin of HVDC constraints

Constraints on the admissible injections of the HVDC links occur as linear and nonlinearequality and inequality constraints on the dynamic states, algebraic states and controlinputs of the model (2.14)-(2.16).

Constraints on AC quantities are typically formulated using average values over agiven time interval. For the control of electromechanical oscillations the time intervalof interest is on the order of one second [KRK91]. States with faster transients, like inthe HVDC converter, are modeled with their steady state value as constant parameter.States with slower transients, like generator’s mechanical power, are modeled with theinstantaneous value as constant parameter.


2.2.1.1 Equality constraints

The equivalent circuit model of the HVDC link used for the constraint derivation isdepicted in Fig. 2.2. It is a simplified version of an analytical model used for dynamicsimulation of HVDC links [DLL03]. The converter station series reactance X includesthe AC filters. The ohmic losses of the DC line are parameterized with the resistorRDC .

As for the electromechanical power system model, the AC voltages and currents,

{Vh,i, Vc,i, Ih,i} ∈ C×C× C i = 1, 2 , (2.22)

are modeled as complex phasors in a reference frame rotating at grid frequency. Theyhave to satisfy the voltage balance equation

Vc,i = Vh,i + jXIh,i i = 1, 2 . (2.23)

Non sinusoidal AC quantities and imbalances between the three AC phases are notconsidered for the constraint derivation.

The DC voltages and current,

{VDC,1, VDC,2, IDC} ∈ R×R×R , (2.24)

have to satisfy the steady state voltage balance

VDC,1 = VDC,2 +RDCIDC . (2.25)

Transients in the DC line are not considered for the constraint derivation.

The two AC-DC converters connect the HVDC to the surrounding AC system andresult in two active power balance constraints,

PAC,i = ℜ(Vc,iI∗h,i) = UDC,i · IDC,i = PDC,i i = 1, 2 , (2.26)

with IDC,1 = −IDC and IDC,2 = IDC . Converter losses are not explicitly considered forthe constraint derivation.

2.2.1.2 Inequality constraints

The inequality constraints include physical and security constraints of the convertersystem and the DC link.

The AC currents through the converter injected by the HVDC link are limited inmagnitude,

|Ih,i| ≤ IAC i = 1, 2 , (2.27)


The DC voltage at the converter provides an upper bound on the magnitude of thevoltage at the AC terminal,

|Vc,i| = |Vh,i + jX · Ih,i| ≤ kDCVDC,i i = 1, 2 , (2.28)

where kDC is a constant that depends on the specific converter technology, for instancethe modulation type used.

Finally, the DC-current of the HVDC is constrained by

|IDC | ≤ IDC . (2.29)

Further constraints, that can be added using affine bounds on currents, voltages orpowers in the system, include

• upper and lower bounds on the AC voltage level at other buses of the powersystem

• upper bounds on the rate of change of the DC current

• upper bounds on the resulting current flow in the AC transmission lines

The examples for the constraint illustration use the following parameters,

IAC = 1.2p.u. (2.30)

IDC = 0.7p.u. (2.31)

X = 0.24p.u. (2.32)

RDC = 0.04p.u. (2.33)

kDC = 1.2 , (2.34)

(2.35)

for a base power of 900MVA. For the illustrations of the constraints set, the AC andDC voltage levels of

|V h| = 0.961p.u. (2.36)

V DC = 1p.u. (2.37)

are used, corresponding to the nominal bus voltage obtained for the two area systemdefined in Section 3.3.3 and the base DC voltage level of ±80kV of ABB’s M1 converter[ABB12].

2.2.1.3 Constraint parameterization

As stated in Section 2.1.2, the HVDC injections can be specified either using the cur-rents Ih,i or the powers Sh,i. Regarding the constraints (2.26)-(2.29) and assuming aconstant DC Voltage VDC,i, one obtains no clear preference of either parameterization:


• Constraint (2.26): convex in Sh,i, non convex in Ih,i (quadratic equality)

• Constraint (2.27): convex in Ih,i, non convex in Sh,i (implicitly defined via (2.4))

• Constraint (2.28): convex in Ih,i, non convex in Sh,i (implicitly defined via (2.4))

• Constraint (2.29): convex in Sh,i, non convex in Ih,i (indefinite quadratic inequal-ity)

Using Ih,i to parameterize the HVDC injections would translate to express all currents ina global rotating reference frame, known as αβ-frame [MBB08]. This is more convenientfor the analysis of simple example system as (2.18)-(2.21) which can then be representedby ordinary instead or algebraic differential equations.

However, HVDC converter control schemes are typically tracking power references[DLL03] or current references in a local dq-frame, that decouples the active an reactivepower components of the current [CBB10]. As a result, classic HVDC feasibility chartsare usually provided using the real and imaginary components of the power Sh,i toparameterize the HVDC injections. The following derivation of the set of admissibleHVDC injections is also carried out in the power domain.

2.2.2 HVDC terminal at a constant voltage bus

2.2.2.1 Assumptions

A basic approximation of the set of admissible HVDC injections is obtained with theset up depicted in Fig. 2.3. Each terminal is considered independently, connected to aconstant voltage bus. This is a reasonable assumption if the HVDC link is connectedto a strong power grid and does not significantly alter the AC power flow in the system.

Denote the constant AC and DC voltage at the HVDC terminal as

Vh = V h ∈ C (2.38)

VDC = V DC ∈ R (2.39)

and the active and reactive power components injected by the HVDC as

{Ph, Qh} ∈ R× R , Sh = Ph + jQh . (2.40)

2.2.2.2 Constraint derivation

The inequality constraints (2.27)-(2.29) can be expressed in the power domain as fol-lows.


Vh = const.

large

Grid

large

Grid

Ih

Figure 2.3: HVDC link injecting into constant voltage bus

Equation (2.27) is expressed in the power domain as

|Ih| ≤ IAC

↔

∣∣∣∣∣

S∗h

V∗h

∣∣∣∣∣≤ IAC

↔ P 2h +Q2

h ≤ (IAC |V h|)2 , (2.41)

which is a disc centered at the origin.


|Vh + jX · Ih| ≤ kDCVDC

↔ |Vh + jX ·S∗h

V∗h

| ≤ kDCV DC

↔ ||V h|2 + jX ·S∗

h| ≤ kDCV DC |V h|

↔ (|V h|2 +XQh)

2 +X2P 2h ≤ (kDCV DC |V h|)

2

↔

(

|V h|2

X+Qh

)2

+ P 2h ≤

(kDCV DC |V h|)2

X2, (2.42)

which is a disc with the center shifted into the negative Qh direction. Note that thepeak admissible reactive power Qh,max occurs for Ph = 0 and is a concave function ofthe grid voltage level |Vh|,

Qh,max =kDCV DC |V h| − |V h|

2

X, (2.43)

illustrated in Fig. 2.4. A simple maximization of (2.43) shows that the highest peakreactive power is reached at a grid voltage level of

|V h| =kDCV DC

2, (2.44)

indicated by a point in Fig. 2.4, and decreases for larger grid voltage levels. Fig. 2.5illustrates the upper limit on the reactive power (2.42) in the variables (Ph, Qh), fordifferent grid voltage levels.


0 0.2 0.4 0.6 0.8 1 1.2−0.5

0

0.5

1

1.5

2

|Vh|

Qh,

max

VDC

= .95 p.u.

VDC

= 1 p.u.

VDC

= 1.05 p.u.

Figure 2.4: HVDC link injecting into constant voltage bus: Peak reactive power

Qh,max as a function of the grid voltage level |V h|, for different DC

voltages VDC .



|IDC| ≤ IDC

↔ |PDCV DC

| ≤ IDC

↔ |Ph| ≤IDCV DC

, (2.45)

which is a symmetric polyhedral bound centered at the origin.

Fig. 2.6 illustrates the inequality constraints (2.41), (2.42) and (2.45), along withtheir intersection. All three constraints represent simple convex sets in the variables(Ph, Qh). The disc constraints can be approximated arbitrarily close by polyhedragenerated from sampling points on the circle. The intersection of the three polyhedraand the plotting was performed using the Multi-Parametric Toolbox [KGBM04].

2.2.3 HVDC terminal in an AC network

2.2.3.1 Assumptions

In small power systems or weakly connected areas of large power systems, the constantgrid voltage assumption used in Section 2.2.2 is not valid. For long HVDC links thetwo terminals’ injections can still be considered independently but locally affect the ACpower flow and voltage levels. Fig. 2.7 shows the corresponding set up of an HVDClink injecting into a finite AC grid, using the two area system defined in Section 3.3.3.

While the DC voltage at the HVDC terminal is considered as constant,

VDC = V DC (2.46)

the AC grid voltage level at the terminal, defined in (2.10) as

Vh = Mh1Vg︸︷︷︸

Vh0

+Mh2Ih (2.47)

has a constant component Vh0 = Mh1Vg and a current dependent component Mh2Ih.

The power injection at the terminal

Sh = VhI∗h = Vh0I

∗h

︸︷︷︸

Sh0

+Mh2|Ih|2 (2.48)

has a component Sh0 = Vh0I∗h that is linear in the current, and a component Mh2|Ih|

2

that is quadratic in the current. The total power injection, expressed in terms of Sh0,is given by

Sh = Sh0 +Mh2|Ih|2 (2.49)


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

P

Q

Bound on Vh

Vg = 0.9

Vg = 0.95

Vg = 1

Vg = 1.05

Figure 2.5: HVDC link injecting into constant voltage bus: Reactive power limit

for different grid voltage levels.


−2 −1 0 1 2−2

−1

0

1

2

P

Q

Bound on Vh

−2 −1 0 1 2−2

−1

0

1

2

P

Q

Bound on Ih

−2 −1 0 1 2−2

−1

0

1

2

P

Q

Bound on IDC

−2 −1 0 1 2−2

−1

0

1

2

P

Q

All constraints

Figure 2.6: HVDC link injecting into constant voltage bus: Summary of individ-

ual constraints (yellow) and resulting set of admissible HVDC powers

(green).


G1

G2

G3

G4

Vh

large

Grid

Ih

Figure 2.7: HVDC link injecting into two area system

Sh = Sh0 +Mh2

|Vh0|2|Sh0|

2 . (2.50)

For the illustrations of the constraints set, the nominal AC grid voltage level resultingfrom the four generators uses

|Vh0| = |Mh1Vg| = 0.961p.u. , (2.51)

corresponding to the nominal operating point of the two area system.

2.2.3.2 Constraint derivation

The inequality constraints (2.27)-(2.29) can be expressed in the power variables {Ph, Qh}

as follows.

Equation (2.27) is expressed in the power domain in two steps. First, the currentbound implies a bound on Sh0,

|Ih| ≤ IAC

↔ |Sh0| ≤ Sh0,max = |Vh0|IAC , (2.52)

which is a disc centered at the origin of the complex power plane. In the complexpower plane, the set containing Sh is a norm dependent transformation (2.50) of thedisc containing Sh0. Theorem A.4, formally stated and proven in Section A.1 in theAppendix, shows that Sh then also lies in a convex set, whose boundary can be calcu-lated analytically. For the specific system in Fig. 2.7, the admissible powers respectingthe current bound (2.27) lie on a disc shifted from the origin, as illustrated in the topleft plot of Fig. 2.8.

Equation (2.28) is expressed in the power domain in two steps. First, the voltagebound implies a bound on Sh0,

|Vh + jX · Ih| ≤ kDCVDC


↔ |Vh0 + (Mh2 + jX) · Ih| ≤ kDCVDC

↔

∣∣∣∣∣Vh0 + (Mh2 + jX) ·

S∗h0

V ∗h0

∣∣∣∣∣≤ kDCVDC

↔

∣∣∣∣∣

Mh2 + jX

V ∗h0

∣∣∣∣∣

∣∣∣∣∣

Vh0V∗h0

Mh2 + jX+ S∗

h0

∣∣∣∣∣≤ kDCVDC

↔

∣∣∣∣∣Sh0 −

|Vh0|2

jX −M∗h2

∣∣∣∣∣≤kDCVDC |Vh0|

|Mh2 + jX|, (2.53)

which is a disc, centered at

Sh0,c =|Vh0|

2

jX −M∗h2

. (2.54)

The set of admissible total power injections Sh respecting the voltage bound (2.28)is derived from (2.53) using Theorem A.4. In the complex power plane, the result isa convex set, bounded by a piecewise continuous curve formed by an ellipsoidal and aparabolic segment. It is illustrated in the top right plot of Fig. 2.8.

Since the assumptions on the DC system did not change compared to Section 2.2.2,the expression of equation (2.29) in the power domain is identical to (2.45). Thesymmetric polyhedral bound is illustrated in the bottom left plot of Fig. 2.8.

The bottom right plot Fig. 2.8 illustrates the intersection of the three constraints onSh. All three constraints represent convex sets in the variables (Ph, Qh), which can beapproximated arbitrarily close by polyhedra.

2.2.4 HVDC links in an AC network

2.2.4.1 Assumptions

The previous sections considered each of the two HVDC terminals seperately. Depend-ing on the system parameters, the coupling of the terminals through the AC systemand the DC link change the shape of the constraint set. For instance, for longer DClinks with large power ratings, the voltage drop across the DC line can not be neglected.Fig. 2.9 shows the corresponding set up of an HVDC link integrated in the two areasystem, defined in Section 3.3.3.

The DC voltage at the HVDC terminals is no longer considered as constant. Instead,the voltage equation

VDC,1 = VDC,2 + IDCRDC , (2.55)

with the rated DC voltage

max(VDC,1, VDC,2) = V DC (2.56)


−2 −1 0 1 2−2

−1

0

1

2

P

Q

Bound on Ih

−2 −1 0 1 2−2

−1

0

1

2

P

Q

Bound on Vh

−2 −1 0 1 2−2

−1

0

1

2

P

Q

Bound on IDC

−2 −1 0 1 2−2

−1

0

1

2

P

QAll constraints

Figure 2.8: External HVDC link injecting into two area system: Summary of indi-

vidual constraints (yellow) and resulting set of admissible HVDC powers

(green).

G1

G2

G3

G4

Vh,1 Vh,2

Ih,1 Ih,2

Figure 2.9: HVDC link in the two area system


captures the coupling through the DC system. The AC grid voltage at the HVDCterminals is defined in the network equations (2.10) as

Vh,1 = Vh0,1 +M1,1Ih,1 +M1,2Ih,2 (2.57)

Vh,2 = Vh0,2 +M2,1Ih,1 +M2,2Ih,2 (2.58)

where Vh0,i denotes the i’th element of the complex vector Mh1Vg and Mi,j denotes theelement in the i’th row and j’th column of the complex matrix Mh2.

The power injection at the left terminal is given by

Sh,1 = Vh,1I∗h,1 = Vh0,1I

∗h,1 +M1,1|Ih,1|

2

︸︷︷︸

Sh,1,loc

+M1,2Ih,2I∗h,1

︸︷︷︸

Sh,1,ext

(2.59)

Sh,2 = Vh,2I∗h,2 = Vh0,2I

∗h,2 +M2,2|Ih,2|

2

︸︷︷︸

Sh,2,loc

+M2,1Ih,1I∗h,2

︸︷︷︸

Sh,2,ext

. (2.60)

The first component Sh,i,loc depends only on the local currents injected at terminal i.The second component Sh,i,ext depends on both the local and the external currents fromthe other terminal, capturing the coupling through the AC system.

With the DC voltage equality constraint (2.55), the HVDC link has three free vari-ables to parameterize the HVDC injections, chosen as the reactive HVDC powers ateach terminal and the active HVDC power across the DC link,

{Ph,1, Qh,1, Qh,2} ∈ R× R× R . (2.61)

The inequality constraints (2.27)-(2.29) can be expressed in the power variables (2.61)in two steps as follows. First, the mixed terms Sh,i,ext are neglected and the set of admis-sible HVDC injections Sh,i,loc are derived separately for each terminal. The derivationis similar to Section 2.2.3, but takes into account the coupling throught the DC linkwith a variable DC voltage. Secondly, the magnitude of the terms Sh,i,ext is boundedand used to compute an outer and inner approximation of the set of admissible HVDCinjections.

2.2.4.2 Constraint derivation with variable DC voltage

The admissible HVDC power injections Sh,i as in (2.59) are first derived neglecting theAC system coupling Sh,i,ext. For the remaining complex power component

Sh,i,loc = Vh0,iI∗h,i

︸︷︷︸

Sh0,i

+Mi,i|Ih,i|2 = Sh0,i +

Mi,i

|Vh0,i|2|Sh0,i|

2 , (2.62)


the assumptions are identical to the approach in Section 2.2.3, except that the constantDC voltage (2.46) is replaced by the DC coupling equations (2.55)-(2.56). Dependingon the sign of the active power flow, the DC voltage expression has two cases,

VDC,i =

{

V DC PDC,i ≤ 0VDC,low(V DC , PDC,i) PDC,i > 0 ,

(2.63)

where the function VDC,low is computed as follows. Since PDC,i = Ph,i > 0, one has

PDC,i = VDC,iIDC,i (2.64)

PDC,i = VDC,iVDC,j − VDC,i

RDC

(j 6= i) (2.65)

PDC,i = VDC,iV DC − VDC,i

RDC(2.66)

0 = V 2DC,i − V DCVDC,i + PDC,iRDC (2.67)

VDC,i =V DC

2+

√√√√V

2DC

4− PDC,iRDC =: VDC,low(V DC , PDC,i) . (2.68)

The negative solution of the quadratic equation corresponds to the zero voltage solutionVDC,low(V DC , 0) = 0, which is not relevant during regular operation of the DC link.

For the constant DC voltage case with Ph,i ≤ 0 in (2.63), the entire constraintderivation of Section 2.2.3 can be directly applied. For the variable DC voltage case,Ph,i > 0, the constraint derivation needs to be adjusted as follows.

The current bound (2.27) does not depend on VDC and is expressed as a disc con-straint in the power domain, as in Section 2.2.3. It is illustrated in the top left plot ofFig. 2.11.

The voltage bound (2.28) can be expressed in the power domain by first following asimilar procedure as Section 2.2.3. This leads to a modification of expression (2.53),

∣∣∣∣∣Sh0,i −

|Vh0,i|2

jX −M∗i,i

∣∣∣∣∣≤kDCVDC,low(V DC , PDC,i)|Vh0,i|

|Mi,i + jX|, (2.69)

whose right side now includes the function VDC,low as in (2.68), which is monotonicallydecreasing in PDC,i. Rearranging terms and squaring twice, equation (2.69) can bereformulated as quartic inequality in the real and imaginary parts of Sh0,i. To computethe admissible set of Sh,i,loc as in (2.62), the analytical expression given in TheoremA.4 is therefore no longer applicable, since it requires Sh0,i to lie in a disc. However,the bound on Sh,i,loc can still be constructed for specific active power values

PDC,i = Ph,i,loc = ℜ(Sh,i,loc) = P > 0 (2.70)


and the corresponding DC voltage

VDC,i = VDC,low(V DC , P ) . (2.71)

With a fixed DC voltage, the admissible set of Sh,i,loc can now be computed usingTheorem A.4, but is only valid at the chosen active power

Ph,i,loc = P , (2.72)

yielding an upper and lower bound on the reactive power,

Qmin(P ) ≤ Qh,i,loc ≤ Qmax(P ) . (2.73)

Fig. 2.10 shows the set of admissible injections Sh,i,loc, respecting the voltage bound(2.28). For negative Ph,i,loc, the constraints are identical to the single terminal con-straints in Section 2.2.3. For positive Ph,i,loc, the parabolic lower reactive power boundis not affected by the variable DC voltage bound in (2.69), since it originates fromthe unconstrained minimizer (see Section A.1). The ellipsoidal upper bound decreasesfor growing Ph,i,loc, due to the negative monotonicity of VDC,low, thereby ensuring theconvexity of the resulting set.

The DC current bound (2.29) leads to an upper and lower bound on the activepower. Due to the resistive losses in the DC link, (2.55)-(2.56), the bound is no longersymmetric,

−IDCV DC ≤ Ph,i,loc ≤ IDC(V DC − IDCRDC) . (2.74)

The polyhedral bound is illustrated in the bottom left plot of Fig. 2.11.

The bottom right plot Fig. 2.11 illustrates the intersection of the three constraintson Sh,1,loc. All three constraints represent convex sets in the variables (Ph,1,loc, Qh,1,loc),which can be approximated arbitrarily close by polyhedra.

2.2.4.3 Constraint derivation with coupled AC power injections

The second term contributing to the HVDC power injections Sh,i at the terminalsi ∈ {1, 2} as in (2.59)-(2.60) is the mixed AC power injection Sh,i,ext,

Sh,1,ext = M1,2Ih,2I∗h,1 (2.75)

Sh,2,ext = M2,1Ih,1I∗h,2 . (2.76)

Neglecting the contribution of the terms Sh,i,ext corresponds to the assumption

|M1,2| ≪ |M1,1| and |M2,1| ≪ |M2,2| , (2.77)


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

P1

Q1

Figure 2.10: HVDC link in the two area system with DC coupling: Set of admis-

sible power injections Sh,1,loc respecting voltage bound (2.28). Region

with Ph,1,loc ≤ 0 (yellow) and with Ph,1,loc > 0 (green). Sampled ac-

tive powers P (dashed lines). Power bounds for a fixed DC voltage

VDC,low(V DC , P ) (blue solid lines). Upper and lower reactive bounds

at the sampled active power P (blue squares).


