control strategies for vsc-hvdc links in weak ac...

UPTEC F 18067

Examensarbete 30 hpJanuari 2019

Control Strategies for VSC-HVDC links in Weak AC Systems

Erik Björklund

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Control Strategies for VSC-HVDC links in Weak ACSystems

Erik Björklund

In this master thesis control systems for a voltage-source converter HVDCconnected to weak ac networks are investigated. HVDC stands for high voltage directcurrent and is a way to transfer power in the electrical power system. A HVDC usesdirect current (dc) instead of alternate current (ac) to transfer power, which requirestransformation between ac and dc since most power grids are ac networks. TheHVDC uses converters to transform ac to dc and dc to ac and the converter requiresa control system. A complete control system of a voltage-source converter HVDCcontains many different parts. The part investigated in this thesis is the active powercontrol. Different structures containing PID-controllers have been tested andevaluated with respect to stability and performance using control theory. The impactof weak ac networks has been evaluated in regards to the different control structures.The investigations have been conducted using mainly steady-state simulations. Basedon the simulation and analyzes of the simulation results a promising control structurehas been obtained and suggested for further investigation.

ISSN: 1401-5757, UPTEC F 18067Examinator: Tomas Nyberg Ämnesgranskare: Hans Rosth Handledare: Pär Samuelsson

Populärvetenskaplig sammanfattning

Mycket av det vi idag tar för givet i vardagen bygger på att vi kan använda nästintill

obegränsat med elektricitet. Vi kan göra det tack vare elnätet som kopplar sam-

man elkonsumenter och elproducenter. Förmågan av producera elektricitet på en

plats och konsumera den på en annan är grundläggande för att dagens samhälle

ska fungera. Denna transport av elektricitet är en form av energiöverföring. Stora

energiöverföringar är kopplade till energiförluster, vilket är något oönskat då den

energin går till spillo. Högspänningsöverföringar med likström, vilket är vad HVDC

är, är ett sätt att transportera stora mängder eekt över stora avstånd med relativt

små förluster, vilket är viktigt för att inte slösa på energi. För att kunna göra det

krävs stabila reglersystem som kan sköta överföreningarna på ett bra sätt. Efter-

som elnätet idag till största del bygger på växelströmsnät så måste man omvandla

växelström till likström och vice versa för att kunna använda HVDC och det är

reglersystemet en viktig del av. En del av växelströmsnäten som nns kan anses

vara svaga medan andra kan anses vara starka. Att ett nät är svagt innebär väldigt

förenklat att man med små störningar, som inte skulle ha någon större inverkan

på ett starkt nät, kan få oönskade problem. Att designa ett bra kontrollsystem till

en HVDC kopplat till ett svagt elnät är därför viktigt. Detta examensarbete har

undersökt hur man ska designa ett kontrollsystem då man har en HVDC kopplat

till svaga elnät.

I

Acknowledgements

Thanks to...

• Hans Rosth at Uppsala University

Erik Björklund, Uppsala, October 2018

II

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aim of the Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Layout of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory 2

2.1 HVDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 HVDC in General . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 HVDC Transmission Using Forced-Commutated Voltage-Source Con-

verters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Weak AC Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Control Methods for VSC-HVDC Systems . . . . . . . . . . . . . . . . . 6

2.3.1 Vector Current Control . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 Power-Synchronization Control . . . . . . . . . . . . . . . . . . . 82.3.3 Abc-αβ-dq Transformation . . . . . . . . . . . . . . . . . . . . . . 92.3.4 Power Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4.1 PID-Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Methods to Investigate Performance and Stability of a System . . 11

3 The Control Structures Investigated 14

3.1 Cascaded Structure A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Cascaded Structure B with Damping in Last Controller . . . . . . . . . . 163.3 Cascaded Structure C with Feed Forward and Damping in the Last Controller 173.4 Structure D with Feed Forward Without Cascaded Structure . . . . . . . 18

4 Method 18

4.1 PSCAD Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Tuning of Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.1 Tuning Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2.2 Tuning of Structure A . . . . . . . . . . . . . . . . . . . . . . . . 224.2.3 Tuning of Structure B and C . . . . . . . . . . . . . . . . . . . . 234.2.4 Tuning of Structure D . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Dierent Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Dierent Capacitances on the DC Side of the HVDC . . . . . . . . . . . 244.5 Sensitivity Analysis with Varying Frequency . . . . . . . . . . . . . . . . 24

5 Results 25

5.1 Tuning of the Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Dierent Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.1 Structure A∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.2 Structure A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2.3 Structure B and C . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2.4 Structure D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

III

5.3 Dierent Capacitances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.4 Sensitivity Analysis with Respect to Frequency Changes . . . . . . . . . 39

6 Discussion 40

7 Conclusions and Future Work 42

8 References 44

Appendices 45

A Nyquist plots for all control loops 45

A.1 Structure A∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.2 Structure A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.3 Structure B and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.4 Structure D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

B Qualitatively the Eect of Dierent Parameter Values 51

C Abc-αβ-dq Transformation 57

IV

Abbreviations

ac alternating current

dc direct current

HVDC high-voltage direct-current

IGBT insulated-gate bipolar transistor

Im imaginary

LCC line-commutated converter

LP low pass

MMC modular-multilevel converter

PCC point-of-common-coupling

PID proportional-integral-derivative

PLL phase-locked loop

p.u. per unit

Re real

SCC short-circuit capacity

SCR short-circuit ratio

VSC voltage-source converter

V

1 Introduction

Today it is possible to generate electrical power in one location and consume it in anotherlocation. It is an essential part of the energy system we use today. Thanks to thatpossibility, homes and industries can use electrical energy produced in places far fromwhere the consumption takes place. A key part of this is the electrical power transmissionsystem which can transfer power over long distances with relatively low losses. One wayto achieve an ecient power transmission is the use of HVDC (high voltage direct current)transmission which transfer power using direct current.

1.1 Background

HVDC transmission is a way to transfer large amounts of power over long distances withlow losses and a way to connect dierent ac networks that are otherwise incompatible.One of the challenges when constructing a HVDC connection are the ac networks con-nected in each end of the HVDC link. When connecting a dc power line to an ac network,a converter to transform the dc current to an ac current or vice versa is needed. Theconverter design depends on the specic conditions for the HVDC link. One property ofgreat importance is the characteristics of the ac network. The properties of an ac networkcan be dierent in many ways, but very simplied an ac network can be categorized asstrong or weak. The properties of the ac network aects the design of the control systemin the converter. Especially if an ac network can be described as weak, as described inSection 2.2, it presents challenges for the control system in terms of stability. With theincrease of renewable energy sources, such as wind power and solar power, the inertia inthe power system decreases [1] [2]. Low inertia in the power system tends to create weakernetworks and due to this the demand for stable control systems for converters connectedto weak networks increases. One of the newer technologies for HVDC converters is thevoltage-source converter (VSC). The VSC HVDC technology is better suited to handleweak ac networks in comparison with older converter technologies, the reasons will bedescribed in Section 2. Despite the advantage over older technologies the need for stablecontrol systems remains the same.