−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

P1

Q1

Bound on Ih

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

P1

Q1

Bound on Vh

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

P1

Q1

Bound on IDC

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

P1

Q1

All constraints

Figure 2.11: HVDC link in the two area system with DC coupling: Summary of

individual constraints (yellow) and resulting set of admissible HVDC

powers Sh,1,loc as in (2.59) (green).


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

P

Q1

Figure 2.12: HVDC link in the two area system: Boundary of the admissible HVDC

powers Sh,1,loc = Ph,1,loc + jQh,1,loc (dashed line). Inner (blue) and

outer (yellow) approximation of the set of admissible HVDC powers

Sh,1 = Ph,1 + jQh,1.

which is only valid if the two HVDC terminals connect distant areas of the AC system.

Due to the bound on the current injections, (2.27), one has

|Sh,1,ext| ≤ |M1,2|I2AC = ∆S1 (2.78)

|Sh,2,ext| ≤ |M2,1|I2AC = ∆S2 . (2.79)

The ∆Si can be used as additive disturbance on the set of admissible Sh,i,loc. Maximiz-ing or minimizing the extension of the constraint boundary provides an inner and outerconvex approximations of the set of admissible Sh,i. The result for the first terminal,i = 1, is illustrated in Fig. 2.12. The exact boundary of the set of admissible HVDCinjections lies in the yellow region, which has a thickness of 2|M1,2|I

2AC . Injections in

the blue regions are guaranteed to be feasible, regardless of what the reactive power se-lected at the other terminal i = 2 is. The procedure of using external current injectionsas disturbance also works for multiple HVDC links. The approximation shown in thissection becomes more conservative, as the HVDC portion of the overall transmissioncapacity in the power system increases.


2.2.4.4 Constraint derivation with coupled DC powers

To formulate the constraints set of the three HVDC injection variables (2.61), the DCsystem coupling of the active HVDC powers

Ph,1 = PDC,1 and Ph,2 = PDC,2 (2.80)

needs to be taken into account. For a given PDC,2 > 0, one can compute PDC,1 asfollows.

PDC,1 = VDC,1IDC,1 = V DCIDC,1 (2.81)

PDC,2 = VDC,2IDC,2 = −VDC,2IDC,1 (2.82)

PDC,2 =

V DC

2+

√√√√V

2DC

4−RDCPDC,2

(

−PDC,1V DC

)

(2.83)

0 = P 2DC,1RDC + PDC,1V

2DC + PDC,2V

2DC (2.84)

PDC,1 = −V

2DC

2RDC+

√√√√ V

4DC

4R2DC

−PDC,2V

2DC

RDC. (2.85)

The power expression uses the positive solution of the quadratic equation to satisfy thezero power balance

PDC,1 = 0 ↔ PDC,2 = 0 . (2.86)

Similarly, for PDC,2 ≤ 0, one obtains

0 = P 2DC,2RDC + PDC,2V

2DC + PDC,1V

2DC (2.87)

PDC,1 = −PDC,2 −P 2DC,2RDC

V2DC

. (2.88)

In total, the DC power coupling equation is therefore given by

PDC,1(PDC,2) =

−PDC,2 −P 2DC,2

RDC

V2

DC

PDC,2 ≤ 0

− V2

DC

2RDC+√

V4

DC

4R2DC

−PDC,2V

2

DC

RDCPDC,2 > 0

. (2.89)

Transformation (2.89) transforms the power constraints of the right terminal, com-puted in the (Ph,2, Qh,2) domain, to the (Ph,1, Qh,2) domain. Fig. 2.13 illustrates theconstraints of the right terminal after the transfomation. Since the constraint sets ofthe two HVDC terminals are now expressed in the same active power variable Ph,1, theycan be combined to characterize the full set of admissible HVDC injections. Fig. 2.14shows the full constraint set, in the three variables {Ph,1, Qh,1, Qh,2} parameterizing theadmissible HVDC injection.


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

P

Q2

Figure 2.13: HVDC link in the two area system: Boundary of the admissible HVDC

powers Ph,1,loc + jQh,2,loc (dashed line) after the DC power transfor-

mation (2.89). Inner (blue) and outer (yellow) approximation of the

set of admissible HVDC powers Ph,1 + jQh,2.


−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

P

Kundur System − feasible HVDC injections

Q1

Q2

Figure 2.14: HVDC link in the two area system: Inner approximation of the set of

admissible HVDC powers {Ph,1, Qh,1, Qh,2}.


2.2.5 Constraint derivation from local power system parameters

The previous sections formulated the HVDC constraints assuming the knowledge ofall AC grid parameters of the network equations (2.10). However, the constraints canalso be formulated using only local parameters from the AC buses where the HVDCterminals are connected.

2.2.5.1 HVDC terminal at a constant voltage bus

In Section 2.2.2, the voltage equation is

Vh = V h . (2.90)

The only required AC parameter estimate for the HVDC constraint derivation is thelocal bus voltage level, |V h|.

2.2.5.2 HVDC terminal in an AC network

In Section 2.2.3, the voltage equation of the HVDC bus is

Vh = Vh0 +Mh2Ih . (2.91)

The HVDC power constraints can be derived if the following AC system parametersare known:

|Vh0| nominal bus voltage level

SCC = |SSC| short circuit capacity

φSC = arctan

(

ℑ(SSC)

ℜ(SSC)

)

impedance angle

The Thevenin equivalent circuit, satisfying the same equation (2.91) as the AC net-work, is shown in Fig. 2.15. The two coefficients in (2.91), used for the constraintderivation, can be determined as follows. First, Vh,0 represents the nominal HVDCbus voltage when no HVDC injections occur, illustrated in the left half of Fig. 2.16.For the constraint derivation, only the magnitude of the bus voltage, |Vh0|, is required.Secondly, Mh2 = ZSC is the network impedance calculated from the short circuit ca-pacity of the HVDC bus. The short circuit capacity SCC ∈ R is the magnitude of theshort circuit power, defined as the product of the nominal bus voltage level and themagnitude of the short circuit current, illustrated in the right half of Fig. 2.16,

SCC = |SSC | = |Vh,0||ISC| = |Vh,0|∣∣∣∣

Vh,0ZSC

∣∣∣∣ =|Vh,0|

2

|ZSC|. (2.92)


Vh0

ZSC

Vh

Ih

Figure 2.15: Equivalent circuit of an AC network connected to an HVDC terminal.

Vh0

ZSC

Vh = Vh0

Vh0

ZSC

Vh = 0

ISC

Figure 2.16: Equivalent circuits used to compute parameters of AC network con-

nected to HVDC terminal.

This allows to compute the magnitude of the network impedance,

|ZSC| =|Vh,0|

2

SCC. (2.93)

To compute the full network impedance ZSC, it is assumed that there is also someinformation about the phase of the short circuit power, known as the impedance angleor X-R-ratio,

tan(φSC) =ℑ(SSC)

ℜ(SSC). (2.94)

The ratio is either given through standard assumptions or estimated for the specificAC bus connected to the HVDC link. Since

SSC = SSC · ejφSC , (2.95)

this allows to compute the full network impedance as

Mh2 = ZSC =Vh,0ISC

=Vh,0V

∗h,0

S∗SC

=|Vh,0|

2

S∗SC

=|Vh,0|

2

SSC· ejφSC . (2.96)


Vh,1

Ih,1

Z1

V1

Z12

Vh,2

Z2

V2

Ih,2

Figure 2.17: An equivalent circuit of an AC network connected to an HVDC link.

2.2.5.3 HVDC links in an AC network

In Section 2.2.4, the voltage equation of the HVDC buses is

Vh,1 = Vh0,1 +M1,1Ih,1 +M1,2Ih,2 (2.97)

Vh,2 = Vh0,2 +M2,1Ih,1 +M2,2Ih,2 . (2.98)

The HVDC power constraints at the terminals i ∈ {1, 2} can be derived if the followingAC system parameters are known:

|Vh0,i| nominal bus voltage levels

SCCi = |SSC,i| magnitude of the short circuit powers

φSC,i = arctan

(

ℑ(SSC,i)

ℜ(SSC,i)

)

impedance angles

Z12 point to point resistance between the HVDC terminals

Since the terminals of the full HVDC link are connected to two buses coupled throughthe AC system, the constraints can not be derived from a simple Thevenin equivalent.A possible choice of a symmetrical equivalent circuit, satisfying the same equations(2.97)-(2.98) as the AC network, is shown in Fig. 2.17. The circuit essentially consistsof two Thevenin equivalent circuits, coupled by the impedance Z12 The coefficients in(2.97)-(2.98), used for the constraint derivation, can be determined as follows.

First, |Vh0,i| represents the nominal HVDC bus voltage level when no HVDC in-jections occur, illustrated in the left half of Fig. 2.18. Secondly, Mi,i = ZSC,i is the


Vh,1 = Vh0,1

Z1

V1

Z12

Vh,2 = Vh0,2

Z2

V2

Vh,1 = 0ISC,1

Z1

V1

Z12

Z2

V2

Figure 2.18: Equivalent circuits used to compute parameters of AC network con-

nected to HVDC terminal.

network impedance calculated from the short circuit capacity of the HVDC buses. Theshort circuit case for terminal i = 1 is illustrated in the right half of Fig. 2.18. Oneobtains, as in the previous section,

M1,1 = ZSC,1 =Vh0,1

ISC,1=|Vh0,1|

2

SSC1· ejφSC,1 (2.99)

M2,2 = ZSC,2 =Vh0,2

ISC,2=|Vh0,2|

2

SSC2

· ejφSC,2 . (2.100)

For the computation of the coupling term M1,2 = M2,1, it is required to express (2.97)-(2.98) in terms of the circuit parameters depicted in Fig. 2.17,

Vh,1 =(Z12 + Z2)V1 + Z1V2

Σ︸︷︷︸

Vh0,1

+Z1(Z12 + Z2)

Σ︸︷︷︸

M1,1

Ih,1 +Z1Z2

Σ︸︷︷︸

M1,2

Ih,2 (2.101)

Vh,2 =Z2V1 + (Z12 + Z1)V2

Σ︸︷︷︸

Vh0,2

+Z1Z2

Σ︸︷︷︸

M2,1

Ih,1 +Z2(Z12 + Z1)

Σ︸︷︷︸

M2,2

Ih,2 , (2.102)

withΣ = Z1 + Z2 + Z12 . (2.103)

The given point to point resistance between the HVDC terminals and the shortcircuit impedances,

Z12 =Z12(Z1 + Z2)

Σ(2.104)


M1,1 =Z1(Z12 + Z2)

Σ(2.105)

M2,2 =Z2(Z12 + Z1)

Σ, (2.106)

provide 3 nonlinear complex equations in the 3 unknown circuit impedances {Z1, Z2, Z12}.For the constraint derivation, a solution of the nonlinear equations for the circuitimpedances is not required. The remaining unknown parameter of (2.97)-(2.98),

M1,2 = M2,1 =Z1Z2

Σ, (2.107)

can be computed directly from (2.104)-(2.106) as

M1,2 =M1,1 +M2,2 − Z12

2(2.108)

requiring no solution for the parameter of the equivalent circuit. This also shows theindependence of the bus voltage relations (2.97)-(2.98) and the constraint derivationfrom the chosen underlying equivalent circuit representation.

3 Power system control through

HVDC links

This chapter presents a framework to improve the dynamic power system performanceby adjusting the HVDC’s power injections into the AC grid. Two approaches for anautomatic grid controller selecting the HVDC injections are presented:

1. A decentralized linear controller using local measurements from the HVDC buses.

2. A centralized model predictive controller using global measurements from theentire power system .

The control approaches are illustrated with three benchmark power systems:

1. A single generator infinite bus system.

2. A two area system with four generators.

3. A large model of the continental European power network.

Example scenarios include the damping of the power oscillations and the synchroniza-tion after power system transients.

3.1 Control system overview

The full power system control structure considered in this chapter is represented schemat-ically in Fig. 3.1. The control plant is formed by the network of AC generators aspresented in Chapter 2. The HVDC injections change the power flow in the AC net-work in accordance with the network equations (2.10), both locally through the reactivepower injection and over large distances through the active power transmission. Theautomatic grid controller uses the HVDC links as actuators to enhance the dynamicpower system performance based on local and global measurements.

42 3 Power system control through HVDC links

Figure 3.1: Schematic of the full control system: The nominal operating points

of the AC and DC components are chosen by the network operator

using OPF calculations. The power system with AC generators forms

the control plant, the HVDC links serve as actuators. During transients

caused by disturbances, the automatic grid controller adjusts the HVDC

power references to meet the desired performance objectives.

3.2 Grid controller design 43

3.2 Grid controller design

To design a grid controller, this section first discusses suitable performance objectives,which measure how much the power system deviates from a desired behavior. Thesecond part specifies the measurement information assumed to be available, formingthe basis of the control decision. The section then presents two design procedures for anautomatic grid controller, adjusting the HVDC injections during power grid transientsin order to achieve the desired performance objectives.

3.2.1 Power system performance objective

A power system performance measure is a scalar function of the system states expressinghow far the power system deviates from the desired ideal behavior. It can be formulatedas

• logical expression,

• probabilistic expression, or as

• algebraic expression

of the system states. In any case, the grid controller has the objective to minimize theperformance measure and thereby to bring the system to the desired system behavior.This minimization can concern one or multiple time instances of the future systemstate.

Logical expressions characterize hard system limitations. They are either satisfiedor violated, providing no further information to compare and rank gradual changes ofthe power system behavior, which is useful for isolated scenarios with hard physicallimitations, such as the n − 1 stability criterion. Probabilistic expressions typicallycharacterize the expected, average behavior for many realizations of an underlyinguncertainty distributions. They are useful for applications such as generator scheduling,with uncertainty in the future generation-demand balance.

For the case of inter-area oscillations and power system transients after faults, a clearcut off criterion, that can serve as logical objective expression, is not obvious. On theother hand, a stochastic modeling of all potential fault scenarios, as required for theprobabilistic objective expression, is also very difficult. Faults and other changes inthe network occur as discrete events. Consequently, the grid controller steering theHVDC links must not address the expected performance over a broad distribution ofscenarios. Instead, the goal is, to support the power system as much as possible in agiven disturbance situation. To compare and rank different grid controller decisions,the gradual improvement of a given power system scenario is therefore best measured


by an algebraic expression of the system states.

The remaining question concerns the choice of the power system variable contributingto the performance measure. After a disturbance, the power system becomes oftenmore fragile and operates closer to a safety critical regime. The primary goal duringsuch transients ranging from seconds to a few minutes is not a marginal improvementof generation costs, as for Optimal Power Flow problems. Instead, the power systemneeds to be steered to a safer regime as fast as possible. Typical power system variablesthat are affected during faults include, as introduced in Section 2.1,

• the bus voltage angles θi,

• the bus frequencies ωi,

• the bus voltage levels |Vg,i|.

Even without the measurement and estimation question, the choice of a variableof interest, suitable for HVDC based power system control, is limited. The powerflow in the network is characterized by the voltage levels and voltage angles at theindividual buses. An absolute reference for the bus voltage angle is not available butresults from the power through the transmission lines connecting the buses. Whilethe nominal power through the transmission lines allows to estimate a relative anglebetween connected buses, this difference is subject to change. In particular, after adisturbance involving a permanent loss of a power system component, the whole powerflow in the system is affected and the new settling point of the transmission line poweris not known a priori.

The bus frequency is directly measurable and has a global reference identical at eachbus, for instance 50 Hz in Europe and 60 Hz in the United States of America. Ifpower demand exceeds the power generated, the frequency decreases, and vice versa.Keeping the frequency to its nominal value is an important task spread over the entirepower system through a cascaded system know as the generators primary and secondaryfrequency control. It is important to note that in order to affect the system’s frequency,one must change the global active power balance. HVDC links have little effect on thisbalance since they only shift the active power between different parts of the network, ascaptured by the DC power coupling equation (2.89). The absolute frequency is thereforeleft to the generator controls and does not enter the grid controller’s objective.

Instead, the relative frequency difference between two buses connected by an HVDClink is directly affected by the chosen active power transmission. During a transient inthe system frequency, ideally all bus frequencies would simultaneously move accordingto the primary and secondary frequency control. This leads to an objective function


that measures the generators frequency deviations from the average system frequency

ω(t) =

∑ngen

i=1 Hiωi(t)∑

iHi, (3.1)

weighted with the ngen generators inertia constants Hi. The objective function J is thesquared relative frequency error

J(t) =

∑ngen

i=1 Hi(ωi(t)− ω(t))2

∑

iHi, (3.2)

withσ(t) =

√

J(t) (3.3)

measuring the average frequency deviation between the generators of the system.

An alternative objective function, used with some of the simple power system models,is the weighted steady state deviation

Jref(t) =

∑ngen

i=1 Hi(Kθ(θi(t)− θi,ref)2 +Kω(ωi(t)− ωref)2)∑

iHi, (3.4)

which can only be applied if the reference values θi,ref and ωref are known. Unless statedotherwise, it is assumed that

Kθ = Kω = 1 . (3.5)

3.2.2 Measurement and estimator system

Two control scenarios are considered in this chapter. A decentralized control approach

assumes only frequency measurements from the two AC buses connected to the HVDClink and will be used for the grid controller presented in Section 3.2.3. A centralized

control approach assumes available measurements from all over the power system usedfor the grid controller presented in Section 3.2.4.

Based on system wide PMU measurements, an estimate of the global dynamic powersystem state can be obtained using Wide Area Measurement System (WAMS), whichare typically used on a slower time scale for power system monitoring, but can alsooperate on a faster time scale for power system control [DLRCTP10]. Estimating thevoltage phase angle and frequency based on time domain measurements is a well studiedproblem, see [TLTB05,Sve01,YBJM08,Jov03] and [PT08] with the references therein.In this chapter, the estimated bus voltage angle and frequency at bus i and at time tkare modeled as perturbed and delayed estimates

θi(tk) = θi(tk − τk) + eθi,k , (3.6)


ωi(tk) = ωi(tk − τk) + eωi,k , (3.7)

or more general,x(tk) = x(tk − τk) + ex,k . (3.8)

The estimates are available at discrete times tk where the time intervals tk−tk−1 are notrequired to be constant in size, accounting for informations losses on the communicationchannel. The time delay τk of the estimate is assumed to be known through the timestamp of the data. The estimation of a grid voltage phasor is sufficiently fast for thecontrol of transients such as inter-area oscillations [BGL+10].

The implementation of the control approach also requires the communication of themeasurements to the controller and of the control signal to the HVDC terminals. Asimilar model as (3.6)-(3.7) therefore represents the delay due to the computation andcommunication of the control signal uref.

For most simulation examples, the centralized control approach focuses on the under-lying control problem, assuming the dynamic state of the power system model (2.14)-(2.16) to be directly measurable and neglecting the estimation error and delay. Unlessstated otherwise, it is therefore assumed that x(tk) = x(tk) with τk = 0 and ex,k = 0

3.2.3 Local damping control

A classical approach to HVDC based power system control is the design of a controllerto damp specific oscillatory modes of the system, based on local measurements at theHVDC terminal [SA93,Eri08]. The optimal controller tuning depends on the topologyof the grid, the HVDC location in the network and the oscillatory modes of interest.Changes in either of these parameters require an adjustment of the controller, or haveto be accounted for with a sufficiently conservative tuning.

The local damping controller changes the power injections of the HVDC link basedon measurements at the converter terminals and will be used for comparison to theHVDC link operated with a centralized grid controller. In this study, the controllerof [Eri08] is used, which chooses the HVDC’s active power adjustments ∆P with aPD-controller and a low pass filter. The only required measurement is the difference ofthe frequencies (ω1, ω2) at the HVDC’s two converter stations. The resulting controllertransfer function is

∆P =(

KP +sKD

1 + sTD

)

· (ω1 − ω2) . (3.9)

The gains of the PD controller are tuning parameters and are selected as KP = 150,KD = 20 and TD = 0.05 s. The tuning was done manually for the simulation examplesin this chapter, the values of [Eri08] yielded the best results.


3.2.4 Coordinated Model Predictive Control

The local damping controller presented in section Section 3.2.3 requires careful tuningfor different operating conditions or changing system parameters. In contrast, a ModelPredictive Control (MPC) scheme can react to changes in the system model withoutadditional tuning [Mac01], as proposed for power systems with HVDC in [FMLM11].

For the HVDC based power system control, the MPC approach operates as follows:

1. Obtain an estimate x of the system model’s dynamic state x.

2. Obtain a prediction model of the power system dynamics (2.14)-(2.16).

3. Select an appropriate HVDC injection by solving an optimization problem mini-mizing the performance objective (3.2) from Section 3.2.1 over a fixed predictionhorizon.

4. Apply the chosen HVDC injection to the power system for a fixed time interval.

5. Repeat the procedure at a fixed sampling rate.

The approach is also known as receding horizon control.

3.2.4.1 Linear power system prediction model

To formulate a tractable MPC problem, a linear prediction model of the power gridand all relevant components is required.

Linearization To obtain a linear system model, the dynamic model of the plant andthe actuator system (2.14)-(2.16) is linearized. Since the dynamic equilibrium point ofpower system undergoes constant changes due to fluctuations in the power balance andis not available a priori, the linearization is performed around the current operatingpoint (x0, z0, u0) with the corresponding time derivative f0 = f(x0, u0) and the outputvalue y0 = h(x0, z0, u0). For consistency with Fig. 3.1, it is assumed that the modeledHVDC dynamics are lumped into the power system model and that u = uref. Theresult is the model

˙x = Ax+Bu+ f0 , (3.10)

y = Cx+Du , (3.11)

wherex ≈ x− x0 , y ≈ y − y0 (3.12)

denote the linearized state and output deviation when the power system is near theoperating point after a small input change

u = u− u0 . (3.13)


Note that the system is not linearized around an equilibrium point of the system. Thecontrolled linearized system is not necessarily steered to the origin. In fact, duringtransients, the operating point, i.e. the load flow of the system, undergoes continueschanges through the generator frequency control. In addition, the system’s operatingpoint is also directly affected by the adjusted HVDC injections u.

Discretization For a discrete time control problem formulation, the system is dis-cretized with a fixed sampling time T . With the input signals u being constant overthe sampling interval, the equivalent formulation of (3.10)-(3.11) is

xk+1 = Axk + Buk + f0 , (3.14)

yk = Cxk +Duk . (3.15)

The index k denotes the value of the signal at time tk, i.e. xk = x(tk). The discretetime matrices are computed using the matrix exponential [Loa78],

[

A B f0

0 0 0

]

= exp

([

A B f0

0 0 0

]

·T

)

, (3.16)

with the zeros denoting all-zero matrices of appropriate size to render the resultingmatrix quadratic. A simple approximation of the discretized model is obtained withthe Euler forward method,

A = I + AT B = BT . (3.17)

Expression of the Objective function It is assumed that the system output y, definedin (2.16), consists of the power systems generator frequencies,

y =[

ω1, ω2, ... , ωngen

]T. (3.18)

Substituting the expressions (3.11)-(3.12), the objective function (3.2) at a time in-stance tk can then be rewritten as a quadratic matrix expression

J(tk) ≈ J(xk, uk) = [Cxk +Duk + y0]TQ[Cxk +Duk + y0] (3.19)

where

Q = (I −M)Tdiag(h)(I −M) , (3.20)

M = [h, h, ..., h]T , (3.21)

h =[H1, ..., Hngen

]T∑

iHi

. (3.22)


3.2.4.2 MPC problem formulation

Quadratic programming formulation of the MPC scheme At each sampling timestep k∗, an MPC-based grid controller first obtains a measurement of the current systemstates and inputs

x0 = x(tk∗) z0 = z(tk∗) u0 = u(tk∗) , (3.23)

determines the linear discrete time power system model (3.14)-(3.15), and solves thequadratic optimization problem

minuk∗ ,...,uk∗+N−1

k∗+N−1∑

k=k∗

J(xk, uk) + ψ(xN) (3.24)

s.t.

xk∗ = 0 , (3.25)

∀k ∈ {k∗, k∗ + 1, ..., k∗ +N − 1}

xk+1 = Axk + Buk + f0 , (3.26)

Huk ≤ K , (3.27)

dmin ≤ uk+1 − uk ≤ dmax , (3.28)

with the quadratic stage cost function J(xk, uk) and the matrices {A, B, f0} as inSection 3.2.4.1. At the beginning of the horizon, the system is at the linearizationpoint with zero deviation, xk∗ = 0. The inequality (3.27) characterizes a polyhedral setof admissible HVDC injections. A simple bound is given by the box constraints

umin ≤ uk ≤ umax , (3.29)

with the upper and lower magnitude limitation of the HVDC converters. A moreelaborate constraint set is given by a polyhedral approximation of the convex HVDCinjection constraints as derived in Section 2.2. An example for single HVDC link withthree degrees of freedom is depicted in Fig. 2.14. In addition, (3.27) can includebounds on the AC bus voltage level, the transmission line loads and the mechanicalpower drawn from the generators. The inequality (3.28) forms a simple rate constrainton the input adjustments of the HVDC power.

The optimization (3.24) is performed over a horizon of N time steps, thereby alsoconsidering the future behavior of the system. The adjustment of the HVDC powerinjection that best enhances the power system is then given by uk∗, the first elementof the resulting optimizer sequence. The HVDC’s updated power injection reference

∀t ∈ (tk∗ , tk∗+1] u(t) = u0 + uk∗ (3.30)


is used over the next sampling interval, a new measurement is taken and the procedurerepeated. The terminal state cost ψ(xN ) is chosen as quadratic function that ensuresclosed loop stability of the MPC scheme.

Closed loop stability of the MPC scheme For the applications in this chapter,the linearized power system models of the form (3.14)-(3.15) are open loop stable.Furthermore, the HVDC constraints (3.27) do not depend on the system state x andalways contain the origin of the u space. This allows to ensure the closed loop stabilityof the MPC scheme by properly selecting the terminal cost function ψ( · ) and withoutadditional constraints on the terminal state xN [MRRS00].

A sufficient condition for closed loop stability of the open loop stable linearizedsystem (3.14)-(3.15) is given if the terminal cost function ψ( · ) satisfies the Lyapunovcondition,

∀x ∈ Rnx : ψ(Ax+ f0)− ψ(x) < −J(x, 0) . (3.31)

Defining the zero input equilibrium point

x = Ax+ f0 → x = (I − A)−1f0 (3.32)

and the relative state∆x = x− x (3.33)

gives the equivalent condition

∀∆x ∈ Rnx : ψ(A∆x+ x)− ψ(∆x+ x) < −J(∆x+ x, 0) . (3.34)

For AC power systems, an equilibrium is only obtained if all frequencies forming thesteady state output vector y converge to the same value, ω ∈ Rn, i.e.

y = y0 + Cx = ω· [1, 1, ..., 1]T . (3.35)

The steady state output vector y has no contribution to the cost function J(x, 0) definedin (3.19), since

(I −M)y = (I −M)[1, 1, ..., 1]T ω = [0, 0, ..., 0]T . (3.36)

The expression of the cost function J(x, 0) then simplifies to

J(x, 0) = (Cx+ y0)TQ(Cx+ y0) (3.37)

= (C∆x+ Cx+ y0)TQ(C∆x+ Cx+ y0) (3.38)

= (C∆x+ y)TQ(C∆x+ y) (3.39)

= (C∆x+ y)T (I −M)Tdiag(h)(I −M)(C∆x + y) (3.40)


= (C∆x)T (I −M)Tdiag(h)(I −M)(C∆x) (3.41)

= (C∆x)TQ(C∆x) . (3.42)

Selecting a quadratic terminal cost function of the form

ψ(x) = (x− x)TP (x− x) (3.43)

allows to rewrite the stability condition (3.31) as

∀ ∆x ∈ Rnx : ∆xT ATPA∆x−∆xTP∆x ≤ −∆xTCTQC∆x , (3.44)

which is equivalent to the Lyapunov inequality

↔ ATPA− P ≤ −CTQC (3.45)

in the matrix variable P . After the solution of (3.45), the final expression of theterminal state cost is

ψ(xN ) = (xN − x)TP (xN − x) . (3.46)

The selected terminal state cost ensures closed loop stability when the MPC schemeis applied to the linearized approximation of the power system. However, for thesimulation of the benchmark case studies in Chapter 3.3, the MPC scheme is appliedto power system models with nonlinear dynamics of the form (2.14)-(2.16). Whileclosed loop stability is not guaranteed for this setup, no instability was observed forthe simulated scenarios, which include large power system transients. Terminal costfunctions and terminal state constraint providing stability guarantees for MPC schemesapplied to nonlinear systems are hard to compute in general. For an introduction tononlinear MPC see, for instance, [FA02].

Time delay compensation with the MPC scheme As discussed in Section 3.3.3.5,the communication of power system measurements and control signal introduces delays.Since the communicated signals are assumed to have a time stamp, the delays are knownand can be naturally incorporated in the MPC scheme. The measurement signals andprediction model parameters available at time step k∗ are denoted by {x0, z0, u0} and{A, B, f0}. The measurements have a known delay of dm,k∗ time steps, the controlsignal has a known delay of dc,k∗ time steps. The delay can be accounted for in theMPC formulation by propagating the system dynamics by dm,k∗ +dc,k∗ time steps beforethe optimization is started, resulting in the quadratic program formulation

minuk∗+dc,k∗

,...,uk∗+N−1

k∗+N−1∑

k=k∗

J(xk, uk) + ψ(xN) (3.47)


s.t.

xk∗−dm,k∗= 0 , (3.48)

∀k ∈ {k∗ − dm,k∗, k∗ − dm,k∗ + 1, ..., k∗ +N − 1}

xk+1 = Axk + Buk + f0 , (3.49)

Huk ≤ K , (3.50)

dmin ≤ uk+1 − uk ≤ dmax , (3.51)

where{uk∗−dm,k∗

, ..., uk∗+dc,k∗ −1} (3.52)

are the known input signals that can be extracted from the optimizer sequence of theprevious MPC iteration. The modification corresponds to an augmentation of the statespace of the linear prediction model to store the past control inputs applied during thedelay interval (see Chapter 2.5 in [Mac01]).

The benchmark case studies in Chapter 3.3 have been simulated without any de-lay compensation measures. Experiments with artificial delays introduced during thesimulation show that the MPC scheme without delay compensation measures is stillrobust to delays of a few hundred milliseconds.

Computational effort of the MPC scheme The duration to solve the quadraticprogram (3.24)-(3.28) depends on the specific problem and the computational hardware.Specialized algorithms exploiting the structure of quadratic problems derived fromMPC problems allow the efficient solution of the MPC problem while providing apriori bounds on the solution time [Ric12,Dom13].

3.3 Simulation of power system benchmark systems 53

3.3 Simulation of power system benchmark systems

This section demonstrates the HVDC based grid controller during transients in differentpower system examples.

3.3.1 Overview of the simulation examples

Three power system examples of different complexity are used to demonstrate theHVDC based grid control during transients.

• Single machine infinite bus system. This model is simple enough to allow aclosed form solution of the algebraic network constraints (2.15) and to illustratethe transient behavior of all power system variables.

• Two area power system with four generators. This model illustrates the auto-matic grid controller applied to a power system with multiple dynamic modes.For comparison, the grid controller is also formulated with the alternative param-eterizations of the control input u, using the AC currents injected by the HVDClinks.

• European power system. This benchmark system is derived from a model used fordynamical studies in the continental ENTSO-E power system. It is parameterizedusing a snapshot of the real load distribution in the ENTSO-E grid and extendedwith three existing and planned HVDC connections.

Two HVDC link models of different complexity are used for the formulation of thecontrol problem and the power system simulation.

• HVDC injection model. This model neglects all dynamics of the DC line andassumes a perfect response to steps in the power injection references. It is usedfor simple problem formulations to illustrate the potential of the HVDC basedgrid control approach.

• HVDC average dynamic model. This model includes a dynamic model of theHVDC’s AC and DC inductances as well as the cascaded controller to track theHVDC references. The modulated converter voltages are modeled with theiraverage AC signals. It is the main model used for simulation and control designpurposes.

Three types of stress scenarios are used to test the power systems with a gridcontroller manipulating the HVDC injections.

• Symmetrical AC line faults.


• Loss of AC generators, loads and transmission lines.

• Generic disturbances of the generator’s rotor angles and frequencies.

3.3.2 Single machine infinite bus system

3.3.2.1 Power system model

Bus 1 Bus 2

Glarge

Grid

Figure 3.2: Topology of the single machine infinite bus system

The single machine infinite bus system is illustrated in Fig. 3.2. It consists of a faultyAC link and a parallel HVDC link connecting a generator to a large grid, modeled asinfinite bus. The system is based on a model by Kundur ( [Kun93], Example 13.1),used for transient stability studies.

The network model of the test case consists of two network AC buses and one gen-erator bus. The resulting signals and the grid controller interacting with the low levelHVDC controllers are shown in Fig. 3.3.

The power system model for both control and simulation purposes captures themain electromechanical dynamics to illustrate in detail all steps of the grid controlprocedure. The HVDC injections are parameterized with active and reactive powervariables and the dynamic power system equations form a set of differential algebraicequations (DAE), (2.14)-(2.15). However, since the network is simple enough, with asingle relevant HVDC terminal, the dynamic equations can be converted to a set ofnonlinear ordinary differential equations (ODE),

x = f(x, u) , (3.53)

as will be shown in the following. This enables further analysis of the system dynamics,equilibria and the closed form expression of the linearized power system equations.

Formulation as ODE model Fig. 3.4 shows the equivalent circuit representationconsidered. |V1| and θ1 are the RMS value and the angle of the generator voltage. |V2|

and θ2 are the voltage RMS value and its angle at Bus 1, |V3| and θ3 are the voltageRMS value and its angle at Bus 2. X1 is the combined generator and transformer


grid-areacontrol

G grid

HVDC control

Figure 3.3: Single machine infinite bus system: Overview of the system structure.

The system actuator is the HVDC. Power system flow variables and

control signals are featured by plain arrows. Measurements and infor-

mation flows are featured by dashed arrows.

V1/θ1

X1

V2/θ2

S

X2

V3/θ3

Figure 3.4: Single machine infinite bus system: Equivalent circuit representation.

reactance, X2 is the reactance of the AC line, all resistances are neglected. The VSC-HVDC is only modeled at one side of the link as constrained current source at Bus 1,the other side is neglected due to the assumption of an infinite bus.

The variables |V1|, X1 and θ3 are assumed to be constant. The parameters X2 and|V3| are mostly constant as well but change abruptly during faults. The generator anglesatisfies the swing equation

2Hθ1 = Pm −|V1||V2|

X1sin(θ1 − θ2) (3.54)

with the rotor inertia constant H and the mechanical generator power Pm, both as-sumed to be constant. Note that all simplifying assumptions are solely made for the


sake of simplicity of this illustrative example and can be overcome in a more complexmodel.

The power S = P +jQ injected by the VSC-HVDC is selected by the controller to bedesigned. Given the values of (|V1|, θ1, |V3|, θ3, P,Q) the remaining variables (|V2|, θ2)are uniquely determined. Using the notation of (3.53) this results in a system with twostates and two inputs:

x = [θ1, θ1]T u = [P,Q]T (3.55)

The two variables (|V2|, θ2) can be computed from the other variables (|V1|, θ1, |V3|, θ3, P,Q),that are either a dynamic state, a constant or imposed externally, as follows. Using thephasor notation

Vk = |Vk|ejθk = Vk,re + jVk,im k = 1, 2, 3 (3.56)

the equality of powers at Bus 1 is

V2

(

V1 − V2

jX1

)∗

+ S = V2

(

V2 − V3

jX2

)∗

(3.57)

Carrying out the multiplications and separating real and imaginary part, one obtainsthe two equations

0 = c2V2,re − c1V2,im +Q− |V2|2(

1

X1+

1

X2

)

(3.58)

0 = c1V2,re + c2V2,im − P (3.59)

with the coefficients

c1 = −(V1,im

X1+ V3,im

X2

)

(3.60)

c2 = −(V1,re

X1+ V3,re

X2

)

(3.61)

The second equation, (3.59), can be solved as

V2,re =P

c1−c2

c1V2,im (3.62)

The RMS value of the bus voltage can be expressed as

|V2|2 = V 2

2,re + V 22,im

=(P

c1

)2

−2c2P

c21

V2,im +

(

1 +c2

2

c21

)

V 22,im (3.63)

which allows to rewrite (3.58) as a quadratic equation

0 = V 22,im +

a5a3 − a2

a6a3︸︷︷︸

p

V2,im +a4a3 − a1

a6a3︸︷︷︸

q

(3.64)


with the coefficients

a1 = Q+ c2Pc1

(3.65)

a2 = −c1 −c2

2

c1(3.66)

a3 =(

1X1

+ 1X2

)

(3.67)

a4 =(Pc1

)2(3.68)

a5 = −2c2Pc2

1

(3.69)

a6 =(

1 + c22

c21

)

(3.70)

This gives the remaining unknown as solution of (3.64):

V2,im = −p

2+

√

p2

4− q (3.71)

Note that only the larger solution is physically meaningful since the other one resultsin the zero voltage solution V2 = 0 for P = Q = 0.

Finally, one can use all expressions (3.60) - (3.71) to obtain a closed form expressionof the voltage at Bus 1 :

V2 = g1(θ1, P,Q) =√

V 22,re + V 2

2,im (3.72)

θ2 = g2(θ1, P,Q) = arctan

(

V2,im

V2,re

)

(3.73)

which allows to write the nonlinear power system equations

d

dt

(

θ1

θ1

)

=

(

θ1|V1|· g1(θ1,P,Q)

X1sin(θ1 − g2(θ1, P,Q))

)

(3.74)

Model linearization For the use of a linear model predictive controller it is necessaryto have a linear discrete prediction model. To this effect, the nonlinear ODE (3.74) islinearized about an operating point (xss, uss). It is convenient to use a single equilibriumpoint for the linearization to obtain an LTI-system. For an equilibrium point holds therelation f(xss, uss) = 0.

In order to uniquely define the steady state power flow, the following two choices aremade:

1. The mechanical power Pm transmitted is equally distributed between the AC linkand the DC link.

2. The bus voltage level |V ss2 | is proportional to |V ss

3 |, namely |V ss2 | = 1.1 · |V ss

3 |.


Any other operating point, for instance obtained using a static power flow optimization,can also be used. Using the assumptions and the standard definitions for AC power,one obtains

θss2 = θss3 + arcsinPmX2

2|V ss2 ||V

ss3 |

(3.75)

θss1 = θss2 + arcsinPmX2

|V ss1 ||V

ss2 |

(3.76)

Sss = P ss + jQss = V ss2 (

V ss2 − V

ss1

jX1+V ss

2 − Vss

3

jX2)∗ (3.77)

Definingxss = [θss1 , 0]T uss = [P ss, Qss]T (3.78)

andx = xss + x u = uss + u (3.79)

one obtains the linearized power system equations

d

dtx = Ax+Bu (3.80)

with the Jacobian A and the input matrix B defined as

A =∂f

∂x(xss, uss) B =

∂f

∂u(xss, uss) (3.81)

The expressions of A and B for this example are easily computed from (3.74).

3.3.2.2 HVDC model

The HVDC model used for the synthesis of the MPC based grid controller and theclosed loop power system simulation is a simple injection model. The model assumesthat the active and reactive power reference values of the HVDC links are immediatlyachieved.

3.3.2.3 Simulation of an AC line fault

Fault scenario The system presented in Section 3.3.2.1 is simulated with a symmet-rical three phase fault on the AC transmission line, near the bus of the left HVDCterminal. The system parameters during the fault scenario are given in Tab. 3.1. Notehow during the 250ms long fault, the values |V3| and X2 change from the viewpointof the generator. The parameter X2 also permanently increases after the fault due toa partial loss of the transmission line . The effect of the grid-area controller setting


Table 3.1: Parameter of the power system in [p.u.]

time [s] |V1| X1 X2 |V3|

0 ≤ t < 1 1.16 0.3 0.5 0.9

1 ≤ t < 1.25 1.16 0.3 0.15 0

1.25 ≤ t < 3 1.16 0.3 1 0.9

the references for the HVDC is studied without any damper windings or stabilizingcontrollers so Pm and |V1| are taken as constants.

For the system of only a single generator, the operating point for a given set of pa-rameters can be calculated analytically as in (3.76). This allows to provide an absolutereference value of both dynamical states to the grid controller and to use the referencedeviation (3.4) as objective function for the MPC problem formulation presented inSection 3.2.4. In the shown simulation results, the MPC based grid controller, uses asampling time of T = 20ms and a prediction horizon N = 25. The input constraints aremodeled as box constraint in the P -Q-plane by setting |P | ≤ 0.5 and |Q| ≤ 0.5. Thelinearized power system equations, derived in Section 3.3.2.1, remain constant until theend of the fault. When the fault is cleared and the line loss is communicated after 250ms, the linearization is updated with the new line parameter X2.

Power system simulation results Figures 3.5 to 3.8 show the time trajectories ofthe power system states and parameters. Figures 3.5 and 3.6 show how |V3| and X3

change as seen from the generator while |V1| and X1 remains unchanged during theentire simulation.

During the fault the bus voltage level |V2| drops and is supported by the reactivepower injections selected by the grid controller. The generator starts to accelerate,see θ1 in Fig. 3.7. The grid controller eventually sets the power references to themaximum permissible values, see Fig. 3.8, to support the bus voltage and to keep therotors angular error ∆θ1 as small as possible.

When the fault is cleared after 250 ms, the controller modulates both P and Q todrive the rotor angles to the new steady state values corresponding to the new value ofX3.

Without damping control, the power system given by (3.74) has small stabilitymargins, leading to instability immediately after the fault or by increasing oscilla-tions [Kun93]. The proposed control scheme effectively stabilizes the system usingboth the active and reactive power controllability while respecting the constraints ofthe HVDC. Note that stabilization is achieved without large reserves in transmission


0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

[seconds]

[p.u

.]

|V

1|

|V2|

|V3|

Figure 3.5: Voltage rms values: during transient bus voltage drops and is recovered

capacity since the active power transmitted through the VSC-HVDC initially reachesalready 90 % of its maximum value.


0 0.5 1 1.5 2 2.5 30.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

[seconds]

[p.u

.]

X

1

X2

Figure 3.6: Network parameters vary during and after fault impedance changes


0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

80

90

100

[seconds]

[deg

ree]

θ

1

θ2

θ3

Figure 3.7: Voltage angles: after fault grid-area controller brings generator angle

to new equilibrium


0 0.5 1 1.5 2 2.5 3−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

[seconds]

[p.u

.]