1.2 Aim of the Project

The aim of this master thesis is to investigate how a power control system for an HVDCstation connected to very weak ac grids may be designed. More specically four dierentcontrol structures will be evaluated. The dierent control structures will be evaluatedbased on stability and dynamic performance in steady state operation. There are dierentparameters that can be taken into accunt when evaluating the performance and stabilitybut in this master thesis the focus will be on the active power transfer, which will beclaried in Section 2. The aim is also to get a general understanding of how the dierentcontrol structures are aected by changes in the ac networks connected to the HVDC.

1.3 Method

The investigations of the dierent control structures are mainly based on simulationsin PSCAD, a software for simulation of power systems. In order to investigate the four

1

dierent control structures in a relevant way, the structures, which are described in Section3, were tuned with an ac network equivalent with the same approach for each structure.The stability and performance of the dierent structures were investigated with Nyquistplots and step responses. Also the sensitivity of the four dierent control structures withrespect to changes in the ac network was investigated by changing the network propertiesin the simulations. The approach is to compare the structures with one and other, notnecessarily tune the control structures to an optimal case which can be used in a realsituation.

1.4 Layout of the report

This is a short overview of the report with the dierent sections and what the sectionscontain.

• TheoryThis section describes the theory of HVDC links, weak ac-systems and PID-controllers.VSC HVDC is described in more detail and the physics behind the control methodsis explained. The characteristics and the implications of weak ac systems are alsopresented. The control theory of the controller is also described.

• The Control Structures InvestigatedThis section presents the dierent investigated control structures. Block diagramsfor each control structure are shown and the control principles of the structures aredescribed.

• Method

This section describes the investigation. Simulations and the setups are presentedand explained.

• ResultsThis section contains the results from the simulations which are described in theMethod section. The results are presented with tables and gures.

• DiscussionHere the results are discussed and analyzed.

• Conclusion and Future Work

In this section the conclusions based on the results are presented along with sug-gestions for future work.

2 Theory

In this section the theory of HVDC links, weak ac-systems and PID-controllers is pre-sented. Voltage-source converters(VSC) HVDC is described in more detail and the physicsbehind the control methods are explained. The characteristics and the implications ofweak ac systems are also presented. The HVDC link and the ac systems together formsthe power system which is to be controlled. The control theory of the controller is alsodescribed.

2

2.1 HVDC

This section describes the general aspects of HVDC and describes two dierent convertertechnologies.

2.1.1 HVDC in General

HVDC is short for high voltage direct current. It is transmission of power using directcurrent (dc). In many transmission lines three-phase alternating current (ac) is used fordierent reasons. According to [3], one of the advantages with ac transmissions is thepossibility to easy change voltage levels using transformers. Another reason is that acmotors are cheaper and more robust than dc motors. Therefore ac transmissions are usedmore frequently in the power grid today. However, dc transmission with HVDC has somefeatures that makes it more feasible for certain applications and the use of HVDC in theworld has increased. Some of the benets are listed in [3]:

• Power transmission over long distances using cables. Because of the physical struc-ture of cables they have higher capacitances than overhead line so the reactive partof the ac current becomes large due to the large part of reactive impedance in thecable. Because of this reactive compensating equipment has to be installed for ca-bles of moderate length (50km) [4]. With i.e. submarine cables this is unpracticalor expensive. With dc current this is not a problem since there are no reactive partof the current.

• When doing bulk-power transmission over long distances HVDC is more feasiblethan ac transmissions [4]. The dc transmission needs fewer lines, 2 instead of 3,and dc transmission does not have the same stability limitations as ac transmissionover long distances.

• When connecting two ac systems with ac transmission lines the two ac systems mustbe synchronized with respect to the nominal frequency. With HVDC links there isno such requirement since the dc current is constant in steady state operation.

2.1.2 HVDC Transmission Using Forced-Commutated Voltage-Source Con-

verters

In Figure 1 an overview of a HVDC link is presented. It is a dc transmission line betweentwo ac networks. At each end of the dc transmission is a converter station which convertac voltage uac to dc voltage Udc.

ac networkconverter

stationDC cable

converter

stationac network

uac uac

−Udc

Udc

−Udc

Udc

Figure 1: An overview of a HVDC link.

For HVDC transmissions there exist two major converter technologies [3]. One is a con-verter technology based on thyristor valves. These are called line-commutated converters

3

(LCC). This technology has been successful and has been used since the late 1960s. Butthe LCC-HVDC has several weaknesses. One problem is that the LCC always consumesreactive power which is approximately 50 − 60% of the active power. Another prob-lem is commutation failures at the inverter station typically caused by disturbances inthe ac system. The LCC is based on thyristor valves and a commutation failure meansthat the valves are not conducting in the correct sequence. Commutation failures are acommon occurrence in LCC-HVDC but repeated commutation failures could force theHVDC link to trip [5]. These problems can be solved but there is a third more funda-mental problem. The LCC-HVDC technology relies on the strength of the ac systemconnected to the HVDC link. If it is a strong ac system this is not a problem. But ifthe ac system is weak, which is explained in Section 2.2, the ac voltage of the networkcannot be considered to be xed. This can cause problems. Another newer convertertechnology is the voltage-source converter [6]. Voltage-source converters (VSCs) makeuse of self-commutating switches e.g. insulated-gate bipolar transistors (IGBTs), whichcan be turned on and o freely. The basic principle is that the IGBTs are switched onand o in order to create an ac voltage from a dc voltage. The IGBTs can be arranged indierent structures. One of these structures is the modular multi-level converter (MMC)structure [3]. The MMC structure is built up of multiple modules. The layout of one ofthese modules is shown in Figure 2a, it consists of two IGBTs and one capacitance. Thevoltage dierence between point A and B can have dierent values and it depends onwhether the IGBTs are conducting or not.

A

B

(a) Detailed layout of a module in the

MMC structure.

A

B

(b) Simplied representation of the

module in Figure 2a.

Figure 2: Layout of one module in the MMC structure and a simplied representationof the same module.

The individual modules, known as cells, are arranged in a structure with two arms as inFigure 3. The structure in Figure 3 represents one phase. In each station there are threesuch structures, one for each phase. When the lower switch is on in Figure 2, the capacitoris inserted in the circuit with the corresponding voltage drop. If the upper switch is on,the capacitor is bypassed with no voltage drop. Depending on how many capacitancesthat are inserted the voltage in point C will be dierent. An inserted capacitance isthe same as an inserted cell. If no cells are inserted in the upper arm in Figure 3 thevoltage in point C will be the same as the positive dc voltage, if the voltage drop overthe inductance is neglected. If all the cells in the lower arm are inserted the voltage in

4

point C will be the negative dc voltage, if the voltage over the inductance is neglected.

C

A

B

A

B

A

B

A

BA

BA

B

Udc

−Udcuconv uvn uvp

Figure 3: MMC structure for one phase.

If the cells in the MMC structure are inserted and bypassed in the right sequence avoltage source is created at the end of both arms in point C. This voltage source canbe controlled in relation to the xed network voltage. This is used when controlling thepower transfer in the converter and is explained in Section 2.3.