PQ|S|

Figure 3.8: VSC-HVDC powers: grid-area controller mitigates disturbance while

keeping powers within VSC-HVDC limits


G1

G2

G3

G4

Figure 3.9: Two area test system with one HVDC link. AC links (lines) and loads

(arrows) modeled as complex impedances, generators (Gi) modeled as

second order model. The lost line is marked with a cross.

3.3.3 Two area power system


The two area system consists of four generators in two areas, coupled by a weak ACline. The system has been often used as test system in the literature and a detaileddescription with all necessary parameters is given in Example 12.6 of [Kun93]. Asmodification, an HVDC link is placed in parallel to the weak AC links, as illustratedin Fig. 3.9. The 220 km long HVDC cable link has a rated power of 100 MW and aDC voltage rating of ±80 kV. The system is disturbed by the loss of one AC line.

The original example system in [Kun93] was used to illustrate the effectiveness ofPower System Stabilizers and other control actions acting on the generators, to dampinter-area oscillations and ensure transient stability. For the studies in this chapter, allsuch control actions are neglected to study the effect of the HVDC with the proposedcontrol scheme independently.

3.3.3.2 HVDC model

The two area system is used to study two parameterizations of the HVDC injections.

First, the HVDC injections chosen by the grid controller are parameterized usingthe AC powers injected at the terminals. Here, a dynamic model is used to representthe HVDC link. The model uses a hierarchical control structure [CBB10, DLL03],illustrated in Fig. 3.10 and Fig. 3.11. The signals generated by the converter switchesare modeled using the averaged sinusoidal values. This has the advantage, that thesimulation model can be directly linearized to obtain the prediction model of the HVDCdynamics.

The tuning is selected using the parameters of ABB’s M9 HVDC-light converter[ABB12] as reference, with a rated power of 1216 MW and a DC voltage reference of


Udc,2,ref = 320 kV. Fig. 3.12 shows a typical dynamic response of a 100 km HVDC linewith the selected tuning, following a reference step in P1,ref and Q1,ref.

Secondly, the HVDC injections chosen by the grid controller are parameterized usingthe AC currents injected at the terminals. Here, a pure injection model is used torepresent the HVDC link, both for control design and simulation purposes.

u s,1 u t,1 u c,1 u s,2u t,2u c,2i 1 i 2

Udc,1 Udc,2Idc,1 Idc,2

Icable

Icable

Pac,1 = Pdc,1 Pac,2 = Pdc,2

Md1Mq

1 Md2Idc,2

VSCVSCXtrafoXtrafo XreactorXreactor RreactorRreactor

Cdc

Cdc

Cdc

Cdc

Ldc

Ldc

Rdc

Rdc

Figure 3.10: Physical model of the HVDC link.

3.3.3.3 Simulation of an AC line loss

Fig. 3.13 shows the resulting frequency oscillations of the four generators. For a morereadable comparison, Fig. 3.14 shows the average grid frequency defined in (3.1) andthe average frequency deviation defined in (3.3). If the HVDC links are operatedwith constant power and voltage references, the frequency level is maintained by thegovernors, but undergoes strong oscillations. It can be seen that the oscillation dampingis improved by using the HVDC links flexibility with the local damping controller,(3.9). The global MPC-based grid controller, with a sampling time of T = 0.1 s anda prediction horizon of N = 10, further improves the damping and quickly brings thefour generators to a common frequency. The corresponding adjusted power injectionsare depicted in Fig. 3.15.

3.3.3.4 Sensitivity to state constraints

The MPC problem formulation in Section 3.2.4.2 is always feasible if the inequality con-straints only bound the system input u. This is the case for instance for the limitationson the HVDC injections presented in Section 2.2.

Constraints on the system state x occur for instance with limits on the maximumrotor angle difference and temporary rotor speed deviations between different parts ofthe network. Such constraints include bounds similar to the equal area criterion, toavoid transient instability due to the nonlinearties of the electromechanical generatorequations (see Section 2.1.2.2 and [MBB08]).


P1,ref

Q1,ref Q2,ref

Udc,2,ref

Udc,2

iq1,ref

id2,ref

id2id1

iq1

Q1 Q2

P1

id1,ref

Idc,2

Md1 Md

2

Mq1

Pcontroller

iq1

controller

Udc

controller

Rectifier Inverter

Q1

controllerQ2

controllerid1

controller

id2controller

Figure 3.11: Control scheme of the HVDC link: Controller blocks with saturated

PI controllers (white boxes), physical converter models (grey boxes),

measurement signals (dashed arrows) and manipulated variables (solid

arrows).

For illustration of the state constraints, a simple power system model of the two areasystem is considered, using second order generator models and HVDC current injectionmodels. The stage cost of the MPC objective function is chosen similar to the objective(3.4) as reference deviation

Jk = (x− xss)TQ(x− xss) + (u− uss)

TR(u− uss) , (3.82)

with Q = I and R = 0.1 · I of appropriate dimensions. This controls the system tothe steady state

xss = [θTss, θTss]

T (3.83)

θss = [0,−0.0168,−0.2998,−0.3235]T (3.84)

θss = [0, 0, 0, 0]T , (3.85)

with the steady state injections

uss = [Issh,1, Issh,2]

T (3.86)

Issh,1 = 2.4017− j· 0.2561 (3.87)

Issh,2 = −1.7471 + j· 1.5707 . (3.88)

The initial state of the first generator angle is perturbed by 0.1 rad, all other statesstart at their steady state value. The predictive controller has to stabilize the system,while respecting the state and input constraints. The input constraints on the HVDC

3.3 Simulation of power system benchmark systems 67P

ower

(MW

)P

ower

(Mva

r)

Volt

age

(p.u

.)

time (s)

-0.1

-0.1

-0.1

0

0

0

0.1

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.4

0.4

0.4

1000

500

600

400

200

0

0

1

1.005

0.995

Figure 3.12: VSC-HVDC link of 100 km length: Active power P1 (top plot), reac-

tive power Q1 (center) and DC voltage Udc,2 (bottom plot). Simulated

response (blue) following reference step changes (red).

68 3 Power system control through HVDC linksF

requ

ency

(Hz)

time (s)

60.4

60.4

60.4

60.2

60.2

60.2

60.0

60.0

60.0

59.8

59.8

59.8

0

0

0

5

5

5

10

10

10

15

15

15

20

20

20

25

25

25

Figure 3.13: Two area test system with one HVDC link: Frequency trajectories of

the four generators. HVDC active power and voltage reference kept

constant (top), damping controller using local measurements (center),

and MPC-based grid controller using global measurements (bottom).

3.3 Simulation of power system benchmark systems 69replacemenA

ver

age

freq

uen

cy

(Hz)

Fre

qu

ency

dev

iati

on

(mH

z)

time (s)

60.3

60.2

60.1

60.0

59.9

59.8

0.1

0.2

0.3

0.4

00

0

5

5

10

10

15

15

20

20

25

25

Figure 3.14: Two area test system with one HVDC link: Average frequency ω (top)

and average frequency deviation σ (bottom). HVDC active power and

voltage reference kept constant (thin red), damping controller using

local measurements (dashed green), and MPC-based grid controller

using global measurements (bold blue).


Pow

er(M

W,M

var)

time (s)

-50

0

50

100

0 5 10 15 20 25

Figure 3.15: Two area test system with one HVDC link: Power injections of the

HVDC link, manipulated by the global MPC-based grid controller.

Active power (bold), reactive power at rectifier terminal (thin) and

reactive power at inverter terminal (dashed).


currents use a simple bound the real and imaginary part of the complex current:

|Ireh,i| ≤ 2.9 |I imh,i | ≤ 2.9 i = 1, 2 (3.89)

The state constraints bound the deviation of the generator angles from their steadystate value,

|θi − θss,i| ≤ δmax (3.90)

for different maximum angular deviations

δmax ∈ {1 rad, 0.2 rad, 0.1 rad} . (3.91)

The controller time step is 20 ms, which also equals the interval of the grid mea-surements. Fig. 3.16 shows the resulting generator angles’ trajectories for a bound ofδmax = 1 rad. It can be seen that after initial transients, the angles (bold lines) arerecovered and converge to the equilibrium (dashed lines). Fig. 3.17 shows that theMPC grid controller respects the prescribed input bounds.

A tightening of the hard state constraints typically leads to an increase of the controlaction and can eventually lead to infeasibility of the MPC optimization problem. Fig.3.18 shows the angular deviation of generator 1 from the steady state θss,1 = 0 using thethree different angular deviation bounds δmax. The smallest deviation bound equals theinitial deviation and can not be further decreased for this initial disturbance. The MPCbased grid controller foresees the angular trajectories approaching these bounds andexploits the available margins as much as possible with regard to the chosen objectivefunction. With tighter state constraints, the transient of HVDC injections become lesssmooth with more rapid adjustments, as seen in Fig. 3.19 for the active power throughthe HVDC line.

3.3.3.5 Sensitivity to measurement delay

For the simple two area system model in Section 3.3.3.4, the estimate of the powersystem states are assumed to be available without disturbance or delay. To illustratethe effect of communication delays, a delay of

τk ∈ {0, 20, 100} ms (3.92)

is introduced to the system, using the notation of (3.6)-(3.7) in .

The resulting angular deviation of the first generator for the three different com-munication delays and a maximum angular deviation δmax = 1 is shown in Fig. 3.20.The amplitudes of the first swings increase as the delay τk becomes longer, but theperformance decreases gracefully and the oscillations are damped out.


0 1 2 3 4 5 6 7 8 9 10−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

[sec]

[rad

]

Figure 3.16: Simplified two area system, Generator angles: The HVDC power in-

jections are adjusted using the grid control scheme to keep the angles

(bold) within limits and bring them to an equilibrium (dashed). Com-

pare to Fig. 1 with constant HVDC power injections.


0 0.5 1 1.5 2 2.5 31.8

2

2.2

2.4

2.6

2.8

3

[sec]

[p.u

.]

Figure 3.17: Simplified two area system, Real component of the AC-current injected

by the left inverter: During transient, HVDC AC-current reference

(bold) is kept within limits (2.9 p.u., horizontal line).


0 0.5 1 1.5 2 2.5 3−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

[sec]

[rad

]

Figure 3.18: Simplified two area system, Angle of generator 1 for different controller

constraints: During transient, generator angle deviation respects pre-

scribed tolerances (horizontal lines) of δmax = 1 rad (bold), 0.2 rad

(thin solid) and 0.1 rad (dashed)


0 0.5 1 1.5 2 2.5 31.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

[sec]

[p.u

.]

Figure 3.19: Simplified two area system, Active power through HVDC line for two

different controller constraints: A larger tolerance of the generator

angle (δmax = 1 rad) allows a smoother transient of the active power

transmitted (bold). Compare to the active power transient (dashed)

for a small tolerance of the generator angle (δmax = 0.1 rad).


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

[sec]

[rad

]

Figure 3.20: Simplified two area system, Angle of generator 1 for different measure-

ment delays: No delay (bold), 20 ms (thin solid), 100 ms (dashed).

With increasing delay, performance degrades gracefully.


Figure 3.21: Topology of the reduced European AC Power System (thin lines) with

four VSC-HVDC links (bold lines). The simulated scenarios are the

loss of a power plant (triangle), loss of a load (circle) and loss of an

AC line (rectangle).

3.3.4 European power system


The power system used for this studies is a reduced dynamic model of the ENTSO-Econtinental grid, developed in [Haa06].

Power system topology The power system topology is illustrated in Fig. 3.21. Itconsists of 74 buses, each connected to an aggregated power plant and a load. Twodifferent types of generation, thermal units and hydro storage units, are installed in thesystem. The nodes are connected by 131 AC transmission lines with a nominal voltageof 380 kV.


10

8

6

4

2

00 500 1000 1500 2000 2500

Line power (MW)

Nu

mb

erof

lin

es

Figure 3.22: Nominal load of the 131 AC lines in the European power system:

histogram in intervals of 25 MW.

The initial operating point for the simulation of the ENTSO-E system described inSection 3.3.4 is given by of the simulated stress scenarios is derived from measurementsof the ENTSO-E grid, with a total installed generation capacity of about 350 GW.About one quarter of the total load demand of about 260 GW is exchanged throughthe AC transmission system. The distribution of the line loads is illustrated in Fig.3.22.

Power plant and load models The power plant models consist of an electrical partand a mechanical part [Haa06]. The electrical model comprises the automatic voltageregulator (AVR) and the electrical part of the synchronous generator. The mechanicalmodel comprises the primary frequency control, the turbine model and the mechanicalpart of the synchronous generator. The steam power plant model with seven dynamicstates includes a three staged turbine model and a steam circuit. It provides themechanical torque to the turbine and can be controlled by the steam valve. The hydrostorage power plant model with four dynamic states contains a reservoir with a heightdifference to the turbine. The water flows via a penstock gallery to the turbine. Themechanical torque of the turbine is controlled by the gate valve. The synchronousgenerators are modeled with the standard nonlinear 6th order model [Kun93,MBB08].Each generator is equipped with an AVR and a governor, modeled with four and threedynamic states respectively. The ENTSO-E grid model of [Haa06] is well damped,with PSS at all generators and with dynamic load models. To investigate the HVDC-based damping in a more challenging scenario with stronger power oscillations, no PSSare incorporated in the model. Dynamic load models can both improve and deterioratethe power oscillation damping [MH95]. Loads are represented by a static model, asconstant impedance loads [Kun93].


Table 3.2: Parameters of the VSC-HVDC links

Placement Name Length Rated Power DC Voltage

France to SpainHVDC1

67 km1216 MW

±320 kVHVDC2 1216 MW

North to SouthHVDC3 700 km 1216 MW ±320 kV

of Germany

Italy to Greece HVDC4 316 km 1216 MW ±320 kV

HVDC placement Four bipolar VSC-HVDC links with a DC voltage of ±320 kVare placed in the European system, as illustrated in Fig. 3.21. The placing usesplanned VSC-HVDC links or already existing current source converter based HVDC(CSC-HVDC) corridors. The first two parallel links connect the South of France withthe North-East of Spain. This connection is based on the France-Spain ElectricalInterconnection (inelfe) [Fra], to be commissioned in 2014. The inelfe transmissionlinks use a multilevel converter arrangement. This results in smaller filter parametersand a faster dynamic response compared to the PWM-based converters used for thesimulations in this chapter. The links are 67 km long and have a rated power of1216 MW each. The second VSC-HVDC link connects the North of Germany with theSouth. It transfers power generated by offshore wind farms in the North to load centersin the South. The link has a rated power of 1216 MW and a length of 700 km. Thethird link connects the South of Italy with Greece. A single pole CSC-HVDC link witha power rating of 500 MW already connects Italy with Greece [GRG+02]. In this study,the CSC-HVDC connection is replaced with a bipolar VSC-HVDC link with a powerrating of 1216 MW. The undersea and land cables have a total length of 316 km. Anoverview of the parameters of the VSC-HVDC links is given in Table 3.2. Note thatthe total installed DC transmission capacity is relatively small, less then 2 percent ofthe total load demand and less then 8 percent of the nominal load exchange.

3.3.4.2 HVDC model

The HVDC model for control design and simulation purposes is identical to the averagedHVDC model, presented in Section 3.3.3.2.


3.3.4.3 Implementation of the network simulation

This section presents the simulation examples of the ENTSO-E power system presentedin Section 3.3.4. The simulations are carried out in MATLAB using the simulationframework of [TDB08]. The MPC algorithm, presented in Section 3.2.4.2, was alsoimplemented in MATLAB. It uses the software Gurobi [Gur13] to solve the quadraticoptimization problems of the MPC formulation (3.24). The simulations include the lossof a large power plant, a load and an AC transmission line. In all cases, the disturbancetakes place at 100 ms.

3.3.4.4 Loss of a large power plant

Consider the loss of a power plant in the northern part of France, as illustrated in Fig.3.21. The power plant has a pre-fault operating point of 1100 MW active power. Fig.3.23 shows the frequency trajectories of the generators that remain active after thedisturbance. It can be seen that without a supervising grid controller, all frequenciesinitially decrease and result in an instable oscillation. The damping can be improvedusing a local damping controller, but remains unstable. The global MPC-based gridcontroller provides an effective way to stabilize and damp the oscillations. For a morereadable comparison, Fig. 3.24 shows the average grid frequency defined in (3.1) andthe average frequency deviation defined in (3.3). Furthermore, Fig. 3.25 illustrates thepower exchanged through the AC network at two internal frontiers of the ENTSO-Enetwork. The oscillations become particularly apparent near the boundaries of thenetwork. The power trajectories of the four HVDC links are depicted in Fig. 3.26. Thetwo parallel HVDC links, HVDC1 and HVDC2, have the exact same behavior, andare plotted in the same figure. For all simulations of the European system, the Modelpredictive control (MPC)-based grid controller uses a sampling time of T = 0.5 s anda prediction horizon of N = 10. The absolute power injection limits are 0.9 p.u. foractive and 0.5 p.u. for reactive power. Similarly, the power rate constraint is 0.1 p.u.per sampling time, for both active and reactive power. It can be seen in Fig. 3.26 thatthese limitations are not violated.

3.3.4.5 Loss of a large load

Consider the loss of a 500 MW load in the center of Spain, depicted in Fig. 3.21,which accounts for about 10 percent of the load at this node. Fig. 3.27 and Fig.3.28 illustrate the resulting strong oscillations. Using the local damping controller, thesystem can be stabilized, but still oscillates. With the global MPC-based grid controllerthe installed HVDC links effectively damp the oscillations and guide the system to a

3.3 Simulation of power system benchmark systems 81F

requ

ency

(Hz)

time (s)

50.05

50.05

50.05

50.00

50.00

50.00

49.95

49.95

49.95

49.90

49.90

49.90

0

0

0

5

5

5

10

10

10

15

15

15

20

20

20

25

25

25

30

30

30

35

35

35

40

40

40

45

45

45

50

50

50

Figure 3.23: Loss of a large power plant: Frequency trajectories of the generators.

HVDC active power and voltage reference kept constant (top), damp-

ing controller using local measurements (center), and MPC-based grid

controller using global measurements (bottom).

82 3 Power system control through HVDC linksreplacemen

Aver

age

freq

uen

cy

(Hz)

Fre

qu

ency

dev

iati

on

(mH

z)

time (s)

50.00

49.99

49.98

49.97

49.96

49.95

0.1

0.2

0.3

0.4

00

0

5

5

10

10

15

15

20

20

25

25

30

30

35

35

40

40

45

45

50

50

Figure 3.24: Loss of a large power plant: Average frequency ω (top) and average

frequency deviation σ (bottom). HVDC active power and voltage ref-

erence kept constant (thin red), damping controller using local mea-

surements (dashed green), and MPC-based grid controller using global

measurements (bold blue).

3.3 Simulation of power system benchmark systems 83P

ower

(MW

)P

ower

(MW

)

time (s)

600

400

200

-60

-80

-100

-120

-140

-160

-180

-200

-400

0

0

0

5

5

10

10

15

15

20

20

25

25

30

30

35

35

40

40

45

45

50

50

Figure 3.25: Loss of a large power plant: Active power flow through the AC lines

into Spain coming from France (top), and into Greece (bottom). Tra-

jectories with HVDC active power and voltage reference kept con-

stant (thin red), damping controller using local measurements (dashed

green), and MPC-based grid controller using global measurements

(bold blue).

new equilibrium. Fig. 3.29 shows that the main contribution for the oscillation comesfrom the manipulation of the power references of HVDC1 and HVDC2, the two parallelHVDC links between France and Spain. Again, the active and the reactive powers canbe seen to respect the HVDC limitations of 0.9 p.u. and 0.5 p.u. .

3.3.4.6 Loss of an AC line

Consider a partial loss of a critical AC line between Switzerland and Italy, depicted inFig. 3.21. Fig. 3.30 and Fig. 3.31 illustrate the oscillations following this topologychange. It is assumed that the MPC controller has knowledge of the line loss at thesampling time following the disturbance. It can be seen that the damping of this


Pow

er(M

W,M

var)

time (s)

0

0

0

0

00

2

2

2

4

4

4

6

6

6

8

8

8

10

10

10

12

12

12

14

14

14

16

16

16

18

18

18

20

20

20

-500

1000

1000

500

500

-200

400

200

Figure 3.26: Loss of a large power plant: Power injections of the four HVDC links,

manipulated by the global MPC controller. HVDC1 and HVDC2

(top) have the same power injections, HVDC3 (center), HVDC4 (bot-

tom). Active power injection (bold), reactive power at rectifier termi-

nal (thin) and reactive power at inverter terminal (dashed).

oscillation takes longer than in the other scenarios and requires the excitation of somefaster dynamic modes. However, the amplitude of the oscillations is much smaller. Asshown in Fig. 3.32, the control action is equally distributed among all HVDC lines.

3.4 Conclusion

This chapter presented an approach for stabilizing transients in meshed AC powergrids using HVDC links in conjunction with a supervising MPC-based grid controller.Equality and inequality constraints on the states of the power grid and the HVDC linkare naturally included in the problem formulation.

The proposed control objective focuses on the relative frequency error between the

3.4 Conclusion 85A

ver

age

freq

uen

cy

(Hz)

Fre

qu

ency

dev

iati

on

(mH

z)

time (s)

50.025

50.020

50.015

50.010

50.005

50.000

0.6

0.5

0.4

0.3

0.2

0.1

00

0

5

5

10

10

15

15

20

20

25

25

30

30

35

35

40

40

45

45

50

50

Figure 3.27: Loss of a large load: Average frequency ω (top) and average frequency

deviation σ (bottom). HVDC active power and voltage reference

kept constant (thin red), damping controller using local measurements

(dashed green), and MPC-based grid controller using global measure-

ments (bold blue).

generators of the AC network. The grid controller is demonstrated with a single ma-chine infinite bus system, a two are power system and with a large ENTSO-E model.

Tested in power networks without PSS and other active damping components, theMPC-based grid controller can efficiently damp grid oscillations and stabilize the gridtransients. This is achieved with a relatively small installed DC transmission capacity,illustrating the large potential of HVDC-based power oscillation damping.

The results indicate that the obtained results are relatively insensitive to delays inthe communication channels of the measurement and control signals. Further studiesshould investigate the sensitivity of the result to uncertainty in the measurement valuesand the control model.


Pow

er(M

W)

Pow

er(M

W)

time (s)

-50

-100

-150

-200

-200

-250

-400

-600

200

400

0

0

0

0

5

5

10

10

15

15

20

20

25

25

30

30

35

35

40

40

45

45

50

50

Figure 3.28: Loss of a large load: Active power flow through the AC lines into Spain

coming from France (top), and into Greece (bottom). Trajectories

with HVDC active power and voltage reference kept constant (thin

red), damping controller using local measurements (dashed green), and

MPC-based grid controller using global measurements (bold blue).

3.4 Conclusion 87P

ower

(MW

,Mva

r)

time (s)

00

00

0

0

2

2

2

4

4

4

6

6

6

8

8

8

10

10

10

12

12

12

14

14

14

16

16

16

18

18

18

20

20

20

1000

600

500

500

-500

-1000

400

200

Figure 3.29: Loss of a large load: Power injections of the four HVDC links, manip-

ulated by the global MPC controller. HVDC1 and HVDC2 (top) have

the same power injections, HVDC3 (center), HVDC4 (bottom). Ac-

tive power injection (bold), reactive power at rectifier terminal (thin)

and reactive power at inverter terminal (dashed).


Aver

age

freq

uen

cy

(Hz)

Fre

qu

ency

dev

iati

on

(mH

z)

time (s)

50.015

50.010

50.005

50.000

49.995

0.4

0.3

0.2

0.1

00

0

5

5

10

10

15

15

20

20

25

25

30

30

35

35

40

40

45

45

50

50

Figure 3.30: Loss of an AC line: Average frequency ω (top) and average frequency

deviation σ (bottom). HVDC active power and voltage reference



ments (bold blue).

3.4 Conclusion 89P

ower

(MW

)P

ower

(MW

)

time (s)

500

450

400

350

300

250

200

-100

-110

-120

-130

-140

-1500

0

5

5

10

10

15

15

20

20

25

25

30

30

35

35

40

40

45

45

50

50

Figure 3.31: Loss of an AC line: Active power flow through the AC lines into Spain

coming from France (top), and into Greece (bottom). Trajectories

with HVDC active power and voltage reference kept constant (thin

red), damping controller using local measurements (dashed green), and

MPC-based grid controller using global measurements (bold blue).


Pow

er(M

W,M

var)

time (s)

00

0

0

00

2

2

2

4

4

4

6

6

6

8

8

8

10

10

10

12

12

12

14

14

14

16

16

16

18

18

18

20

20

20

1000

1000

1000

500

500

500

-500

Figure 3.32: Loss of an AC line: Power injections of the four HVDC links, manip-

ulated by the global MPC controller. HVDC1 and HVDC2 (top) have

the same power injections, HVDC3 (center), HVDC4 (bottom). Ac-

tive power injection (bold), reactive power at rectifier terminal (thin)

and reactive power at inverter terminal (dashed).

4 Placement of HVDC links in power

systems

This chapter presents an algorithm for the selection of good HVDC locations withregard to the dynamic power system performance. The chapter is structured in twoparts. First, a general dynamic performance measure for the constrained control oflinear systems is presented. Secondly, the performance measure is applied to evaluateand rank candidate locations for new HVDC links in AC power systems.

4.1 Introduction

4.1.1 Planning of power systems with HVDC links

The planning of the future power system creates many engineering challenges, includingthe incorporation of novel power sources, distribution networks, protection systems, in-formation infrastructure and security systems. On a non technical side, it also involvesmulti-disciplinary challenges like the economic organization and regulation of powermarkets and the definition of legal and political agreements between transmission sys-tem operators and national parties. Economic and social considerations imply thatthe future power system can not be a complete redesign, but results from a gradualtransition of the current traditional AC network. Due to availability requirements, newelements in the power system have to be carefully evaluated, also regarding their effecton the system during critical disturbance situations such as faults, transients or powersystem oscillations. The assessment of the impact on the power system dynamics causedby newly placed HVDC links and other power system components is the main topic ofthis chapter. FACTS and in particular voltage source converter HVDC links have theadvantage that they can actively control the active and reactive power injections intothe power grid at each of the terminals. This can be exploited to enhance the dynamicpower system performance during transients, for instance by modulating the HVDC’spower injections in order to damp power oscillations [SA93,FMLM11,PAMM11].

92 4 Placement of HVDC links in power systems

4.1.2 General performance measures for actuator selection

Actuator selection is a critical step for the design of systems since it determines thefundamental properties of the resulting control problem. Choosing and placing anactuator is a multi-objective optimization with constraints from different domains, suchas physical system limitations, performance specifications and economic requirements.Problems from power system control involve a large number of dynamic states, at leasttwo to six for every generator modeled in the grid [Kun93]. They are also subjectto constraints that have to be taken into account when designing the control schemesselecting the actuator setpoint. The first part of this chapter shows how to assess theperformance of these control schemes for different actuator selections and disturbancescenarios.

In linear control theory, the traditional approach to the comparison of different actu-ators of unconstrained systems involves the computation of the controllability matrix orthe controllability gramian and a rank comparison [SP05]. While the approach can beapplied to relatively large systems, it only gives a limited information, e.g. the dimen-sion of the controllable subspace, for systems without nonlinearities and no constraints.For constrained nonlinear systems with disturbances and control inputs from a boundedset, reachability analysis can be used to determine the set of controllable states. Sincethe set computations involve the solution of partial differential equations [MBT05] orpolyhedral algebra [RKML06], the approach is limited to low dimensional systems.Furthermore, it is desired to compute a performance metric, that typically involves themaximization of a scalar function over the controllable set.

The approach presented in this chapter is suitable for large linear systems, subjectto symmetric constraints on the states, the input and the initial state, restricted toellipsoidal sets. Using results from [BEFB94], the problem can then be reformulatedas semidefinite program (SDP), including a scalar value that measures the achievableactuator performance. The approach is tractable and represents an alternative betweenthe simple computation of the controllability matrix and a full reachability analysis.

4.1.3 Power system performance measures for HVDC selection

There are several approaches to the dynamic analysis of power systems after a dis-turbance. The traditional approach usually involves extensive simulations of differentdisturbance scenarios [TDB08]. While the simulation model can be very detailed, itis not possible to cover all disturbance situations and critical transients occurring inthe system may be missed. Alternatively, small signal analysis of a linearized systemmodel allows to determine the local behavior during small disturbances [Kun93]. This

4.2 Performance evaluation of actuators 93

approach works for generic small disturbances but does not evaluate the global behav-ior where the limitations of the system are met. Finally, an approach deriving globalstability certificates for the nonlinear model of unconstrained pure AC power systemshas been presented in [DB12].

In order to use any of the previous approaches for the assessment of possible HVDCcandidate locations, a performance measure needs to be defined and computed for eachof the candidates. Furthermore, the measure should reflect the fact that during largedisturbance scenarios, some components of the power system may operate at their lim-itations such as physical limits and saturations for security reasons. A grid controlscheme for HVDC links and other FACTS devices that uses a linearized approxima-tion of the system and captures the limitations was presented in Section 3.2.4 andin [FMLM11]. The approach is based on Model Predictive Control, a discrete timecontrol approach, solving a quadratic program (QP) to optimize a cost function, mea-suring the obtainable performance of the system. Since the outcome of the optimizationdepends on the dynamic state of the system and is recomputed at a sampling rate of50 Hz, the function is only known pointwise and can not be directly used as a generalgrid performance measure.

In this chapter, a general grid performance measure is defined and used as decisioncriteria for the placement of HVDC links in two AC system examples with a givenrange of initial disturbances and system limitations.

4.2 Performance evaluation of actuators

This section defines the performance measure used to evaluate different actuators andgives the interpretation for power grids controlled using HVDC links.

4.2.1 Problem statement

The performance of a constrained linear system with a given actuator is the solutionof the min-max optimization problem

J = minK

maxx(0)∈X0

∫ ∞

0z(t)TMz(t)dt (4.1)

s.t. x(t) = Ax(t) +Bu(t) (4.2)

z(t) = Cx(t) (4.3)

u(t) = Kx(t) (4.4)

∀x(0) ∈ X0 , ∀t : (4.5)


0 = Heq,xx(t) +Heq,uu(t) (4.6)

u(t) ∈ U (4.7)

where

X0 = {x ∈ Rnx : xTExx ≤ 1} (4.8)

U = {u ∈ Rnu : uTEuu ≤ 1} . (4.9)

The objective is the integral performance of the signals of interest, z, weighted withthe symmetric positive semidefinite matrix M = MMT ≥ 0. To obtain a tractableproblem formulation, the control input cannot be chosen freely as a function of timebut is restricted to a linear feedback policy (4.4). Using the feedback gain K, the goalis to minimize the worst case performance over the set of initial states X0. The systemequations for the given actuator are defined with the matrices A,B,C of appropriatedimensions. As a constraint, the input and the state have to satisfy the equalityconstraints defined by the matrices Heq,x and Heq,u. To leave a degree of freedom forthe control input, Heq,u must have less rows than columns. Furthermore, it is requiredthat all possible input trajectories remain in the set U. The ellipsoidal sets are definedusing the symmetric positive definite matrices Ex > 0 and Eu > 0.

4.2.2 Problem formulation using LMIs

This section presents an LMI formulation for the computation of the performancemeasure J , defined in (4.1)-(4.9). Throughout the section, results for linear matrix in-equalities [BEFB94] are applied. The same symbol is used to denote scalar inequalitiesand semidefiniteness constraints. For matrices, A ≥ B states that the matrix A−B ispositive semidefinite, with no eigenvalues in the open left half of the complex plane.

4.2.3 Worst case performance formulation

It is clear that the value of the integral in (4.1) depends on the initial dynamic stateof the power system, x(0). More precisely, if the grid controller is chosen as linearfeedback controller (4.4) and the matrix A + BK is stable, the value of the integralbecomes ∫ ∞

0z(t)TMz(t)dt = x(0)TPx(0) (4.10)

with the symmetric matrix P > 0 defined as the solution of the Lyapunov equation

P (A+BK) + (A+BK)TP + CTMC = 0 . (4.11)


If the choice of the controller K is known, P can be computed and the value of theintegral can be calculated for any given initial dynamic state x(0). For the applica-tion to the linearized power system, this means that the performance obtained for aspecific angular or frequency disturbance can be computed immediately using a simplequadratic function and without any simulation of the system trajectories.

To evaluate the worst possible case that yields the biggest cost (4.1), it is necessaryto compute the value of the integral not only for a specific initial state but insteadto determine the largest value that can occur for all possible initial states x(0) in theellipsoid X0. Since Ex > 0 one can compute

Ex = EET , E = V D1/2 (4.12)

with the columns of V containing the eigenvectors of Ex and D1/2 containing the squareroot of the eigenvalues of Ex on its diagonal. Then the transformation

x = E−T x (4.13)

converts the ellipsoid X0 in the x space into a unit ball in the x space and the largestvalue of the integral can be computed as an eigenvalue problem

maxx∈X0

xTPx = maxx:xT x≤1

xT E−1PE−T x

= λmax(E−1PE−T ) . (4.14)

4.2.4 Handling of equality constraints

To incorporate the equality constraints

Heq,xx(t) +Heq,uu(t) = 0 (4.15)

into the model, a change of variables

u = −Heq,uHeq,xx+Neq,uv (4.16)

can be performed using the pseudoinverse and the nullspace of the matrix Heq,u,

Heq,u = pinv(Heq,u) = HTeq,u(Heq,uH

Teq,u)

−1 (4.17)

Neq,u = null(Heq,u) . (4.18)

The columns of Neq,u span the linear space that is mapped to the origin throughmultiplication with Heq,u. Since

Heq,uHeq,u = I and Heq,uNeq,u = 0 , (4.19)


the constraint (4.15) will always be satisfied for inputs of the form (4.16). Moreover,the system will change to

x = Ax+Bu (4.20)

= (A− BHeq,uHeq,x)x+BNeq,uv (4.21)

= Ax+Bv . (4.22)

If Heq,u has neq linearly independent rows, the new input v has nv = nu−neq elements.

4.2.5 Handling of input inequality constraints

The constraints on the control input (4.7) are stated in ellipsoidal form,

∀x(0) ∈ X0, ∀t : u(t)TEuu(t) ≤ 1 . (4.23)

where Eu > 0. A different interpretation of the matrix P solving (4.11) is that of aninvariant set. Once the dynamic state at time t1 satisfies the ellipsoidal constraint

x(t1)TPx(t1) ≤ k (4.24)

for some k > 0, it will also satisfy this constraint for all t > t1. Using (4.14) and (4.24),one can conclude, that all trajectories x(t) starting in the set X0 satisfy the inequality

x(t)TPx(t) ≤ λmax(E−1PE−T ) . (4.25)

To satisfy (4.23), the linear feedback u = Kx has to satisfy

KTEuK ≤P

λmax(E−1PE−T )(4.26)

4.2.6 LMI formulation

The LMI formulation to determine the closed loop performance as in (4.1) incorporatesall constraints derived in the previous sections. All equalities have therefore beenremoved from the problem using the procedure in Section 4.2.4 and the matrices A andB have been computed using (4.22). The feedback controller of the remaining inputsis denoted by v = Kx. It is useful to formulate the problem in terms of Q = P−1 > 0.In this case (4.11) becomes

(A+BK)Q+Q(A +BK)T +QCTMCQ = 0 . (4.27)


Replacing M = MMT , relaxing the equality as inequality and using the Schur comple-ment, (4.27) becomes

(

(AQ+BY ) + (AQ+BY )T QCTM

MTCQ −I

)

≤ 0, (4.28)

where Y = KQ makes the constraint linear in the optimization variables Q and Y . Toformulate the objective

minK

maxx(0)∈X0

x(0)TPx(0) (4.29)

using (4.14), introduce the upper bound

λmax(E−1PE−T ) = λmax(E−1Q−1E−T ) ≤1

s(4.30)

with s ∈ R, s > 0. This bound is equivalent to

Q−1 ≤EET

s=Exs

(4.31)

andQ− sE−1

x ≥ 0 . (4.32)

Minimizing λmax is achieved by maximizing s in the objective of the semidefinite pro-gram.

Finally, (4.26) can be written as

Y TEuY ≤ Qs , (4.33)

which is equivalent to(

Q Y T

Y sE−1u

)

≥ 0 (4.34)

Hence, the semidefinite program to solve (4.1)-(4.7) is

maxs>0,Q>0,Y

s (4.35)

s.t. (4.28), (4.32), (4.34)

The linear control law u = Kx can be extracted as

K = −Heq,uHeq,x +Neq,uY Q−1 (4.36)

with Heq,x specified, Heq,u and Neq,u computed as in Section 4.2.4, and Y and Q fromthe solution of (4.35). The worst case performance J as in (4.1) is given by 1/s.


��

��

Figure 4.1: Two decision criteria J1 and J2 for different HVDC placements (dots).

Placements on the Pareto frontier (solid line) can not decrease one cri-

teria without increasing another. Other HVDC placements (gray dots)

are Pareto inefficient. Minimizing along a specific direction (orthogonal

to the dashed line) returns an optimal HVDC placement.

4.3 Optimal placement of HVDC links

4.3.1 HVDC placement as multi-objective optimization

The installation of new power system components, in particular HVDC links, is costlyand requires a careful comparison of the competing decision criteria. Typically this isdone using Multiple Criteria Optimization [EG02]. The decision criteria include theprice of the installation of the HVDC, the potential economic gain on the power market,a general measure of political or legal feasibility of the installation and a measure ofthe grid performance during transients.

Fig. 4.1 illustrates the principle tradeoff between competing objectives. Decisions aretypically chosen somewhere on the Pareto efficiency curve, an example using a simplelinear decision rule is shown in the figure. This chapter focuses on nobj = 2 objectives.First, a performance measure of the power grid evaluates how effectively the HVDClink under consideration can be used for power oscillation damping. As a secondaryobjective, an economic measure is the variable cost of a potential HVDC link locations,given by the length of the DC line.

4.3 Optimal placement of HVDC links 99

4.3.2 Power grid control using HVDC

This section formulates the control model arising when using the HVDC for poweroscillation damping. After defining the constrained power system model, a performancemeasure is introduced, which reflects how good the control scheme works.

The dynamic equations of the power system model include the generator dynamicsas well as the dynamics of local generator controls and are linearized around the currentoperating condition [MBB08],

x(t) = Ax(t) +Bu(t) (4.37)

z(t) = Cx(t) . (4.38)

The dynamic state x(t) ∈ Rnx consists of the states vectors of the ngen generators,including the deviation of the rotor angle, ∆δ, and the deviation of the frequencies,∆ω. The vector z(t) contains the signals of interest, a subset or a combination of thedynamic system state. The manipulated variables u(t) ∈ Rnu are the reference valuesof the actively controlled power system components. For the examples in this chapter,the injections of a single HVDC link are parameterized using

u(t) = [Il,α, Ir,α, Il,β, Ir,β]T , (4.39)

the αβ-phasor representation of the HVDC currents at the left and at the right termi-nal.Furthermore, the input is subject to constraints

u(t) ∈ U = {u ∈ Rnu : uTEuu ≤ 1} (4.40)

The ellipsoidal input constraint U describes the operating range of the HVDC linksthat can be used for power system damping. It originates from the active and reactivepower limits of the HVDC links, as well as bounds of the AC system such as line loadlimitations.

The initial statex(0) ∈ X0 = {x ∈ Rnx : xTExx ≤ 1} (4.41)

is assumed to lie in an ellipsoidal set X0, describing the worst expected system stateafter a disturbance. The set is generated by bounds on the initial frequency deviationof each of the generators, |∆ω| ≤ ω, and bounds on initial the rotor angle deviations,|∆δ| ≤ δ. Using different bounds for different parts of the system, the set X0 canbe used to describe weaker or more unreliable parts of the power grid that should beconsidered for the placement of HVDC links.

To use the HVDC link for power system stabilization and damping, the controlledvariables u(t) are chosen based on the current dynamic state of the system x(t). In


practice, the dynamic state of the power system is not directly measured but needsto be estimated using Wide Area Measurement Systems. However, for a comparativeevaluation of different HVDC locations the controlled variables are assumed to be givenby a linear feedback,

u(t) = Kx(t) . (4.42)

The controller K should respect the constraints (4.40) for all expected initial states(4.41).

The performance objective of the power system control problem is formulated as anintegral of a quadratic function of the signals of interest,

Jperf =∫ ∞

0z(t)TMz(t)dt , (4.43)

where the matrix M weights the signals of interest. For all HVDC placement examples,the signals of interest are chosen to be the generator frequency deviations

z(t) = ∆ω(t) = [∆ω1(t), ...,∆ωngen(t)]T ∈ Rngen

, (4.44)

and are weighted with

M = diag(H1, H2, ..., Hngen) . (4.45)

according to the generator inertias Hi.

4.3.3 Performance assessment of HVDC locations

The selection of the optimal feedback gain (4.42) requires to solve the following opti-mization problem

J∗perf = min

Kmaxx(0)∈X0

Jperf (4.46)

s.t. (4.37), (4.38), (4.40), (4.42), (4.43) .

In words, the optimal closed loop performance measure J∗perf is the best achievable

power system performance (4.43) for worst expected initial disturbance in X0. Thevalue of J∗

perf will differ depending on the selected HVDC location. The smaller thevalue of J∗

perf, the better the HVDC location is suited for power system control.

The computation of the grid performance measure (4.46) is a min-max problem anddifficult to compute directly. However, the assumption of a quadratic performanceintegral (4.43), linear system dynamics (4.37), ellipsoidal input constraints (4.40) andellipsoidal bounds on the initial state (4.41) allows to convert problem (4.46) to an

4.3 Optimal placement of HVDC links 101

equivalent formulation as a semidefinite program (SDP). The SDP is a convex opti-mization problem over matrix variables, where the original problem constraints areconverted to linear matrix inequalities (LMIs). The details of the conversion from(4.46) to an SDP are given in [FM13a], based on basic LMI results given for instancein [BEFB94]. The approach also allows to incorporate equality constraints on x and u,which becomes necessary when the power system model (4.37) also involves algebraicconstraints. The result without equality constraints is summarized as follows.

A positive scalar s is introduced, whose inverse serves as upper bound to the perfor-mance integral

Jperf ≤1

s. (4.47)

As shown in Section 4.2.6, the min-max problem (4.46) can be reformulated as thesemidefinite program,

1

J∗perf

= maxs>0,Q>0,Y

s (4.48)

s.t.