2.2 Weak AC Network

According to reference [7] an ac system can be dened as weak in two dierent aspectsfor a HVDC. The rst is that the ac system can have a high impedance relative theHVDC at the connection point. The second is that the ac systems mechanical inertia islow compared to the HVDC. Both of these conditions can make the ac system weak in acontrol perspective. In a high impedance system the impedance of the network seen fromthe HVDC is high. With a high impedance the ac system becomes sensitive to powerchanges from the HVDC. This is often measured by the short-circuit ratio (SCR). TheSCR is the ratio between the ac-system short circuit capacity and the rated power of theHVDC. According to [7] the SCR is dened as

SCR =SacPdcN

(1)

where Sac (VA) is the short-circuit capacity of the ac system and PdcN (W) is the rateddc power of the HVDC. A SCR of less than 3.0 is regarded as a weak system. At verylow SCR the short-circuit capacity is not a very good measure of how weak a ac systemis. For very weak ac networks it is more important to investigate the impedance seen

5

from the converter in a HVDC link. In an ac system with low or non-existing inertia thesystem has limited or no rotating mass in the system which contribute to the inertia. Asystem with a high amount of rotating mass has a high amount of energy stored in therotating masses and that energy storage dampens sudden changes in the ac system. Anexample of this is when a HVDC link is powering an islanded system with few rotatingdevices in the system. For LCC-HVDC a denition similar to SCR has been proposed tobe used. In [7] the eective inertia constant Hdc is dened as

Hdc =Hac

PdcN(2)

where Hac (MW ·s) is the total rotational inertia of the ac system. For a conventionalLCC-HVDC an eective inertia constant, Hdc, of at least 2.0s to 3.0s is required [8]. AVSC-HVDC does not have the same requirement since it can produce its own ac voltageindependent of the ac system but still a system with low inertia is less stable than asystem with high inertia.

2.3 Control Methods for VSC-HVDC Systems

In Section 2.1.2 it is explained how the HVDC convert dc to ac using IGBTs. Basicallya voltage source is modulated. This modulated voltage source has a certain amplitudeand phase which can be controlled. It is by controlling the modulated voltage sourcethe power transfer is controlled, which will be described in this section. In Figure 4 asimplied representation of one side of the HVDC link and an ac system is shown. In thegure the voltage uv can be seen as the modulated voltage source created by the MMCstructure and the IGBT in Figure 4 represents the entire MMC structure. uv is the meanconverter voltage and uv = (uvp − uvn)/2 where uvp and uvn are dened in Figure 3.

P,Q, iv

Xconv

upcc uv

unet

Xnet

Figure 4: A simplied representation of HVDC converter. The IGBT represents theMMC structure which convert the dc voltage to ac voltage. uv is the meanconverter voltage and upcc is the voltage where the converter is connectedto the ac network. To the left a simple network representation with volt-age source unet and network reactance Xnet is shown. Xconv is the converterreactance, including transformer and phase reactors. The resistances in thecircuit are neglected. The current iv is the reactor current.

6

The transfer of active power P and reactive power Q between unet and uv presented inFigure 4 can be described as

P =UnetUv sin θ

X(3)

Q =U2net − UnetUv cos θ

X(4)

where Unet is the amplitude of unet, Uv is the amplitude of uv and θ is the angle betweenthe two voltages unet and uv in Figure 4. X is the combined reactance's in Figure 4,X = Xnet + Xconv. So if unet can be seen as xed the modulation of uv determinesP and Q. If the phase angle θ is small then sin θ ≈ θ and cos θ ≈ 1. By doing theseapproximations it can be seen that the active power P is mainly controlled by altering theangle θ and the reactive power Q by altering the amplitude of uv in relation to unet. So Pand Q can be controlled individually by modulation of the voltage uv. The direction of theactive and reactive power transfer are dened in Figure 4. It is a simplied representationof the ac system with only one ideal ac voltage source, in real cases there are often multiplesources in an ac network and they are not ideal voltage sources. But the principle withthe amplitude and phase angle is the same. In Figure 5a the phasor diagrams illustratehow the transfer of active power P depends on the angle θ and in Figure 5b the phasordiagrams illustrate how the reactive power depends on the amplitude of uv.

unet = uv

P = 0

unetuv

−θ

P > 0

unetuv

θ

P < 0

(a) Phasor diagrams of how the active power P depends on the angle θ.

unet = uv

Q = 0

unetuv

Q > 0

unet

uv

Q < 0

(b) Phasor diagrams of how the reactive power Q depends on the amplitude

of uv.

Figure 5: Phasor diagrams of how the active and reactive power depend on the ampli-tude and phase of uv

There are dierent ways to determine how to modulate uv. The physical principles behind

7

the control of the power transfer are the angle θ and the amplitude dierence betweenuv and unet as shown in Figure 4. But the phase shift and amplitude corresponding toa desired power ow can be achieved by the control system in dierent ways. Two ofthose methods are vector current control and power-synchronization control describedin Section 2.3.1 and 2.3.2. Power-synchronization control is a better control method forweak networks, which is explained further in Section 2.3.1 and 2.3.2.

2.3.1 Vector Current Control

The vector current control is a method, described in [9] and [10], which can be used fora VSC-HVDC system. As described with Figure 4 and Equation 3 and 4 the physicalprinciple used for controlling the power transfer is to control the voltage uv. With vectorcurrent control the phase angle and amplitude dierence are controlled by controlling thecurrent through the inductances. If the voltage drop over the inductance is controlledthe phase and amplitude between the two voltages uv and unet are controlled. If thereactance's Xnet and Xconv in Figure 4 are replaced with X as in Equation 3 and 4 therelation between unet and uv can be expressed as

uv(t) = unet(t)− jXiv(t) (5)

where j is the square root of −1 and iv is the reactor current dened in Figure 4. jX isthe impedance of the inductance in Figure 4, so the expression comes from Ohm's law.uv, unet and iv can be complex and phase shifted. The dierence between the voltagesuv and unet can be controlled by controlling the current. This is a good method if thevoltage unet can be considered xed and the ac system has low impedance, which meansX is close to Xconv. It works well when the ac system connected to the converter can beconsidered as a strong ac network. In Section 2.2 weak ac systems are described and onecharacteristics of a weak ac system is high network impedance. For a weak ac systemin Figure 4 the network impedance Xnet can be considered to be large. If the networkimpedance is large a current through the impedance Xnet will give rise to a high voltagedrop over the impedance. Small changes in the current will give large voltage changeswhich will decrease the stability of the system. Therefore this method is not ideal forcases with weak ac systems.

2.3.2 Power-Synchronization Control

An alternative method to vector current control is the power-synchronization controlwhich is described in [3]. This method also uses the same equations as vector currentcontrol, Equation 3 and 4. But instead of controlling the current through the induc-tances, and by that the voltage drop over inductances, this method directly controls thevoltage drop over the inductances Xnet and Xconv by adjusting uv. This method is notas fast as the vector current control method but it is better suited for situations withhigh impedance ac systems. If the voltage drop over the inductance is controlled directlythe phase shift and amplitude dierence between the two voltages uv and unet can becontrolled directly. This control method is however not well suited for ac systems withlow network impedance. Power-synchronization control does not, as mentioned, directly

8

control the current through the converter. Because of this the current through the in-ductances in Figure 4 is not limited which means the power transfer only depends on thevoltages. If Xnet is high, X in Equation 3 and 4 is high which means changes in P andQ are damped when U1, U2 and θ are changed. If the same changes in U1, U2 and θ aremade and X is low this will give rise to big changes in P and Q. This could possiblycreate oscillations and instability. Therefore this method is better suited for cases withweak ac systems.

2.3.3 Abc-αβ-dq Transformation

The VSC in Figure 4 is a one phase representation of the VSC. In a converter there arethree sets of this, one for each phase. Each phase has a value on uv which needs to becontrolled. The three uv in each phase can be described as

uv,a = Uv sin(ωt)uv,b = Uv sin(ωt− 2π

3)

uv,c = Uv sin(ωt− 4π3

)(6)

where the Uv is the amplitude of the respective uv and ω (rad/s) is the angular frequencyof uv. The relation between angular frequency ω and frequency f is ω = 2πf . If thethree phases are perfectly symmetric, which often can be assumed, Uv is equal for eachphase. The indices a, b and c are for the three phases. These three ac quantities canbe transformed to two dc quantities via αβ-transformation (also known as the Clarketransformation) and dq -transformation. The details of the transformations are describedin Appendix C. After the transformation two dc quantities ud and uq are obtained inthe dq-frame. To have two constant dc quantities in the dq-frame is an advantage whencontrolling of the ac voltages. On constant dc voltages PID-controllers can be applied inorder to attenuate steady state errors.

2.3.4 Power Balance

As described in Section 2.3 the VSC can control both active power P and reactive powerQ independently by changing θ and Uv in Equation 3 and 4. θ and Uv can be controlledseparately, within physical limitations, in a HVDC VSC station. The converters haveseparate control systems for active and reactive power transfer. With a full HVDC VSClink the active power control have two dierent modes. The link has two converterstations, one on each side of the link. The active power ows in one direction andin this situation the two dierent stations must be in dierent control modes. Bothstations control the active power P transfer through the station. One of the stations,named station A, have a power reference to fulll which means it deliver that amount ofpower from the converter. When power goes through the HVDC link it means current isgoing through the dc lines which have a certain resistance which means there are lossesaccording to P = I2/R. If the other station, station B, also push the same amount ofpower through the converter it means that the sum of the power going into and out fromthe HVDC is either negative or positive. If more power is going into the link the dcvoltage will raise and if more power is going out of the link the dc voltage will drop.Since there are always losses in the dc lines, both stations cannot transfer exactly the

9

same amount of power through the converters at the same time. To avoid this situationone of the stations control the dc voltage. The dc voltage can be seen as a measure ofthe energy stored in the link. The station in dc voltage control measures the dc voltageand modify its power reference so the dc voltage remains at the reference. So one stationshould be in active power control and one should be in dc voltage control. In Figure 6a simple HVDC is presented with two converters, two network equivalents and dc linesbetween them. It is the same as in Figure 4 but with both converters and a simplied acside. If the active power is owing from left to right in the gure the station to the leftmust send a larger amount of power to the dc lines than the station to the right takesfrom the dc lines due to the losses in the lines.

P1, Q1

X1

uv,1

unet

R1

R2

P2, Q2

X2

uv,2

unet

Figure 6: A simplied representation of the HVDC link. On each side there is a con-verter. Simple network equivalents with voltage source unet and a inductanceX on each side. The inductances X1 and X2 represent both the networkreactance Xnet and the converter reactance Xconv.

2.4 Control Theory

In this section the theory around the control aspect of the problem is described. In theprevious theory sections the power system to be controlled has been presented. In generala control system with feedback can be described as in Figure 7 with a reference signal,r, that is compared with a measured signal, y, and the control error, e, is sent into thecontroller and an input signal, u, to the system gives an output, y, from the system.

Controller Systemu(t)+r(t) e(t) y(t)

−

y(t)

Figure 7: A general picture of a control structure with a controller and a system withfeedback.

The main part of the control theory concerns the controllers used in the converter and thetools used when tuning these controllers. The controllers used in the HVDC converterare PID-controllers and in order to investigate and tune these controllers Nyquist plotshave been used.

10

2.4.1 PID-Controllers

PID-controllers are linear controllers that consists of a proportional part, integral partand derivative part. PID-controllers are described in [11] and [12]. If the controller inFigure 7 is a full PID-controller the equation for the controller is

u(t) = Kpe(t) +KI

∫ t

0

e(τ)dτ +KDd

dte(t) (7)

in the time domain. KP is the proportional constant, KI is the integral constant and KD

is the derivative constant. If Equation 7 is transformed into the frequency domain withLaplace transformation the transfer function F (s) becomes

F (s) = KP +KI

s+KDs. (8)

The PID-controllers can be combined into dierent controllers with some of the partsi.e. PI-controller or PD-controller. Dierent parts if the PID-controller contribute withdierent properties to a controller. Simplied the proportional part responds to changesin the system, the integral part reduces the static control error and the derivative partadds damping to the system.

In a PID-controller the D-part is in many cases not implemented as an ideal deriva-tive part. Since the D-part works on the derivative of the control error it would besensitive to fast changes of the error even if the amplitude of the changes is small. Thiswould make the controller sensitive to noise. One way of realizing the D-part is to use alow pass lter to add an extra pole in the D-part so

F (s) = KP +KI

s+

KDs

αKDs+ 1. (9)

where α determines how close to ideal derivation the D-part is. By adding an extrapole the sensitivity to i.e. noise is limited. The dierent constants in Equation 9 can berewritten in terms of other constants and there are dierent ways of implementing thecontroller.

2.4.2 Methods to Investigate Performance and Stability of a System

In Figure 7 a control system with controller, system and negative feedback is presented.The stability and speed of such a system can be investigated by examining the open loopgain and phase shift for dierent frequencies. In Figure 7 the system is closed because ithas feedback. The properties of the closed loop system can be determined by examiningthe open loop system. In Section 2.4.1 the PID-controllers are described as linear. If thesystem can be considered as linear, a frequency response can be made using sinusoidalinput signals in order to determine gain and phase shift. In Figure 8 an open loop systemin presented with an input and output signal.

11

Systeminput signal output signal

Figure 8: An open loop system.

If a sinusoidal signal is set as input signal in Figure 8, the output will be a sinusoidalsignal with a gain and phase shifted as

sin(ωt)→ A sin(ωt+ ϕ). (10)

where ω is the angular frequency of the sinusoidal, A is the open loop gain and ϕ is thephase shift of the system at angular frequency ω. So for each frequency the gain andphase shift can be measured for the open loop system by injection of a known sinusoidalsignal.

If a system investigated has a closed loop, the open loop can be investigated accordingto Figure 9. In order to determine the open loop gain and phase shift for a closed systema sinusoidal signal can be injected as in Figure 9.

System+r(t) e(t) y(t)

+

+sin(ωt)

−

Figure 9: A general picture of how a sinusoidal signal can be injected in order to inves-tigate the open loop gain and phase shift for a system.

If the reference signal r(t) is set to a constant value rfixed, the input signal e(t) to thesystem will be e(t) = rfixed − (sin(ωt) + y(t)) where y(t) = a + A2 sin(ωt + ϕ2) where ais a constant. This means e(t) = rfixed − a + A1 sin(ωt + ϕ1) where the sinus term is asuperposition of the injected sinus and the sinus term in y(t). If the constants a and rfixedare removed from e(t) and y(t), there will be a sinusoidal input signal and a sinusoidaloutput signal as in Figure 8 and the gain and phase shift for that specic frequency forthe open loop system can be obtained by comparing the sinus parts of e(t) and y(t). Thiscan be repeated for dierent angular frequencies ω. Information about the open loopsystem can be used in order to investigate the closed loop system. A system as shown inFigure 7 can be investigated for stability using the frequency response of the open loopsystem. The open loop gain A and open loop phase shift ϕ can be plotted as

z = Aeiϕ (11)

in a plot with real and imaginary axis. The plot is known as a Nyquist plot. It can beused in order to determine whether the closed loop system is stable or not by using theNyquist criterion. According to the simplied criterion, the system investigated is stableas long as the Nyquist curve does not encircle the point -1 on the real axis [11]. In Figure

12

10 an example of the Nyquist plot is shown were the curve is z from Equation 11. Thecurve does not encircle the point -1 on the real axis.