0 ≥

(

(AQ+BY ) + (AQ+BY )T QCT M

MTCQ −I

)

,

0 ≤ Q− sE−1x .

0 ≤

(

Q Y T

Y sE−1u

)

,

with the scalar optimization variable s and matrix optimization variables Q and Y .The inequality signs on matrices denote semidefiniteness constraints.

All variables besides s, Q and Y are constant and can be computed from the originalproblem data. The matrices Ex and Eu are used to define the ellipsoidal sets X0 andU. The matrices A, B and C are the state space matrices (4.37). The matrix M comesfrom the factorization M = MMT of the performance weight (4.43).

The solution of the SDP (4.48) is computed using the Matlab toolbox Yalmip [L04]in combination with the solver SDPT3 [TTT99].

4.3.4 Placement algorithm for a single HVDC link

The dynamic performance in a power system during transients can serve as one decisioncriteria for the planning of newly constructed HVDC links. Other criteria includeeconomic as well as political considerations. In a power network with nbus buses that


can be connected to either terminal of the new HVDC link there are

ncand,1 =nbus(nbus − 1)

2(4.49)

candidate locations where the new HVDC link could be placed. If there are no fur-ther restrictions on the placement, each of these candidate locations can be evaluatedregarding the best achievable performance during transients by solving the problemdefined in (4.1)-(4.7). The solution of the SDP formulation (4.48) provides the perfor-mance measure J = 1/s that can serve as decision criteria to compare different HVDCcandidate locations.

Each of the ncand,1 candidate locations alters the power flow and the controllableinputs of the power system and corresponds to different system matrices

(Ai, Bi) i = 1, ..., ncand,1 . (4.50)

To choose the best location for the addition of a single HVDC link, it it suffices tosolve (4.48) for each pair (Ai, Bi) and to compute the resulting performance values ofJi. The smallest value

i∗ = arg miniJi (4.51)

yields the optimal HVDC placement with respect to the dynamic performance measure(4.1).

4.3.5 Placement algorithm for multiple HVDC link

In principal, the evaluation of the effect from multiple HVDC links on the closedloop performance of the power system can be carried out as outlined in Section 4.3.4.However, the number of candidate topologies ncand is a combinatorial number of thenumber of buses, nbus, and HVDC links, nhvdc. If repeated HVDC links are permitted,

ncand =(ncand,1 + nhvdc − 1)!

(ncand,1 − 1)!nhvdc!(4.52)

where ncand,1 denotes the number of candidate location when placing a single HVDClink, as in (4.49).

In order to keep a tractable problem formulation, a heuristic is required to elimi-nate the combinatorial complexity. To this end, a recursive placement is presented aspseudocode in Algorithm 1. At each iteration, the algorithm loops through all non-identical pairs of buses (i, j) as candidate locations for a new HVDC link and computesthe linearized power system model (A,B) using the function ComputeModel. This

4.4 Examples 103

function has the location of the left and right HVDC terminals as argument and car-ries out the linearization of the power system dynamic equations. The algorithm thensolves the corresponding SDP (4.48) and keeps track of the best location computed sofar. In the next iteration, the optimal locations from the previous levels is kept in theupdated index vector Il and Ir. The process is repeated until nhvdc links were placed,requiring a total of

ncand = nhvdc ·ncand,1 (4.53)

topologies to be evaluated.

Algorithm 1 PlaceHVDCs(B, nhvdc)

Require: Set of bus numbers B, number of HVDC links to place nhvdc

Ensure: Set of bus numbers for the left terminal Il, Set of bus numbers for the rightterminal Ir

1: Il = ∅, Ir = ∅2: while nhvdc > 0 do

3: Jmin ←∞

4: for i ∈ B do

5: for j ∈ B, j 6= i do

6: (A,B)← ComputeModel([Il, i], [Ir, j])7: J ← SolveLMI(A,B)8: if J < Jmin then

9: (Jmin, i∗, j∗)← (J, i, j)

10: end if

11: end for

12: end for

13: Il ← [Il, i∗]

14: Ir ← [Ir, j∗]

15: nhvdc ← nhvdc − 116: end while

4.4 Examples

This section first illustrates the general actuator evaluation in LTI systems. It thenapplies the evaluation scheme to candidate HVDC locations in a meshed AC system,using a criterion given by the performance measure J∗

perf, through the solution of (4.46).A secondary criterion for the HVDC selection is the length of the proposed DC links,computed as the distance between the two buses it connects.


4.4.1 Illustration of the LMI based performance evaluation

To illustrate the LMI based approach to the solution of (4.1)-(4.7), consider the simpletwo dimensional test system

x = Ax+Bu (4.54)

z = Cx (4.55)

with

A =

(

−1 2−3 −4

)

, B =

(

0 11 2

)

, C =

(

1 00 1

)

.

Furthermore, the matrices

M =

(

1 00 2

)

, Ex =

(

4 11 1

)

, Eu =

(

0.05 00 0.2

)

,

characterize the objective function, the ellipsoid containing the initial states and theellipsoid of the input constraints. The SDP (4.48) was encoded using YALMIP [L04]and solved using SDPT3 [TTT99]. The solution provides the gain K and the Lyapunovweight P as

K =

(

4.326 −1.811−1.245 −1.523

)

, P =

(

0.224 −0.002−0.002 0.113

)

,

which reduces the performance measure

J = λmax(E−1PE−T ) = 1/s = 0.180 (4.59)

as much as possible. Fig. 4.2 shows that the bound is approached by the integratedcost

J(t) =∫ t

0z(τ)TMz(τ)dτ (4.60)

for some of the randomly sampled initial conditions. The critical initial conditions canbe seen in Fig. 4.3, where the set of initial states X0 is contained inside the invariantset xTPx ≤ 1/s. Also, Fig. 4.4 illustrates that all input trajectories remain inside thespecified boundaries of U.

4.4.2 Placement of HVDC links in the two area power system

The initial motivation of the closed loop performance assessment of constrained linearsystems was the selection of the best location for newly constructed HVDC links ina meshed AC power grid. A benchmark case is the two area power system with four

4.4 Examples 105

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

t

J(t)

Figure 4.2: LMI based control of constrained 2D test system: The integrated cost

J(t) approaches the upper bound 0.180 for some of the randomly sam-

pled initial conditions. This bound was determined tightly and mini-

mized in the SDP (4.48) as 1/s .


−1.5 −1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

x1

x2

Figure 4.3: LMI based control of constrained 2D test system: Specified set of initial

states X0 (dark ellipse), invariant set xTPx ≤ 1/s (light ellipse) and

state trajectories for randomly sampled initial conditions (solid lines).

4.4 Examples 107

−4 −3 −2 −1 0 1 2 3 4

−3

−2

−1

0

1

2

3

u1

u2

Figure 4.4: LMI based control of constrained 2D test system: Specified set of ad-

missible inputs U (dark ellipse) and input trajectories for randomly

sampled initial conditions (solid lines).


12 3 4 5 6

7

G1

G2

G3

G4

Figure 4.5: Test system with four generators and seven buses in two areas [Kun93].

generators and 11 buses, connected through a weak AC link, as presented in Section 12.8of [Kun93] and illustrated in Fig. 4.5. Since four of the buses are transformer buses,there remains a total of nbus = 7 candidate connections for the terminals of new HVDClinks. Without any further constraints on the placement, this results in ncand,1 = 21possible locations for the placement of a single HVDC link.

The nonlinear dynamic equations consist of four second order generator models withthe rotor angle and the frequency deviations (δi,∆ωi), i = 1, ..., 4 as dynamic states.The model is linearized about the current operating point, using the parameters andoperating points given in [Kun93]. The resulting system has eight dynamic states andfor every HVDC link three free inputs, one used to satisfy the active power balanceconstraint. Algorithm 1 is used to repeatedly evaluate the 21 HVDC locations, placinga link of 200 MW active power and 40 MVar reactive power rating. The algorithmimposes a bound on the current injections, uTu ≤ 2. The ellipsoidal set of initial statesX0 is chosen within the bounds |δi| ≤ 0.5 and |∆ωi| ≤ 0.01. The performance measurein (4.1) integrates the equally weighted frequency deviations

∑

i ∆ωi(t)2.

Fig. 4.6 shows the performance measure that is obtained using a single HVDC linkin each of the 21 locations. The square denotes the optimal location that is kept forthe second iteration of Algorithm 1. Note that the general level of the performancemeasure J in Fig. 4.6 decreases as HVDC links are added to the system. Also, thelocation of the peaks change since at each level, the optimal HVDC links from theprevious levels remain in the system and contribute to the rejection of the most criticaldisturbance directions. The final result, depicted in Fig. 4.7, shows the topology of thesystem after three iterations of Algorithm 1. Fig. 4.8 and Fig. 4.9 show the simulationresult of the system with three HVDC links for randomly sampled initial conditions inX0. It can be seen that the trajectories approach the computed performance bound ofJ = 3.705 while respecting the input constraints.

Besides the dynamic performance criterion, each candidate set of HVDC locationyields a simple cost estimate, the total length of the DC links. For such a small system,it is possible to evaluate all combinations of multiple HVDC links, which requires tosolve 21, 231 or 1771 SDPs as in (4.48) to optimally place one, two or three HVDC

4.4 Examples 109

2 4 6 8 10 12 14 16 18 200.5

1

1.5

2

HVDC location

log 10

(J)

1 HVDC link

2 HVDC links

3 HVDC links

Figure 4.6: Recursive placement of three HVDC links in a four generator system:

Performance measure J as in (4.1) for 21 different locations. Optimal

location of the first link (square), the second link (triangle) and third

link (circle).

G1

G2

G3

G4

HVDC 1

HVDC 2HVDC 3

Figure 4.7: Recursive placement of three HVDC links in a four generator test sys-

tem: All bus pairs are candidate links. Each HVDC link added to the

system minimizes (4.51), based on the solution of (4.1)-(4.7).


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

time

Jt

Figure 4.8: LMI based control of power system with 3 HVDC links: The integrated

cost J(t) approaches the upper bound 3.705 for some of the randomly

sampled initial conditions. This bound was determined tightly and

minimized in the SDP (4.48) as 1/s .

4.4 Examples 111

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

time

|u| 2

Figure 4.9: LMI based control of power system with 3 HVDC links: Specified bound

on input norm (dashed line) and input norm trajectories for randomly

sampled initial conditions (solid lines).


Placement strategyBest HVDC locations(multiples HVDCs per location allowed)

J∗perf as in (4.46)

1 link, exhaustive Bus 3-5 15.62 links, exhaustive Bus 1-2, Bus 6-7 4.83 links, exhaustive Bus 1-2, Bus 1-2, Bus 6-7 3.23 links, recursive Bus 3-5, Bus 2-7, Bus 1-2 5.1

Placement strategyWorst HVDC locations(multiples HVDCs per location allowed)

J∗perf as in (4.46)

1 link, exhaustive Bus 6-7 127.42 links, exhaustive Bus 6-7, Bus 6-7 113.93 links, exhaustive Bus 6-7, Bus 6-7, Bus 6-7 110.4

Table 4.1: Four generator test system. Optimal and worst HVDC locations (com-

pare bus numbers in Fig. 4.5) in terms of the performance measure J∗perf.

Initial disturbance equally distributed among the generators.

links. The result is plotted in Fig. 4.10 It can be seen that the HVDC links with thebest performance are not the longest ones. To take the placing decisions for largerproblems, the HVDC links can be placed recursively, one evaluated after the other.For three HVDC links in the four generator test system, this requires to solve only 63(instead of 1771) SDPs. The result shows only a slightly worse performance in termsof the performance measure J∗

perf, as illustrated in Fig. 4.10. The total length of theHVDC links, shown for illustration on the vertical axis, was not part of the selectionprocess of the HVDC locations. Tab. 4.1 shows the corresponding HVDC locations.It can be seen that the single HVDC link is chosen between the two areas on the leftand the right. The two HVDC links are chosen parallel to the local oscillations withinthe two main areas. The third links further strengthens the weak area on the left, orin the recursive placements, strengthens the connection between the two areas. Thereasoning for the selection of these locations seems intuitive for this small example,but is chosen automatically by ranking the performance measures J∗

perf. The table alsocontains the information about the worst HVDC locations, according the performancemeasure J∗

perf. The worst link location is always in the area on the right, which alsooccurs within the best placements. Apparently this link can only be efficiently used fordamping in coordination with other HVDC links in the left area.

Fig. 4.11 to Fig. 4.15 shows the trajectories of the generator rotor angles, relative totheir center of inertia for different choices of HVDC links. The initial states used forthe plots were chosen as the worst possible scenario within X0, thereby reaching theperformance bound computed as J∗

perf.

4.4 Examples 113

0 100 200 300 400 500 600 700 800 9000

20

40

60

80

100

120

140

total HVDC length [km]

J

Figure 4.10: Placement of up to three 200 MW HVDC links in the four generator

test system: Performance measure J∗perf (4.46) as a function of the

total DC line length. One HVDC link (blue), two HVDC links (red),

three HVDC links (green). The optimal placement obtained using

exhaustive search (large circles) and the recursive placement of three

HVDC links (black square).


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time [s]

angl

es [r

ad/s

]

Figure 4.11: Four generator test system with HVDC links used for damping: Rotor

angle trajectory relative to the center of inertia after initial disturbance

and with HVDC damping control.

Result with no HVDC link chosen to minimize J∗perf as in (4.46) via

exhaustive search.

4.4 Examples 115

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time [s]

angl

es [r

ad/s

]




Result with one HVDC link chosen to minimize J∗perf as in (4.46) via

exhaustive search.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time [s]

angl

es [r

ad/s

]




Result with two HVDC link chosen to minimize J∗perf as in (4.46) via

exhaustive search.

4.4 Examples 117

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time [s]

angl

es [r

ad/s

]




Result with three HVDC link chosen to minimize J∗perf as in (4.46) via

exhaustive search.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time [s]

angl

es [r

ad/s

]




Result with three HVDC links, chosen via recursive placement of one

HVDC link at a time, each minimizing J∗perf as in (4.46).

4.4 Examples 119

Figure 4.16: European network model with 74 buses and 130 AC branches

4.4.3 Placement of HVDC links in the European power system

To illustrate the placement algorithm on a large power system, consider a reduced modelof the ENTSO-E power grid [Haa06], illustrated in Fig. 4.16. The model consists ofnbus = 74 buses and with no further constraints imposed on the HVDC locations,yields a total of ncand = 2701 possible HVDC locations. In particular, no candidateHVDC links were discarded that are unfeasible due to economic, geographic or politicalreasons. The load flow of the test case is based on a system snapshot during mid-dayoperation. Since the power transmitted over an HVDC link is small compared to theoverall power generated in the system, the closed loop performance depends stronglyon the location of the HVDC links. The set of initial disturbances X0 is chosen for eachgenerator to be the same as in the two area example, in Section 4.4.2. The bounds onthe active and reactive HVDC power injections of the HVDC are chosen to be 1000MW and 500 MVar. To reduce the computational complexity for solving the SDP


0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

HVDC length [km]

J

Figure 4.17: Placement of one 1000 MW HVDC link in the European test system:

Performance measure J∗perf as in (4.46) as a function of the DC line

length. The ten best placements (red) and the optimal placement

(large circle), obtained using exhaustive search.

in (4.48), balanced model reduction has been used to decrease the dimension of thedynamic system (4.37).

Fig. 4.17 shows the performance measure J∗perf and the DC line length for the place-

ment of a single HVDC link. The corresponding HVDC locations of the ten smallestvalues of the performance measure is shown in Fig. 4.18. The dominant orientation ofthe HVDC locations is along the North-East to South-West direction. The only HVDClink towards Southern Italy ranks as number ten according to the performance mea-sure. Interestingly, the best placement is obtained with a rather short line in SouthernFrance.

Due to the large number of HVDC locations, an exhaustive evaluation of all combi-nations of multiple HVDC links becomes quickly intractable. Therefore the placement

4.4 Examples 121

Figure 4.18: Placement of a first HVDC link in a European network model using

the performance measure J∗perf as in (4.46): The ten best links (bold

blue) and the selected link (bold green).


problem for multiple HVDC links needs to be solved recursively, placing one HVDC linkafter the other. Fixing the optimal HVDC link, the evaluation procedure is repeated toadd a second link, as shown in Fig. 4.19. With the North-East to South-West directiontaken care of through the first HVDC link, the preferable orientation for the secondHVDC link is orthogonal, along the North-West to South-East direction. Adding athird HVDC also yields some suggestions for the East-West orientation, Fig. 4.20.Clearly some of the suggestions are not feasible in practice, but still give some insightsto interesting candidate locations for further investigation during the planning of newHVDC links.

To estimate how strongly the best HVDC link differs from the rest, define the ratio

Jrel,i =J∗

perf,i

mini(J∗perf,i)

, (4.61)

of the sorted grid performance measures J∗perf,i (4.46) for different HVDC locations

i ∈ [1, ..., ncand] and the minimum grid performance measure, corresponding to theselected HVDC location. Fig. 4.21 shows this ratio for the top ten HVDC locations forthe recursive placement of three HVDC links. The top ten HVDC locations differ byless then a factor of 1.4 from the optimal locations. Therefore, the preference of eitherof the top ten HVDC locations is not very strong.

4.5 Conclusion

This chapter presented an efficient approach for the closed loop performance evaluationof a constrained linear system with different actuator choices. The min-max problemthat arises from the actuator evaluation is formulated as an LMI, embedding equalityconstraints and actuator inequality constraints. A heuristic is proposed for the selectionof multiple actuators. The approach is illustrated with the problem of placing HVDClinks in a two area power system test case.

The actuator performance measure has then be applied to the problem of evaluatingHVDC locations in meshed AC systems. The performance measure characterizes thesuitability of a given HVDC link for damping control of the AC network. The perfor-mance measure accounts for HVDC constraints and prior knowledge of expected systemdisturbances and can be efficiently computed using semidefinite programming. In ad-dition, an economic measure characterizes the variable cost of the DC line, measuredby its length.

The evaluation scheme has been applied to a simple four generator test system aswell as a larger model of the European power grid and show a successively improved

4.5 Conclusion 123

Figure 4.19: Placement of a second HVDC link in a European network model using

the performance measure J∗perf as in (4.46): Previously placed link

(bold orange), the ten best links (bold blue) and the selected link

(bold green).


Figure 4.20: Placement of a third HVDC link in a European network model using

the performance measure J∗perf as in (4.46): Previously placed links

(bold orange), the ten best links (bold blue) and the selected link

(bold green).

4.5 Conclusion 125

1 2 3 4 5 6 7 8 9 101

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

HVDC locations (sorted)

J/J*

1 HVDC link

2 HVDC links

3 HVDC links

Figure 4.21: Recursive placement of three HVDC links in a European network

model: Relative performance measure Jrel (4.61) for different HVDC

locations (sorted). At each recursion, the top ten HVDC locations

differ by less than a factor of 1.4 from the optimal HVDC location.


damping of the system. The presented evaluation method with an optimization basedperformance measure can equally be used to evaluate other active control componentsof the power grid, such as FACTS or PSS devices.

5 Outlook

The results on HVDC based power system control presented in this thesis motivatefurther investigations in several directions, that are briefly discussed in this chapter.

5.1 Problems with centralized grid control

The positive simulation results of the European system, for instance the bold bluetrajectories in Fig. 5.1 (a copy of Fig. 3.24 in Section 3.3.4), provide an upper bound onthe HVDC’s potential for oscillation damping and the reduction of relative frequencydeviations. For a more realistic scenario, towards a control scheme that could begradually introduced by a transmission system operator, the starting assumptions ofthe grid control approach need to be revisited.

The assumptions of the centralized grid controller as presented in Section 3.2.4are:

1. A sufficiently detailed nonlinear and linearized power system model is available,capturing all relevant dynamics. Changes in the network topology, generationand demand are known immediately.

2. A full measurement of the dynamic power system state, as defined in the detailedmodel, is also available. No measurement or estimation uncertainties occur.

3. All delays due to the communication channels and the computational time of theMPC optimization can be neglected.

4. All transmission system operators of the large network have to share all availableinformation but also have to accept temporary setpoint changes of HVDC linksin their region, made by an external centralized automatic grid controller.

The technical aspects in assumption 3 are probably the easiest to satisfy. Sophisti-cated implementations of the MPC optimization schemes exist [Dom13,Ric12], expectedcomputational and communication delays can be compensated in the MPC problem for-mulation [Mac01], and the closed loop power system performance seems to deteriorategracefully with unexpected delays (see Section 3.3.3.5).

128 5 Outlookreplacemen

Aver

age

freq

uen

cy

(Hz)

Fre

qu

ency

dev

iati

on

(mH

z)

time (s)

50.00

49.99

49.98

49.97

49.96

49.95

0.1

0.2

0.3

0.4

00

0

5

5

10

10

15

15

20

20

25

25

30

30

35

35

40

40

45

45

50

50

Figure 5.1: Loss of a large power plant: Average frequency ω (top) and average fre-

quency deviation σ (bottom). HVDC active power and voltage reference



ments (bold blue).

5.2 Towards distributed grid control 129

Similarly, the full state measurement assumptions 2 may become tractable in thefuture. The installation of new phasor measurement units (PMU) and the accelerationof data aggregation from supervisory control and data acquisition (SCADA) systemstowards wide area measurement systems (WAMS) give an increasingly accurate andfast snapshot of the power system state.