Re

Im

−1

1/Am

ϕm

Figure 10: Example of Nyquist curve plotted in a real and imaginary plane with gainmargin Am and phase margin ϕm dened and dashed circle shows the unitcircle.

Other information that can be obtained from a Nyquist plot are gain margin and phasemargin of the closed loop system. The gain margin and phase margin are dened inFigure 10. If the gain margin Am is 1 the system is at the stability margin. For theclosed loop system to be stable with respect to Am the gain margin should be bigger than1. The stability limit for the phase margin is at 0. If ϕm > 0 the closed loop systemis stable with respect to ϕm. For the closed loop system to be stable the system musthave both enough phase margin and enough gain margin. An idea of the speed of thesystem can also be obtained from the frequency sweep. The frequency where the Nyquistcurve cross the unit circle is known as the cross over frequency which corresponds to gain1. The bandwidth, which is a frequency, for a system is dened as where the gain of asystem is below 1/

√2 (-3dB) [11]. When the gain for certain frequencies are below 1/

√2

these frequencies are damped. If the gain for higher frequencies decreases, the speed ofthe system will decrease. The bandwidth is often used to determine the speed of a sys-tem. A dierence between the bandwidth and the crossover frequency is on which systemthey are used. The bandwidth is in most cases used for the closed system while the crossover frequency is for the open system. Despite this dierence the same reasoning can beused with the cross over frequency in order to get a basic idea how fast a system is in acomparison with another system.

In section 2.4.1 dierent parts of the PID-controller were described. The proportional,

13

integral and derivative part have dierent eects on the stability margins. If the propor-tional part is increased the cross over frequency increases, in other words the speed, butthe gain margin decreases. If the integral part is increased the system becomes fasterand the steady state error is reduced. But it has an eect of a lag compensator whichdecreases the phase margin. The derivative has a dampening eect on the system whichdecreases overshoot but it makes the system slower. It also has the eect of an leadcompensator which increases the phase margin. If an ideal derivative part is added itmakes the controller sensitive to noise. These dierent terms can be combined to obtaindierent behavior from a controller.

3 The Control Structures Investigated

In this section the dierent investigated control structures are presented. Block diagramsfor each control structure are shown and the control principles of the structures aredescribed. Four dierent structures are investigated in this master thesis. One of thestructures is investigated with dierent values on the parameters in the controllers. InFigure 11 a very simple structure of the controller and the system is presented. TheController block is the part of the controller that is investigated in this master thesis. Inthe end the controller modulates the voltages uv, which is described in Section 2.1.2, butthe parameter that is the output from the part of the controller this thesis investigates is∆f . The parameter ∆f is a deviation of the frequency for the angle θ in uv. Everythingafter this stage in the real controller is considered to be part of the System shown inFigure 11. In Equation 3 and Figure 5a in Section 2.3 it is explained how the angle θbetween uv and unet controls the active power transfer. If the frequency of uv increasesor decreases for a period of time the angle θ will change and with it the active powertransfer. In the complete controller, ∆f is integrated to an angle θadd which is added tothe angle of uv so the angle θ changes. The angle θadd is integrated as

θadd =

∫∆fdt. (12)

If ∆f = 0, the angle between uv and unet remains unchanged, as long as the networkfrequency and base frequency remains the same. Otherwise the angle is changed in orderto decrease or increase the angle θ and thereby change the power transfer. In Section3.1-3.4 the four dierent structures for the Controller block in Figure 11 are presented.The dierent control structures are using dierent ways to compute ∆f in order to reachthe power reference. For simplicity the dierent control structures will be referred to asstructure A-D. The structure in Section 3.1 referred to as structure A, the structure inSection 3.2 referred to as structure B, the structure in Section 3.3 referred to as structureC and the structure in Section 3.4 referred to as structure D.

14

Controller System∆f+Pref e P

−

P

Figure 11: A simple representation of the control problem. A controller which uses thecontrol error of the output power and compute an input signal signal to thesystem in terms of a dierence in frequency

3.1 Cascaded Structure A

The rst structure investigated is presented with block diagrams in Figure 12. It is acascaded structure with several PID-controllers. The purpose with the cascaded structureis to control both the active power P and the dc voltage Udc. More specically to includethe possibility to limit Udc during faults.

P CTRL UDC CTRL GOV+Pref e1 +∆Udc e2 + Pord − e3 ∆f

+

P

−

Udc

Udc,ref

+

UDC

CTRL D

− +

Figure 12: Block diagram of the cascaded structure A, which is one of the controlstructures investigated and referred to as structure A.

The rst controller named P CTRL is a PI-controller with transfer function as Equation7 without the derivative part. The implementation of the PI controller is in such a waythat the transfer function F1(s) is

F1(s) = K1

(1 +

1

T1s

)(13)

where K1 is the proportional constant and it also scales the integrating constant. T1 isa time constant to the integral part which scales the integral part in comparison withthe proportional part. In Equation 9 the controller constants are dierent but the canbe expressed in terms of the constants in Equation 13. F1(s) acts on the control errorof the power. The output from the rst regulator is a contribution to the Udc,ref whichis the dc voltage reference. The second controller in the cascade is the controller UDCCTRL, which consists of two parts, the UDC CTRL and UDC CTRL D. UDC CTRL

15

is a PI-controller which acts on the control error on the dc voltage. UDC CTRL D is aderivative-controller (D-controller) which acts on the measured dc voltage. In Figure 12e2 is the control error on the dc voltage, included the contribution from the rst controller.The transfer function for the controller that acts on e2 is

F2,P I(s) = K2

(1 +

1

T2s

)(14)

which have the same parameters as Equation 13, but not necessarily the same valuesafter the tuning. A proportional constant and a time constant for the integral part. Thecontroller acting on the measured dc voltage consists of derivative part (D-part) whichis implemented with help of a low-pass lter (LP-lter) such that the transfer functionF2,D(s) becomes

F2,D(s) = K2,D

(1− 1

1 + sT2,D

)= K2,D

sT2,D1 + sT2,D

(15)

where K2,D is a proportional constant and T2,D is the time constant of the LP-lter.This transfer function can be rewritten to the D-part in Equation 9, which is describedin Section 2.4.1. F2,P I acts on e2 and F2,D acts on measured dc voltage Udc. The PI-regulator minimizes the control error and the D-part has a dampening eect on the Udc.The D-part adds damping and phase margin. The two contributions are added togetherinto a signal named Pord. The dierence between Pord and the measured power P is theinput signal to the last controller, named GOV, in the cascaded control structure. Thename is short for governor which is a common name for a controller used to control speedof a machine. GOV is a P-controller with transfer function

F3(s) = K3 (16)

where K3 is the proportional constant. The input signal is the control error e3 and theoutput signal is ∆f as described in Section 3. This control structure has feedback onboth the measured power and measured dc voltage. In the investigations there will betwo variants of structure A with two dierent sets of values on the control parameters.The structure referred to as A will be tuned according to the description in Section 4.2.2.The other sets of values will be referred to as structure A∗, which will be the structureA tuned in another way. These values will not be tuned in a specic way. The valuesfor A∗ are chosen because they are interesting. The tuning for A∗ is focused more on thecontrol of the dc voltage than the active power.

3.2 Cascaded Structure B with Damping in Last Controller

The second structure investigated is a cascaded structure similar to the one in describedin Section 3.1. This structure is presented with block diagrams in Figure 13.

16


P

−

Udc

Udc,ref

+

UDC

CTRL D

− +

GOV

D-part

+ +

+

Figure 13: Block diagram of the cascaded structure B with a derivative part in theGOV controller and referred to as structure B.

The dierence from control structure A in Figure 12 is a damping part in GOV, the lastcontroller in the cascade. A D-part has been added which acts on the measured powerP . The transfer function, F3,D, for this addition is

F3,D(s) = K3,DsT3,D

1 + sT3,D(17)

which is the same as Equation 15 with a LP-lter. It has the same eect as well, but itacts on the measured power. The other controllers in the cascade has the same transferfunctions as the cascaded structure in Section 3.1.

3.3 Cascaded Structure C with Feed Forward and Damping in

the Last Controller

The third control structure investigated is a cascaded structure with a feed forward anddamping in the last controller, it is presented in Figure 14.


P

−

Udc

Udc,ref

+

UDC

CTRL D

− +

GOV

D-part

+ +

+−

Figure 14: Block diagram of the cascaded structure C with feed forward and a derivativepart in the GOV controller. Referred to as structure C.

17

The structure in Figure 14 is similar to the control structure in Section 3.2 with onedierence. There is a feed forward term from the power reference Pref directly to the lastcontroller GOV.

3.4 Structure D with Feed Forward Without Cascaded Structure

The fourth control structure is without the cascaded structure and only uses the lastcontroller of the cascade used in the other structures. In addition there is a secondcontroller in parallel as in Figure 15. The purpose of the extra GOV PLL block isto eliminate the static control error. Since GOV only is a proportional controller thecontrol structure does not contain an integrating part. Because of this the controllerGOV PLL, which is a PI-controller, is added. The input to this controller is the angleof the network, θnet. This extra PI-controller estimates the frequency of the ac networkand adds an additional dierence in frequency to ∆fnet. If the network frequency is thenominal frequency (often 50 Hz) the additional term from GOV PLL is zero. Otherwiseit adds on extra frequency dierence in order to compensate for deviation in networkfrequency. In a case without GOV PLL and the frequency of the network not equal tozero a static control error is required to achieve steady state since ∆f must compensatefor the deviation from the nominal frequency. With GOV PLL this is taken care of bythe additional term and ∆f can be zero and thus fullling the power reference. In thisthesis the eects of the GOV PLL dynamics will not be investigated in detail.

GOV

GOV PLL

−Pref e1

θnet

+

∆f

∆fnet

P

+

+

Figure 15: Control structure D without cascade and a PLL in parallel. Referred to asstructure D

4 Method

In this section the method of the work is described. A large part of the investigationshave been done using simulations in PSCAD. PSCAD is a software for simulations ofpower systems. A model of a HVDC link has been used in order to test the dierentcontrol structures described in Section 3. A general description of the PSCAD model andthe simulations is presented in Section 4.1. The investigation itself is described in Section4.2 - 4.5. During the investigation the approach was to tune the control parameters foreach structure using the same criteria for each structure, this is described in Section4.2. For each control structure Nyquist plots were made to investigate the stability

18

and a step response for each structure was made to verify the design of each controller.After the initial tuning the tuned controllers were tested for ac networks with dierentSCC(short circuit capacity) to see how each controller reacted to dierent SCR(shortcircuit ratio), described in Section 4.3. This was a sensitivity analysis. Another changeof the parameters in the circuit was reduced capacitances in the HVDC link model inorder to investigate any dependencies on the capacitances in the system. This is describedin section 4.4. A third test, described in Section 4.5, of the controllers was a variationof the network frequency to investigate sensitivity to frequency changes for each controlstructure. Both the change of SCR and the change of network frequency are related toweak networks described in Section 2.2. For each change of conditions for the controller,Nyquist plots and step responses were made to investigate the changes in stability andspeed for each control structure. For structure A, a second case with other values on thecontrol parameters was investigated as a reference. It is referred to as structure A∗.

4.1 PSCAD Model Description

The PSCAD model used for the simulations is a model of a symmetrical monopole HVDC.The model is for a HVDC with base power 1254 MVA, dc voltage 320 kV and ac voltage400 kV. The nominal frequency in the simulations were 50 Hz. The model includes thefollowing components: power transformers, converter reactors, converter bridges (includ-ing d.c. capacitors), dc reactors, dc cables/over-Head Lines (OHL) and breakers. ThePSCAD model is a very good representaion of a real HVDC-link.

The PSCAD model used in the simulations can be described to consist of ve com-ponents. It is described in Figure 16 with a dc-cable, two converter stations on eachside of the cable and each converter station connected to a network equivalent. Bothstations are connected to a positive and negative dc voltage on one side of the station.On the other side the station is connected to a network equivalent. The voltage upcc inthe connection between station and network are the same as dened in Figure 4.

Network 1 Station 1 DC cable Station 2 Network 2upcc,1 upcc,2

−Udc,1

Udc,1

−Udc,2

Udc,2

Figure 16: A general picture of the HVDC model in PSCAD.

The networks in Figure 16 are represented by a network equivalent. In Figure 17 thenetwork equivalent is presented. Rs and Rp are resistances and L is an inductance. Thevoltage unet is the underlying voltage source and upcc is the voltage at the converter denedin Figure 4. In Figure 4 the resistances are neglected and the total network impedanceis represented by Xnet. In the simulations the network impedance is simulated with thenetwork equivalent in Figure 17. During the investigation the values of the componentsare altered in order to investigate dierent aspects of the controller but the layout remainsthe same.

19

unet

Rs

Rp

L

upcc

Figure 17: The network equivalent in the PSCAD model.

The stations in Figure 16 consist of a circuit and the control system. The circuit containsinductances, resistances, MMC structure and capacitances. If not otherwise stated, thecircuit is not changed during the simulations. As described in Section 2.3.4 the converterstations control both active and reactive power. For both of active power and reactivepower control there are two modes. For the active power it is direct control of the activepower and indirect control via dc voltage control. For the reactive power control thereare direct control of the reactive power and indirect control of the reactive power via acvoltage control.

The two stations was set to dierent modes during the simulations. The constant modesfor each station are listed in Table 1. Station 1 was set to be in dc voltage control and acvoltage control. Station 2 was set to be in active power control and ac voltage control.In addition Station 1 was set to be in vector current control described in Section 2.3.1while Station 2 was set to be in power-synchronization control described in Section 2.3.2.Station 1 was connected to a strong network and Station 2 to a weak network. It isbenecial to run with ac voltage control for weak ac networks since it limits the eect onthe network.

Table 1: Table with the dierent settings in the two stations.