The acquired data can be used to estimate first the parameters, using system iden-tification, and then the dynamic state of a predefined power system model structure.Naturally this always implies a certain violation of assumption 1. It is also uncertain,whether all „relevant dynamics“ that contribute to a transient are captured in themodel. However, through the linearization, the MPC prediction model is already onlyan approximation of the power system model used for simulation. It has been observed,that as long as the prediction models indicates the correct direction of a power systemtransients, the grid controller can select a reasonable injection for the HVDC link. Thisobservation certainly needs further quantitative investigation. The prediction modelsused for the European system in Section 3.3.4 has several thousand states. The ro-bustness investigation of the closed loop performance will therefore likely require anapproach based on sampling of the uncertain plant models. If the model order can besufficiently reduced and the model uncertainty reasonably parameterized and bounded,robust control techniques such as [KBM96] could also become tractable.

The biggest issue for the implementation of a centralized grid controller is likely astrategic and political problem in assumption 4. A centralized grid controller coordi-nating multiple HVDC links, and possibly other devices such as FACTS, requires someautonomy transfered from the individual transmission system operators to a centralcontrol unit. Additionally, the decisions made during critical transients could requirea weaker pursuit of the individual objectives for the greater good of the whole powersystem. In any case, an agreement on the admissible set of free injection adjustmentsof an HVDC link connecting distant areas of the grid requires careful negotiation of allparties involved.

5.2 Towards distributed grid control

A more realistic scenario for HVDC based grid control is a distributed approach, whereindividual HVDC links are controlled more independently, instead of fully centralized.

The opposite extreme case of a fully centralized grid controller is the fully decentral-ized grid controller, presented in Section 3.2.3.

The assumptions of the decentralized damping controller are:

130 5 Outlook

• No information, except the frequency deviations between the two HVDC buses,is available to the controller.

• No system wide measurements, power system model or topology changes areavailable to the controller.

• A single tuning of the damping controller (the PD gains) has to be compatiblewith all changes in the grid topology or grid operating condition.

Due to the simplicity, restricted measurement information, and conservative tuning, thelocal damping controller can only have a limited impact on power system transients.Therefore, the resulting dashed green trajectories, depicted in Fig. 5.1 provide a lowerbound on the HVDC’s potential for oscillation damping and the reduction of relativefrequency deviations.

A compromise between the two extreme cases, the fully centralized and the fullydecentralized grid controller is referred to as a distributed control approach. Typically,distributed controllers are applied to systems whose physics have naturally separatedstructure of agents with specific shared variables, for example fleets of autonomousrobots. The controller and communication structure follows the physical structure,with limited communication between the agents. If the overall control objective is alsoseparable between the agents, the distributed control approach can converge to a globaloptimum. An example of a distributed grid control approach for optimal automaticgeneration control, is given in [VHRW08].

For large power networks with HVDC links, the separable structure of the physicalplant is generally not given. All dynamic variables are instantaneously coupled to oneanother, while some couplings are weaker and some are stronger. For instance, gener-ators close to one another are strongly coupled. However, there is no clear separationof the dynamic states into agents.

As stated in the previous section, the fully centralized communication structurecorresponding to this coupling, is not realistic. Instead, a communication structure is

imposed through technical and organizational constraints, separating the grid controllerinto local control elements.

Fig. 5.2 illustrate a proposed hierarchical distributed power system control structure.The measurement signals of the nh local control blocks include local power systemvariables. For instance, if

yi,loc = [ωi,1, ωi,2, |V0,i,1|, |V0,i,2|]T , (5.1)

the measurements contain the frequency deviations and the nominal grid voltage level{ωi,l, |V0,i,l|} at the left, l = 1, and right, l = 2, HVDC terminal. The local fast gridcontroller can then be used to follow local power system objectives such as oscillation

5.2 Towards distributed grid control 131

damping between the terminals and support of the local bus voltage level. A supervisoryslower grid controller coordinates the different local controllers based on measurementsyglob from all over the power system, for example global frequency measurements,

yglob =[

ω1, ..., ωngen

]

(5.2)

In the case of HVDC links, it is possible to associate the manipulated injectionvariables with specific power system objectives. For instance the active power injectionshave a strong impact on frequency deviations between the two adjacent AC buses,while the reactive power injection have a strong impact on the voltage level of thebus connected to the HVDC terminal (see the simulation example in Section 3.3.2and [Kun93]). Without formal analysis, this suggests a possible selection of

vi = Pi,1,ref (5.3)

ui,ref = [Pi,1,ref, Qi,1,ref, Qi,2,ref]T (5.4)

= [vi, Qi,1,ref, Qi,2,ref]T . (5.5)

Once the variables are chosen and the communication structure with the correspondingupdate rate is known, the control problems for the local and global grid controller canbe formulated.

In a model based control approach, an appropriate control model of the surroundingpower system needs to be specified. For a known linear power system model with giveninput and output signals, standard model reduction techniques can be used to generatelocal prediction models of the power system. This has also been used in Chapter 4to simplify the SDP formulation for the HVDC placement. Alternatives include theidentification of black box models [Lju99] from the local measurements.

The decoupled hierarchical structure shown in Fig. 5.2 is not the only approach to adistributed control structure of the grid controller. However, as a compromise, it usesmore realistic assumptions than the fully centralized control scheme while providingmore coordination than the purely decentralized damping control scheme.

132 5 Outlook

Figure 5.2: Distributed grid controller structure. Fast controller track local objec-

tives using local measurements. Slower controller tracks global objec-

tives using global measurements.

A Appendix

A.1 Norm dependent scaling of discs

A frequently used result in Section 2.2 for the derivation of HVDC constraints is thefollowing result of planar geometry, formulated in Theorem A.4.

Definition A.1 A disc C(x0, r) ⊂ R2 with center point x0 ∈ R2 and radius r ∈ R+ is

a set defined by

C(x0, r) = {x ∈ R2 : (x− x0)T (x− x0) ≤ r2} . (A.1)

Definition A.2 The norm dependent mapping fd with the scaling direction d ∈ R2

is the transformation

fd : R2 → R

2 , fd(x) = x+ dxTx (A.2)

Definition A.3 A distorted disc Y(x0, r, d) ⊂ R2 is a set defined by

Y(x0, r, d) = {y ∈ R2 : ∃x ∈ C(x0, r) s.t. y = fd(x)} . (A.3)

In words, the set Y is constructed by shifting each point in the disc C along the directiond, scaled with its distance to the origin. This allows to formulate the main result ofthis section:

Theorem A.4 A distorted disc Y(x0, r, d) is a convex set. The set boundaries are

defined by one of the following:

1. A single ellipse.

2. A curve with two elements: A segment of an ellipse and a segment of a parabola.

A.1.1 Proof of the number of boundary elements

To construct the distorted disc, the original disc C(x0, r) is parameterized using twoscalar variables

ψ ∈ [0, π] (A.4)

134 A Appendix

t ∈ [0, 1] . (A.5)

Fig. A.1 illustrates the variable ψ, denoting the angle in the ed-coordinate frame. Theaxis d is aligned with the scaling direction d and spans the radius of the disc, i.e.

d =d

‖d‖r ‖d‖ = r dTd = r‖d‖ . (A.6)

The orthogonal axis

e =

[

d2

−d1

]

dT e = 0 (A.7)

completes the frame.

For a given ψ ∈ (0, π), where sin(ψ) 6= 0, the two extremal disc points are

x+(ψ) = x0 + e cos(ψ) + d sin(ψ) (A.8)

x−(ψ) = x0 + e cos(ψ)− d sin(ψ) , (A.9)

The line segment between the extremal points can be parameterized with t ∈ [0, 1] as

xψ(t) = tx+(ψ) + (1− t)x−(ψ) (A.10)

= x−(ψ) + d(2t sin(ψ)) (A.11)

= x0 + e cos(ψ)− d sin(ψ) + d(2t sin(ψ)) . (A.12)

After the transformation, the line segment’s orientation in the direction d is preserved,

yψ(t) = fd(xψ(t)) (A.13)

yψ(t) = xψ(t) + d(xψ(t)Txψ(t)) (A.14)

= x−(ψ) + 2d sin(ψ)t+ ...

d(4r2 sin2(ψ)t2 + 4 sin(ψ)(dTx0 − r2 sin(ψ))t+ ‖x−(ψ)‖2) . (A.15)

The boundary of the distorted disc can now be constructed one line segment at a time,by checking the maximum and minimum extension along the d-direction for each angleψ. These extremal extensions are given by

min0≤t≤1

Jψ(t) and max0≤t≤1

Jψ(t) (A.16)

where

Jψ(t) = dTyψ(t) (A.17)

= dTx−(ψ) + 2r2 sin(ψ)t+ ...

A.1 Norm dependent scaling of discs 135

r‖d‖(4 sin2(ψ)r2t2 + 4 sin(ψ)(dTx0 − r2 sin(ψ))t+ ‖x−(ψ)‖2) . (A.18)

It can be seen that Jψ(t) is a convex quadratic function in t, with the unconstrainedminimum at t∗,

0 =∂Jψ(t)

∂t(t∗) (A.19)

0 = 2r2 sin(ψ) + r‖d‖(8 sin2(ψ)r2t∗ + 4 sin(ψ)(dTx0 − r2 sin(ψ))) (A.20)

0 = r2 + 4r3‖d‖ sin(ψ)t∗ + 2r‖d‖dT (x0 − d sin(ψ)) (A.21)

0 = 1 + 4r‖d‖ sin(ψ)t∗ + 2dT (x0 − d sin(ψ)) , (A.22)

which yields

t∗ =1

2−

1 + 2dTx0

4r‖d‖ sin(ψ)=: t∗(ψ) . (A.23)

For the limit cases ψ = 0 and ψ = 1, the line segment xψ(t) becomes a singularpoint which is mapped to a singular yψ(t). For all other angles in the open intervalψ ∈ (0, π), the denominator of the right term of (A.23) is strictly positive. Therefore,t∗(ψ) is either above or below 0.5, depending only on the sign of the term 1 + 2dTx0

and not on the value of ψ. Consequently, only two of the cases illustrated in Fig. A.2are possible for a given set of parameters {x0, r, d}.

In the following, assume that 2dTx0 ≥ −1. The derivation for 2dTx0 ≤ −1 isanalogous. By inspection of the pictures on the right half of Fig. A.2, one obtains theoptimizers

tmax(ψ) = arg max0≤t≤1

Jψ(t) = 1 (A.24)

tmin(ψ) = arg min0≤t≤1

Jψ(t) = max(t∗(ψ), 0) . (A.25)

The unconstrained minimizer t∗(ψ) is used for the lower bound if

t∗(ψ) > 0 (A.26)

or equivalently, using (A.23), if

sin(ψ) >1 + 2dTx0

2|d‖‖d‖. (A.27)

The truth of inequality (A.27) depends on ψ only if the right side takes values in therange of the sine,

−1 ≤1 + 2dTx0

2|d‖‖d‖≤ 1 . (A.28)

136 A Appendix

The lower bound satisfied by the assumption 2dTx0 ≥ −1. If the upper bound is alsosatisfied, there are two critical angles at which t∗(ψ) = 0

ψ∗1 = arcsin

(

1 + 2dTx0

2r‖d‖

)

(A.29)

ψ∗2 = π − ψ∗

1 . (A.30)

In total, the closed form expression for tmin(ψ) becomes

tmin(ψ) =

t∗(ψ)(

1+2dT x0

2r‖d‖≤ 1

)

∧ (ψ∗1 ≤ ψ ≤ ψ∗

2)

0 else(A.31)

If the first constraint is violated,

1 + 2dTx0

2r‖d‖> 1 , (A.32)

one always has tmin(ψ) = 0, independently of ψ, defining a single lower boundaryelement. Otherwise one has two different lower boundary elements, defined on differentranges of ψ.

A.1.2 Proof of the shape of the boundary elements

Again it is assumed that 2dTx0 ≥ −1. The derivation of the boundary shape for2dTx0 ≤ −1 is analogous.

First, consider the case where

1 + 2dTx0

2r‖d‖> 1 . (A.33)

This gives∀ψ ∈ (0, π) : tmin(ψ) = 0, tmax(ψ) = 1 , (A.34)

and therefore

∀ψ ∈ (0, π) : xψ(tmin) = x−(ψ), xψ(tmax) = x+(ψ) . (A.35)

These points all lie on the boundary of the disc,

(x− x0)T (x− x0) = ‖x− x0‖2 = r2 (A.36)

and satisfyxTx = r2 + 2xTx0 − x

T0 x0 . (A.37)


Figure A.1: Parameterization of the disc points.

138 A Appendix

Figure A.2: Extrema of Jψ(t): 2dTx0 ≤ −1 (top and bottom left half), 2dTx0 ≥ −1

(top and bottom right).


This allows to rewrite the boundary after the transformation fd(.) as affine function,

y = fd(x) (A.38)

= x+ dxTx (A.39)

= x+ d(r2 + 2xTx0 − xT0 x0) (A.40)

= (I + 2dxT0 )︸︷︷︸

A

x+ d(r2 − xT0 x0)︸︷︷︸

b

. (A.41)

The transformed disc forms an ellipse in the y-variables,

‖A−1y + A−1b− x0‖2 = r2 . (A.42)

The degenerate case where A is not invertible corresponds to the limit of a very thinellipse becoming a line segment.

Secondly, consider the case where

1 + 2dTx0

2r‖d‖≤ 1 . (A.43)

This gives

tmin(ψ) =

{

t∗(ψ) ψ∗1 ≤ ψ ≤ ψ∗

2

0 else(A.44)

tmax(ψ) = 1 (A.45)

Again, the maximum extension in d-direction is always reached on the boundary ofthe disc since

xψ(tmax) = xψ(1) = x+(ψ) . (A.46)

For ψ < ψ∗1 and ψ > ψ∗

2 , the minimum extension in d-direction is also reached on theboundary of the disc since then

xψ(tmin) = xψ(0) = x−(ψ) . (A.47)

As before, this results in an ellipsoidal bound of the distorted disc as in (A.42).

For ψ∗1 ≤ ψ ≤ ψ∗

2, the minimum extension in d-direction is reached inside the disc,with

xψ(tmin) = xψ(t∗(ψ)) (A.48)

= t∗(ψ)x+(ψ) + (1− t∗(ψ))x−(ψ) (A.49)

= 2d sin(ψ)t∗(ψ) + x0 + e cos(ψ)− d sin(ψ) (A.50)

140 A Appendix

= 2d sin(ψ)

(

1

2−

1 + 2dTx0

4r‖d‖ sin(ψ)

)

+ x0 + e cos(ψ)− d sin(ψ) (A.51)

= d

(

sin(ψ)−1 + 2dTx0

2r‖d‖

)

+ x0 + e cos(ψ)− d sin(ψ) . (A.52)

The projection onto d,

dTxψ(tmin) = r2

(

sin(ψ)−1 + 2dTx0

2r‖d‖

)

+ dTx0 + dT e cos(ψ)− r2 sin(ψ) (A.53)

= dTx0 − r1 + 2dTx0

2‖d‖(A.54)

= const. . (A.55)

is independent of ψ. Hence the minimizer of Jψ(t) lies on a line orthogonal to thedirection d. Fig. A.3 illustrates the disc with the minimizing line with

x∗ = xψ(t∗(ψ)) (A.56)

and its disc intersections

xa = x−(ψ∗2) (A.57)

= x0 + e cos(ψ∗2)− d sin(ψ∗

2) (A.58)

= x0 + e cos(π − ψ∗1)− d sin(π − ψ∗

1) (A.59)

= x0 − e cos(ψ∗1)− d sin(ψ∗

1) (A.60)

xb = x−(ψ∗1) (A.61)

= x0 + e cos(ψ∗1)− d sin(ψ∗

1) , (A.62)

The minimizing line segment inside the disc can be parameterized with l ∈ [0, 1] as

x∗(l) = lxa + (1− l)xb = (xa − xb)l + xb , (A.63)

and yields the transformed line segment

y∗(l) = fd(x∗(l)) = x∗l + d(x∗(l)Tx∗(l)) (A.64)

= (xa − xb)l + xb + d(‖xa − xb‖2l2 + 2xTb (xa − xb)l + ‖xb‖

2) . (A.65)

By definition, the boundary points with l = 0 and l = 1 lie on the circle boundingthe disc. This ensures continuity of the transformed line segment with the ellipsoidalsegment.

Projecting the transformed line segment onto the de-coordinate frame,

eTy∗(l) = r‖xa − xb‖l + eTxb (A.66)


Figure A.3: Disc with minimizing line segment.

dTy∗(l) = dTxb + r‖d‖(‖xa − xb‖2l2 + 2xTb (xa − xb)l + ‖xb‖

2) , (A.67)

shows that the e component is an affine function in l and the d-component is a convexquadratic function in l. The transformed line segment is therefore a parabola, orientedin the d-direction.

Fig. A.4 illustrates the boundary of the distorted disc with an example. The ellip-soidal and the parabolic curve segments bound the set Y(x0, r, d) It can be seen thatthe parabolic segment is aligned with the d-direction, but is not symmetrical due tothe nonlinear scaling of the distorted disc.

142 A Appendix

−4 −2 0 2 4 6 8

−4

−3

−2

−1

0

1

2

3

4

5

Figure A.4: Disc before transformation (yellow), origin (star), scaling direction d

(arrow), ellipsoidal boundary segment (bold dashed), parabolic bound-

ary segment (bold solid), transformed boundary points ya = fd(xa) and

yb = fd(xb) (red squares)


A.1.3 Proof of the convexity

Again it is assumed that 2dTx0 ≥ −1. The derivation of the boundary shape for2dTx0 ≤ −1 is analogous.

Section A.1.1 and Section A.1.2 showed that the distorted disc Y(x0, r, d) is boundedeither by an ellipse or a piecewise function consisting of an ellipsoidal and a parabolicsegment. If the distorted disc is bounded by a single ellipse, convexity follows.

To show the convexity of the distorted disc with a piecewise boundary, first note thecontinuity of the boundary, ensured by the parameterization (A.63).

Both the ellipsoidal and the parabolic segment are smooth, with strictly positivecurvature and no singularities. If the union of the two boundary segments is smoothin the boundary points (illustrated by a red squares in Fig. A.4),

ya = fd(xa) yb = fd(xb) , (A.68)

the complete boundary curve is smooth, with strictly positive curvature and no singu-larities, thereby defining a convex set [KB86].

The boundary points lie on both the ellipsoidal curve yψ(t) as in (A.15) and theparabolic curve y∗(l) as in (A.65),

ya = yψ(ψ = ψ∗2 , t = 0) = y∗(l = 1) (A.69)

yb = yψ(ψ = ψ∗1 , t = 0) = y∗(l = 0) . (A.70)

For smoothness in the boundary points, the two segments need to have parallel tangentvectors,

∂ya∂ψ

=∂yψ∂ψ

(ψ = ψ∗2 , t = 0) ‖

∂ya∂l

=∂y∗

∂l(l = 1) (A.71)

∂yb∂ψ

=∂yψ∂ψ

(ψ = ψ∗1 , t = 0) ‖

∂yb∂l

=∂y∗

∂l(l = 0) . (A.72)

Using (A.15), one obtains the tangent vector of the ellipsoidal curve at ya,

∂yψ∂ψ

(ψ = ψ∗2 , t = 0) =

∂ (x−(ψ) + d‖x−(ψ))‖2)

∂ψ(ψ = ψ∗

2) (A.73)

= −e sin(ψ∗2)− d cos(ψ∗

2)+

2d(x0 + e cos(ψ∗2)− d sin(ψ∗

2))T (−e sin(ψ∗2)− d cos(ψ∗

2)) (A.74)

= −e sin(ψ∗2)− d cos(ψ∗

2) + 2dxT0 (−e sin(ψ∗2)− d cos(ψ∗

2)) (A.75)

= e(− sin(ψ∗2)) + d(− cos(ψ∗

2) + 2‖d‖

rxT0 (−e sin(ψ∗

2)− d cos(ψ∗2))) (A.76)

144 A Appendix

= e(− sin(ψ∗1)) + d(cos(ψ∗

1) + 2‖d‖


1) + d cos(ψ∗1))) (A.77)

Using (A.65), one obtains the tangent vector of the parabolic curve at ya,

∂y∗

∂l(l = 1) =

∂(x∗(l) + d‖x∗(l)‖2)

∂l(l = 1) = (A.78)

= xa − xb + 2dxTa (xa − xb) (A.79)

= −2e cos(ψ∗1) + 2d(x0 − e cos(ψ∗

1)− d sin(ψ∗1))T (−2e cos(ψ∗

1)) (A.80)

= −2e cos(ψ∗1) + 4d(−xT0 e cos(ψ∗

1) + r2 cos2(ψ∗1)) (A.81)

= e(−2 cos(ψ∗1)) + d(4

‖d‖

r(−xT0 e cos(ψ∗

1) + r2 cos2(ψ∗1))) (A.82)

The two tangent vectors at ya are parallel if they have the same ratio between the dand e components,

dT∂yψ∂ψ

(ψ = ψ∗2 , t = 0)

eT∂yψ∂ψ

(ψ = ψ∗2 , t = 0)

=dT ∂y

∗

∂l(l = 1)

eT ∂y∗

∂l(l = 1)

(A.83)

cos(ψ∗1) + 2‖d‖


1) + d cos(ψ∗1))

− sin(ψ∗1)

=4‖d‖

r(−xT0 e cos(ψ∗

1) + r2 cos2(ψ∗1))

−2 cos(ψ∗1)

.