Station 1 Station 2

dc voltage control active power controlvector current control power-synchronization controlac voltage control ac voltage controlstrong ac network weak ac network

In station 1 and network 1 no changes were made during the simulations, they were setto be constant. It was in station 2 the investigations were conducted. Station 2 was setto be in power-synchronization control and the station was connected to a weak network.Both network 2 and the control system in station 2 were changed during the work.

20

4.2 Tuning of Controller

The dierent control structures that are presented in section 3.1-3.4 were tuned in orderto get a meaningful comparison between them. All four dierent control structures weretuned with the same approach. The structures were tuned according to the followingthree criteria.

• Stability margins. Each loop in each structure should have a minimum of 2.5 ingain margin and 45 in phase margin.

• Since three of the structures are cascaded structures (A, B and C) the secondrequirement was that each inner loop should be faster than the corresponding outerloop. For example the GOV -loop in structure A should be faster than the UDCCTRL-loop.

• The aim of the tuning was to get the control loops as fast as possible while stillfullling the two other criteria.

The approach to the tuning of the cascaded structures was that the innermost controlloop in each structure was tuned rst. Then the tuning was done for the second innermostloop and then the outermost loop. So for the control structures in Section 3.1-3.3, rstthe loop with the GOV block was tuned and then the loop with UDC CTRL and GOVand lastly the outermost loop with P CTRL, UDC CTRL and GOV. For each controlloop the crossover frequency was noted in order to keep the outer loops slower than theinner loops. It was enough that the cross over frequency was higher for the inner loopsthan the outer loops. The general setup for the tuning is described in Section 4.2.1. Howeach loop was tuned is described in Section 4.2.2-4.2.4.

The results with the tuned control parameters are found in Section 5.1, also the Nyquistplots for the four dierent structures and the cross over frequencies can be found there.Only the results for the outermost loop are in the Section 5.1. The Nyquist plots for eachinner loop are presented in Appendix A. The crossover frequencies for each control loopin each controller are also listed. The tuning for each structure is described in Section4.2.2-4.2.4. How the change of each control parameter value changes the Nyquist plotsare qualitatively displayed in Appendix B with Nyquist plots. Once the dierent controlstructures were tuned a step response were made with each control structure to conrmthe results. The step was made from 900 MW to 1000 MW in active power P . The stepresponses can be seen in Section 5.1. After the tuning was complete the control param-eters of the controllers were frozen and other parameters in the circuit were changed toinvestigate the behavior of the dierent controllers when parameters in the power systemwere changed.

4.2.1 Tuning Setup

The tuning of the dierent structures with the mentioned criteria was made using thePSCAD model described in Section 4.1. Network 1 was set to a SCC of 43650 MVA,which corresponds to a SCR of 43.7 and can be considered as a strong network. Station1 connected to this network was set to vector current control of the dc voltage. Network

21

2 was set to 1560 MVA, which can be considered to be weak for this HVDC-link. Station2 connected to the weak network was set to use power-synchronization control. Bothof the stations were set to ac voltage control. Since only the active power transfer isinvestigated the ac voltage control is not interesting to examine since it is connected tothe reactive power.

In order to tune the dierent control structures a steady state condition with 1000 MWactive power transfer was used. The underlying voltage source in the network equivalentfor network 2 was set to 455 V in order to obtain close to 0 Mvar in reactive power trans-fer through the Station 2. In this steady state with the active power P set to 1000 MWand reactive power Q close to 0 Mvar the stability analysis were made. It was alwaysthe innermost loop that was investigated rst. The reference to that loop was frozen to axed value and a sinus was added to the input signal of the loop as described in Section2.4.2. The frequency of the sinus signal was swept from 30 Hz to 0.5 Hz. The input signalto the open loop was compared with the output signal of the open loop and the gain andthe phase shift were obtained and a Nyquist plot could be made. From the plot the gainand phase margins were examined. Also the crossover frequency was noted. From thisplot the control parameters in the controller were changed to make the controller as fastas possible with sucient stability margins. For each loop in each structure Nyquist plotswere made in order to determine the stability of each loop and tune the parameters in it.

4.2.2 Tuning of Structure A

In the control structure in Section 3.1, structure A, the innermost loop consist of the GOVcontroller and its feedback. In order to tune this Pord was set to a constant value whichcorresponds to 1000 MW. Then a sinus was added to the feedback signal P as describedin Section 2.4.2. The signal e3 was compared to the active power output P . The gain andphase shift of the open loop were obtained from the comparison. The control parametersin GOV were changed in order to fulll the criteria stated in Section 4.2. Since GOV isa P-controller it is only one parameter K3 to change. The parameter K3 was increaseduntil the controller reached the gain or phase margin. Once the innermost loop had beentuned this was xed which meant no further changes in the control parameters.

Next control loop was tuned, this loop includes the controllers UDC CTRL and UDCCTRL D. The signals Udc,ref and ∆Udc were frozen and a sinus is added to the feedbacksignal Udc, same as for the innermost loop. This sinus will have an eect on the activepower transfer P . In this thesis the active power output P is compared to Udc withthe added sinus to see how changes of the input to this loop aects the active poweroutput. Since Udc,ref and ∆Udc are frozen they do not contribute to the analysis. Thetransfer functions of UDC CTRL and UDC CTRL D are stated in Equation 14 and 15.In this loop there are four control parameters. When tuning these it was observed whichparameters increased the crossover frequency fastest in comparison with the decreasingstability margins. The parameter K2 had the most eect on the speed so this was in-creased until the system reach the gain margin. Then the time constant to the integralpart, T2, was decreased to further increase the speed. UDC CTRL D is the derivativepart of the controller, this decreases the speed of the loop since the derivative part has an

22

dampening eect on changes of the output from UDC CTRL. This was set so the eecton the controller became low, but the controller was kept for the dampening eect. Thesecond loop was kept slower than the innermost loop and once it was tuned it was frozensame as the innermost loop.

Lastly the outermost loop was tuned in the same way as the previous two loops withfrozen reference signal and added sinus. In this loop the proportional part was increaseduntil the gain margin was reached or until the loop was as fast as the inner loop. Thisapproach gave low gain margin since the proportional part is increased.

4.2.3 Tuning of Structure B and C

Structure B, presented in Figure 13, is very similar to structure A. The only dierence isan additional D-part in the GOV controller which acts on the measured active power P .Because of this the tuning of the innermost loop became somewhat dierent for structureB. In addition from the constant K3, same as for structure A there where two additionalconstants to tune, which are presented in Equation 17. These were tuned in order tominimize the decrease of the cross over frequency while adding as much phase margin aspossible. After this the controller was frozen and the tuning of the other two loops weredone in the same way as for structure A as described in Section 4.2.2. Since Pref is set toa constant in the tuning the Nyquist plot for structure B and C becomes identical andtherefore these structures were tuned to the same control parameter values.

4.2.4 Tuning of Structure D

For the structure D in Figure 15 the tuning was straightforward. The only tuning wasdone for the controller named GOV which is a P-controller. The constant K3 was in-creased until the gain margin was reached.

4.3 Dierent Networks

An important parameter in the system is the ac network. When tuning the controllers asimple network equivalent with innite voltage sources was used. In the tuning, a networkwith SCC, short circuit capacity, of 1560 MVA was used corresponding to a SCR of 1.56.In order to see how the controllers reacted to a dierent network SCC, simulations withother SCC were made. How the short circuit ratio is dened is described in Section 2.2.In the converter used in this thesis PdcN is assumed to be the nominal power which is1000 MW. According to the denition of SCR this gives a SCR of 1.56. When changingSCC the SCR also changes. Three other SCR besides 1.56 were tested, it was SCR equalto 1.25, 1.88 and 2.51. In the test the network equivalent was modied in order to getthe right SCC. In Table 2 the dierent components values are listed for the four dierentSCC including the tuning case.