(A.84)

Using0 < sinψ∗

1 < 1 and 0 < cosψ∗1 < 1 , (A.85)

expression (A.84) can be further simplified to

−cos(ψ∗

1)

sin(ψ∗1)

+ 2‖d‖

rxT0 (e− d

cos(ψ∗1)

sin(ψ∗1)

) = 2‖d‖

r(xT0 e− r

2 cos(ψ∗1)) (A.86)

r

‖d‖(1 + 2xT0 d)

cos(ψ∗1)

sin(ψ∗1)

= 2r2 cos(ψ∗1) (A.87)

sin(ψ∗1) =

1 + 2xT0 d

2r‖d‖, (A.88)

which is equivalent to (A.29), thereby confirming that the tangent vectors of the twocurve segments at ya are parallel. A similar argument can be made for the tangentvectors at yb, proving the convexity of Y(x0, r, d).

Bibliography

[ABB12] ABB, Power Systems – HVDC. HVDC light - It’s time to connect., 2012.Report from www.abb.com.

[Ada12] R. Adapa. High-wire act: Hvdc technology: The state of the art. Power

and Energy Magazine, IEEE, 10(6):18–29, 2012.

[AESMF96] M.E. Aboul-Ela, A.A. Sallam, J.D. McCalley, and A.A. Fouad. Dampingcontroller design for power system oscillations using global signals. IEEE

Transactions on Power Systems, 11(2):767 –773, May 1996.

[AHL+99] U. Axelsson, A. Holm, C. Liljegren, K. Eriksson, and L. Weimers. Got-land hvdc light transmission - world’s first commercial small scale dctransmission. In CIRED Conference, Nice, France, volume 32, 1999.

[And13] G. Andersson. Power System Dynamics and Control. ETH Zurich, 2013.

[BBM13] F. Borrelli, A. Bemporad, and M. Morari. Predictive control for lin-

ear and hybrid systems. In preparation, draft at http://www.mpc.berkeley.edu, 2013.

[BEFB94] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Ma-

trix Inequalities in System and Control Theory, volume 15 of Studies in

Applied Mathematics. SIAM, Philadelphia, PA, June 1994.

[BGL+10] M. Balabin, K. Görner, Y. Li, I. Naumkin, and C. Rehtanz. Evaluationof pmu performance during transients. In Power System Technology

(POWERCON), 2010 International Conference on, 2010.

[BK11] C.D. Barker and N.M. Kirby. Reactive power loading of componentswithin a modular multi-level hvdc vsc converter. In Electrical Power

and Energy Conference (EPEC), 2011 IEEE, pages 86–90. IEEE, 2011.

[CBB10] S. Cole, J. Beerten, and R. Belmans. Generalized Dynamic VSC MTDCModel for Power System Stability Studies. IEEE Transactions on Power

Systems, 25(3):1655 –1662, August 2010.

[CE05] L. Cai and I. Erlich. Simultaneous coordinated tuning of pss and factsdamping controllers in large power systems. Power Systems, IEEE

Transactions on, 20(1):294–300, Feb 2005.

www.abb.com

http://www.mpc.berkeley.edu

http://www.mpc.berkeley.edu

146 Bibliography

[CKA13a] S. Chatzivasileiadis, T. Krause, and G. Andersson. Hvdc line placementfor maximizing social welfare - an analytical approach. In PowerTech

(POWERTECH), 2013 IEEE Grenoble, pages 1–6. IEEE, 2013.

[CKA13b] S. Chatzivasileiadis, T. Krause, and G. Andersson. Supergrid or localnetwork reinforcements, and the value of controllability - an analyticalapproach. In PowerTech (POWERTECH), 2013 IEEE Grenoble, pages1–6. IEEE, 2013.

[DB10] F. Doerfler and F. Bullo. Synchronization of power networks: Networkreduction and effective resistance. In IFAC Workshop on Distributed

Estimation and Control in Networked Systems, pages 197–202, Annecy,France, September 2010.

[DB12] F. Doerfler and F. Bullo. Synchronization and transient stability inpower networks and non-uniform kuramoto oscillators. 50:1616–1642,September 2012.

[DEK75] D.R. Davidson, D.N. Ewart, and L.K. Kirchmayer. Long term dy-namic response of power systems: An analysis of major disturbances.Power Apparatus and Systems, IEEE Transactions on, 94(3):819–826,May 1975.

[DG06] T. Dang and D. Georges. Controllability gramian for optimal placementof power system stabilizers in power systems. In Circuits and Systems,

2006. APCCAS 2006. IEEE Asia Pacific Conference on, pages 1373–1378, Dec 2006.

[DLL03] D.Jovcic, L.A.Lamont, and L.Xu. VSC transmission model for analyticalstudies. In IEEE PES General Meeting, pages 1737–1742, 2003.

[DLRCTP10] J. De La Ree, V. Centeno, J.S. Thorp, and A.G. Phadke. Synchronizedphasor measurement applications in power systems. Smart Grid, IEEE

Transactions on, 1(1):20–27, 2010.

[Dom13] A. Domahidi. Methods and tools for embedded optimization and control.

PhD thesis, ETH Zurich, 2013.

[EG02] M. Ehrgott and X. Gandibleux. Multiple Criteria Optimization.Springer, 2002.

[Eri08] R. Erikkson. Security-centered Coordinated Control in AC/DC Trans-

mission Systems. PhD thesis, KTH Stockholm, 2008.

[FA02] R. Findeisen and F. Allgöwer. An introduction to nonlinear model pre-dictive. In Control, 21st Benelux Meeting on Systems and Control, Vei-

dhoven, pages 1–23, 2002.

Bibliography 147

[Fai12] P. Fairley. Dc versus ac: The second war of currents has already begun.IEEE Power and Energy Magazine, 10(6):102–104, Dec 2012.

[FIDM14] A. Fuchs, M. Imhof, T. Demiray, and M. Morari. Stabilization of LargePower Systems Using VSC-HVDC and Model Predictive Control. IEEE

Transactions on Power Delivery, 29(1):480–488, February 2014.

[FM13a] A. Fuchs and M. Morari. Actuator performance evaluation using LMIsfor optimal HVDC placement. In European Control Conference, Zurich,Switzerland, July 2013.

[FM13b] A. Fuchs and M. Morari. Placement of HVDC links for power grid sta-bilization during transients. In IEEE Powertech, Power System Tech-

nology, pages 1–6, Grenoble, France, June 2013.

[FMLM10] A. Fuchs, S. Mariéthoz, M. Larsson, and M. Morari. Constrained op-timal control of VSC-HVDC for power system enhancement. In IEEE

Powercon, Power System Technology, Hangzhou, China, October 2010.

[FMLM11] A. Fuchs, S. Mariéthoz, M. Larsson, and M. Morari. Grid stabilizationthrough VSC-HVDC using wide area measurements. In IEEE Pow-

ertech, Power System Technology, Trondheim, Norway, June 2011.

[Fra] France-Spain Electrical Interconncetion (inelfe). Project Progress.http://www.inelfe.eu.

[Fra11] C.M. Franck. Hvdc circuit breakers: A review identifying future researchneeds. Power Delivery, IEEE Transactions on, 26(2):998–1007, 2011.

[GB13] J. Glasdam and C.L. Bak. Hvdc connected offshore wind power plants:Review and outlook of current research. 2013.

[GRG+02] A. Giorgi, R. Rendina, G. Georgantzsi, C. Marchiori, G. Patzienza,S. Corsi, C. Pincella, M. Pozzi, K.G. Danielsson, H. Jonasson, A. Orini,and R. Grampa. The Italy-Greece HVDC Link. CIGRE Session, 2002.

[Gur13] Gurobi Optimization, Inc. Gurobi Optimizer Reference Manual, 2013.http://www.gurobi.com.

[Haa06] T. Haase. Anforderungen an eine durch Erneuerbare Energien geprägten

Energieversorgung - Untersuchung des Regelverhaltens von Kraftwerken

und Verbundnetzen. PhD thesis, Universität Rostock, 2006.

[HEMN02] N. Hörle, K. Eriksson, A. Maeland, and T. Nestli. Electrical supply foroffshore installations made possible by use of vsc technology. In Cigre

2002 Conference, Paris, volume 14, pages 15–20, 2002.

[HG08] G. Hug-Glanzmann. Coordinated Power Flow Control to Enhance

Steady-State Security in Power Systems . PhD thesis, ETH Zurich,

http://www.inelfe.eu

http://www.gurobi.com

148 Bibliography

Switzerland, May 2008.

[Ils94] E. Ilstad. World record hvdc submarine cables. Electrical Insulation

Magazine, IEEE, 10(4):64, 1994.

[Jov03] D. Jovcic. Phase locked loop systems for facts. IEEE Transactions on

Power Systems, 18(3):1116–1124, 2003.

[KB86] H. Knörrer and E. Brieskorn. Plane Algebraic Curves. Birkhauser Verlag,1986.

[KBM96] M. Kothare, V. Balakrishnan, and M. Morari. Robust constrainedmodel predictive control using linear matrix inequalities. Automatica,32(10):1361–1379, October 1996.

[KGBM04] M. Kvasnica, P. Grieder, M. Baotic, and M. Morari. Multi-ParametricToolbox (MPT). In HSCC (Hybrid Systems: Computation and Control),pages 448–462, March 2004.

[Kir09] M. Kirik. VSC-HVDC for Long-Term Voltage Stability Improvement.Master’s thesis, KTH, 2009.

[KRK91] M. Klein, G.J. Rogers, and P. Kundur. A fundamental study of inter-areaoscillations in power systems. IEEE Transactions on Power Systems,6(3):914 –921, August 1991.

[KSLP93] M.R. Khaldi, A. K. Sarkar, K.Y. Lee, and Y.M. Park. The modal perfor-mance measure for parameter optimization of power system stabilizers.Energy Conversion, IEEE Transactions on, 8(4):660–666, Dec 1993.

[KSS07] B.K. Kumar, S.N. Singh, and S.C. Srivastava. Placement of facts con-trollers using modal controllability indices to damp out power systemoscillations. Generation, Transmission Distribution, IET, 1(2):209–217,March 2007.

[Kun93] P. Kundur. Power System Stability and Control. McGraw-Hill, Inc.,1993.

[L04] J. Löfberg. Yalmip : A toolbox for modeling and optimization in MAT-LAB. In Proceedings of the CACSD, Taipei, Taiwan, 2004.

[Laz05] L. Lazaridis. Economic Comparison of HVAC and HVDC Solutions forLarge Offshore Wind Farms under Special Consideration of Reliability.Master’s thesis, KTH, 2005.

[LGS08] H. Latorre, M. Ghandhari, and L. Söder. Active and reactive powercontrol of a VSC-HVdc. Electric Power Systems Research, 78:1756–1763,October 2008.

Bibliography 149

[Lju99] L. Ljung. System Identification - Theory For the User. Prentice Hall,1999.

[Loa78] C. Van Loan. Computing integrals involving the matrix exponential.IEEE Transactions on Automatic Control, 23(3):395 –404, 1978.

[Mac01] J. Maciejowski. Predictive Control with Constraints. Prentice Hall, 2001.

[MBB08] J. Machowski, J.W. Bialek, and J.R. Bumby. Power System Dynamics:

Stability and Control. John Wiley & Sons, Ltd., 2008.

[MBT05] I.M. Mitchell, A.M. Bayen, and C.J. Tomlin. A time-dependenthamilton-jacobi formulation of reachable sets for continuous dynamicgames. Automatic Control, IEEE Transactions on, 50(7):947 – 957, july2005.

[MFM14] S. Mariéthoz, A. Fuchs, and M. Morari. A VSC-HVDC DecentralizedModel Predictive Control Scheme for Fast Power Tracking. IEEE Trans-

actions on Power Delivery, 29(1):462–471, February 2014.

[MH95] J.V. Milanovic and I.A. Hiskens. Effects of dynamic load model pa-rameters on damping of oscillations in power systems. Electric Power

Systems Research, 33(1):53 – 61, 1995.

[MNNDSL13] P. Mc Namara, R. R. Negenborn, B. De Schutter, and G. Lightbody. Op-timal coordination of a multiple hvdc link system using centralized anddistributed control. IEEE Transactions on Control Systems Technology,21(2):302 –314, 2013.

[MRRS00] D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Con-strained model predictive control: Stability and optimality. Automatica,pages 789–814, 2000.

[NJV+94] A. Nyman, K. Jaaskelainen, M. Vaitomaa, B. Jansson, and K. Daniels-son. The fenno-skan hvdc link commissioning. Power Delivery, IEEE

Transactions on, 9(1):1–9, 1994.

[Pad90] K.R. Padiyar. HVDC Power Transmission Systems: Technology and

System Interactions. Wiley, 1990.

[PAMM11] R. Preece, A.M. Almutairi, O. Marjanovic, and J.V. Milanovic. Dampingof electromechanical oscillations by vsc-hvdc active power modulationwith supplementary wams based modal lqg controller. In Power and

Energy Society General Meeting, 2011 IEEE, July 2011.

[PT08] A.G. Phadke and J.S. Thorp. Synchronized Phasor Measurements and

Their Applications. Springer, 2008.

[Ree12] G. Reed. Dc technologies - solutions to electric power system advance-

150 Bibliography

ments. Power and Energy Magazine, IEEE, 10(6), 2012.

[Ric12] S. Richter. Computational Complexity Certification of Gradient Methods

for Real-Time Model Predictive Control. . PhD thesis, ETH Zurich, 2012.

[RKML06] S.V. Rakovic, E.C. Kerrigan, D.Q. Mayne, and J. Lygeros. Reachabilityanalysis of discrete-time systems with disturbances. Automatic Control,

IEEE Transactions on, 51(4):546 – 561, april 2006.

[RS08] C. Rehtanz and K. Sengbusch. Lastflusssteuerung zur systemstabil-isierung. In ETG/BDEW-Tutorial Schutz- und Leittechnik, Fulda, Ger-many, November 2008.

[SA93] T. Smed and G. Andersson. Utilizing hvdc to damp power oscillations.Power Delivery, IEEE Transactions on, 8(2):620 –627, apr 1993.

[Sad06] R. Sadikovic. Use of FACTS Devices for Power Fflow Control and

Damping of Oscillations in Power Systems. PhD thesis, ETH Zurich,2006.

[SCSM06] B. Sheng, F. Chataignere, G. Simioli, and Weimin M. Performance veri-fication of the three-gorges - shanghai hvdc thyristor modules. In Power

System Technology, 2006. PowerCon 2006. International Conference on,pages 1–6, Oct 2006.

[SL14] T.H. Summers and J. Lygeros. Optimal Sensor and Actuator Placementin Complex Dynamical Networks. In IFAC World Congress, August2014.

[SP05] S. Skogestad and I. Postlethwaite. Multivariable Feedback Control: Anal-

ysis and Design. John Wiley & Sons, 2005.

[Sve01] J. Svensson. Synchronisation methods for grid-connected voltage sourceconverters. IEEE Proceedings - Generation, Transmission and Distribu-

tion, 149(3):229–235, 2001.

[TDB08] G. Andersson T. Demiray and L. Busarello. Evaluation study for thesimulation of power system transients using dynamic phasor models.August 2008.

[TLTB05] A. Timbus, M. Liserre, R. Teodorescu, and F. Blaabjerg. Synchroniza-tion methods for three phase distributed power generation systems. anoverview and evaluation. In IEEE 36th Conference on Power Electronics

Specialists, pages 2474–2481, Aachen, Germany, 2005.

[TTT99] K.C. Toh, M.J. Todd, and R.H. Tutuncu. Sdpt3 – a matlab softwarepackage for semidefinite programming, version 1.3. pages 545–581, 1999.

[VBF91] V. Vittal, N. Bhatia, and AA Fouad. Analysis of the inter-area mode

phenomenon in power systems following large disturbances. Power Sys-

tems, IEEE Transactions on, 6(4):1515–1521, Nov 1991.

[VHG10] D. Van Hertem and M. Ghandhari. Multi-terminal vsc hvdc for the euro-pean supergrid: Obstacles. Renewable and Sustainable Energy Reviews,14(9):3156–3163, 2010.

[VHRW08] A.N. Venkat, I.A. Hiskens, J.B. Rawlings, and S.J. Wright. DistributedMPC strategies with application to power system automatic generationcontrol. IEEE Transactions on Control Systems Technology, 16(6):1192–1206, November 2008.

[Vou95] C.D. Vournas. Voltage stability and controllability indices for mul-timachine power systems. Power Systems, IEEE Transactions on,10(3):1183–1194, Aug 1995.

[YBJM08] D. Yazdani, A. Bakhshai, G. Joos, and M. Mojiri. A nonlinear adaptivesynchronization technique for grid-connected distributed energy sources.IEEE Transactions on Power Electronics, 23(4):2181–2186, 2008.

[ZDG+96] K. Zhou, J. C. Doyle, K. Glover, et al. Robust and optimal control.Prentice Hall, Upper Saddle River, NJ, 1996.

List of publications

The following publications were written during the author’s PhD studies at the Au-tomatic Control Laboratory, ETH Zurich. The work leading to the publications wasperformed in close collaboration with various co-authors and collaborators. The pub-lications containing material featured in this thesis are referenced by a citation mark.

Publications related to HVDC and power systems

HVDC based control of power system transients

(Chapter 2 and Chapter 3):

• A. Fuchs, M. Imhof, T. Demiray, M. Morari: Stabilization of Large Power Systems

Using VSC-HVDC and Model Predictive Control. IEEE Transactions on PowerDelivery, vol. 29, no. 1, pp. 480-488, Feb. 2014, [FIDM14].

• A. Fuchs, S. Mariéthoz, M. Larsson, M. Morari: Grid stabilization through VSC-

HVDC using wide area measurements. IEEE Powertech, Power System Technol-ogy, Trondheim, Norway, pp. 1-6, Jun. 2011, [FMLM11].

Low level control control of HVDC links

(Chapter 3):

• S. Mariéthoz, A. Fuchs, M. Morari: A VSC-HVDC Decentralized Model Predictive

Control Scheme for Fast Power Tracking. IEEE Transactions on Power Delivery,vol. 29, no. 1, pp. 462-471 , Feb. 2014, [MFM14].

• A. Fuchs, S. Mariéthoz, M. Larsson, M. Morari: Constrained optimal control

of VSC-HVDC for power system enhancement. IEEE Powercon, Power SystemTechnology, Hangzhou, China, pp. 1-8, Oct. 2010, [FMLM10].

154 Bibliography

Placement of HVDC links

(Chapter 4):

• A. Fuchs, M. Morari: Actuator performance evaluation using LMIs for optimal

HVDC placement. European Control Conference, Zurich, Switzerland, pp. 1529-1534, Jul. 2013, [FM13a].

• A. Fuchs, M. Morari: Placement of HVDC links for power grid stabilization during

transients. IEEE Powertech, Power System Technology, Grenoble, France, pp. 1-6, Jun. 2013, [FM13b].

Other publications

The following publications were written during the thesis but are not related HVDCcontrol and power systems.

Efficient evaluation of explicit MPC solutions

• A. Fuchs, D. Axehill, M. Morari: On the evaluation of mp-MIQP solutions. Tech-nical report vol. AUT13-06, arxiv.org/abs/1311.4752 , Nov. 2013.

• A. Fuchs, D. Axehill, M. Morari: On the choice of the Linear Decision Functions

for Point Location in Polytopic Data Sets - Application to Explicit MPC. IEEEConference on Decision and Control, Atlanta, USA, pp. 5283-5288, Dec. 2010.

• A. Fuchs, C.N. Jones, M. Morari: Optimized Decision Trees for Point Location in

Polytopic Data Sets - Application to Explicit MPC. American Control Conference,Baltimore, USA, pp. 5507 - 5512, Jun. 2010.

Computer vision

• U. Hillenbrand, A. Fuchs: An experimental study of four variants of pose clus-

tering from dense range data. Computer Vision and Image Understanding, vol.115, no. 10, pp. 1427-1448, Oct. 2011.

CV Alexander Fuchs

2008 – 2014 ETH Zurich, SwitzerlandDr. Sc. ETH Zürich at the Automatic Control LaboratoryThesis: Coordinated Control of Power Systems with HVDC Links

Advisors: Prof. Dr. Manfred Morari and Prof. Dr. Göran Andersson

Industry Partners: Swissgrid and ABB

2007 – 2008 Dresden University of Technology, GermanyDipl.-Ing. in MechatronicsThesis: Cluster Algorithms for Pose Estimation

Advisors: Prof. Dr. Klaus Janschek and Dr. Ulrich Hillenbrand

2006 – 2007 University of Texas at Austin, USAM.Sc. in Aerospace EngineeringThesis: Robust Control of Satellite Formations

Advisors: Prof. Dr. Glenn Lightsey and Prof. Dr. Maruti Akella

2004 – 2006 Ecole Centrale Paris, FranceIngénieur Centralien (Double diploma)

2002 – 2004 Dresden University of Technology, GermanyPrediploma in Mechatronics

– 2001 Cotta-Gymnasium, Brand-Erbisdorf, GermanyAbitur

coordinated control of power systems with hvdc …...dissertation eth no. 21723 coordinated control...

Documents