23

Table 2: Table with dierent SCC and SCR and what values the components in thenetwork equivalent have in each case.

SCR Voltage on the source [kV] Sac[MVA] Rs[Ω] Rp [Ω] L [H]1.25 490 1254 4.51 2445.35 0.411.56 455 1560 3.62 1965.68 0.331.88 435 1881 3.00 1630.23 0.272.51 418 2508 2.25 1222.67 0.20

The underlying voltage source has to be changed in order to keep the reactive powertransfer close to zero. For each dierent control structure the four dierent SCR weresimulated. Nyquist plots and step responses were made to see which eects the dierentSCC had. The results can be seen in Section 5.2. Also changes in the crossover frequencywere noted.

4.4 Dierent Capacitances on the DC Side of the HVDC

An investigation regarding the sensitivity to changes of the capacitances on the dc sideof the HVDC was also conducted. A hypothesis was that dierent capacitances couldaect the control of the HVDC. There are capacitances in the converters and in thetransmission lines. If these are decreased the energy storage in the HVDC is decreasedand aects time constants on the dc side energy storage since the energy E stored in acapacitor is

E =1

2CU2 (18)

where U is the voltage and C the capacitance. A reduced energy storage could make theHVDC more sensitive to changes. Therefore, as a test, the capacitance was decreasedfor each tuned control structure to see how the dierent structures behaved with lesscapacitance in the system. The capacitance was reduced to 80 % of the initial values bychanging the capacitance in each cell in the converter and by reducing the length of thetransmission line to 80 %. For each control structure Nyquist plots and step responseswere made.

4.5 Sensitivity Analysis with Varying Frequency

Another interesting aspect to investigate is how dierent frequencies of the network am-plies with dierent control structures. This can be related to an ac network with varyingfrequency described in Section 2.2. In order to investigate this the frequency of the volt-age source in the network equivalent is modulated. A sinusoidal signal is added to thenominal frequency which is 50 Hz. The frequency of the sinusoidal is modulated as a low-pass lter, as e−τt. At the same time the power reference is kept at a constant level andthe amplication of the sinusoidal at dierent frequencies of the sinusoidal is observed tosee how dierent control structures amplies dierent frequencies. This was done for allfour control structures with two cases with structure A. Also a reference case where thecontroller is passive was made to see how the circuit itself responded without the controlsystem.

24

5 Results

In this section the results are presented and described. It includes Nyquist plots, stepresponses, control parameters and the related data. The section is divided into sectionswith the results for tuning, dierent networks, dierent capacitances and the sensitivityanalysis with respect to frequency changes. In some of the Nyquist plots there can beseen a small waveform behavior of the curve at low frequencies. The reason for this isthat the frequency sweep is slightly to fast for the output to stabilize for low frequencies.The behavior occurs outside of the unit circle which is not the interesting part of theplot.

5.1 Tuning of the Controllers

The Nyquist plot for the outer loop for each control structure are compared in Figure18. In Table 3 gain margins, phase margins and crossover frequencies from the Nyquistplot are listed. A∗ has both the best gain margin and phase margin, but it is also has atleast a factor two lower crossover frequency. Structure A,B and C have small dierencesin gain margin, phase margin and crossover frequency. B and C has slightly lower gainmargin. Structure D has low gain margin but it has the largest phase margin and it hasa clearly higher crossover frequency. The gain margin for structure D is slightly below2.5 but the dierence is small. The reason for this is that the tuning becomes the samefor the GOV controller for both structure A and D.

25

Figure 18: Nyquist plot with the ve dierent control structures A∗, A, B, C and D.

Table 3: Table with the gain margin, phase margin and crossover frequency for dierentcontrol structures for the outermost control loop.

Structure Gain margin Phase margin [] Crossover frequency [Hz]A 3.60 70.89 4.23B 3.47 62.21 4.28C 3.51 62.19 4.28D 2.48 77.07 5.89A∗ 5.89 76.28 2.23

In Figure 18 and Table 3 the stability of the dierent structures are presented. Thecorresponding step responses and some key values are presented in Figure 19 and Table4. The values in Table 4 are dened as the following. Overshoot is dened as (Pmax −(Pfinal − Pinital))/(Pfinal − Pinital) and is given in %. Rise time Tr is dened as the timeit takes for the step to go from 0.1 of the step to 0.9 of the step. The settling time isdened as the time it takes to get Pfinal = Pfinal ± 0.015(P − Pinitial). To notice is thatstructure D has the shortest rise time, smallest overshoot and shortest settling time. A∗

has a rise time in the middle and second shortest settling time but the overshoot is over

26

17 % which is signicantly higher than the other structures. Structure B and C havelong rise time but low overshoot. A has the second largest overshoot and second shortestrise time.

Figure 19: Step response for the dierent control structures. The step is 900 MW to1000 MW which corresponds to a step from 0.718 to 0.797 in per unit.

Table 4: Table with the rise time, overshot and settling time for dierent control struc-tures for the outermost control loop.

Structure Rise time [ms] Overshoot [%] Settling time [ms]A 48 7.14 1014B 108 5.13 991C 107 3.83 1005D 19 2.38 536A∗ 79 17.20 824

5.2 Dierent Networks

With the dierent control structures set to their tuned values the network equivalent waschanged so the SCR varied between 1.25 and 2.51. The Nyquist plots and step responsesfor each controller are presented in section 5.2.1-5.2.4 and it is described how the stabilityand step responses are eected by the changes to the network equivalent.

27

5.2.1 Structure A∗

In Figure 20 and Table 5 the Nyquist plot and key values in the Nyquist plot are presented.From the gure one can see that for higher SCR the gain margin decreases and the phasemargin increases. For lower SCR the opposite occur, higher gain margin and lower phasemargin. The dierent SCR:s have a very small eect on the crossover frequency.

Figure 20: Nyquist plot with structure A∗ with varying SCR. Higher SCR gives lowergain margin and higher phase margin.

Table 5: Table with the gain margin, phase margin and crossover frequency for structureA∗.

SCR Gain margin Phase margin [] Crossover frequency [Hz]1.25 7.52 68.89 2.211.56 5.89 76.28 2.231.88 4.65 81.15 2.232.51 3.29 85.61 2.16

28

Since the crossover frequency remains almost the same for the dierent SCR:s, the stepresponses are expected to remain the same. In Figure 21 this is exactly what can be seen.In Table 6 key values from the gure is listed. Worth to notice is that the settling timeis much shorter for SCR equal to 1.25 but it is because the undershoot, which is presentin the step response, is just below 1.5 % but as can be seen in the gure the response issimilar to the other ones. There the step responses for the four dierent SCR are plottedand there are very small dierences. So for structure A∗ dierent SCR has an eect onthe stability margins but not on the step responses.

Figure 21: Step responses for structure A∗ with varying SCR.

Table 6: Table with the rise time, overshot and settling time for control structure A∗.

SCR Rise time [ms] Overshoot [%] Settling time [ms]1.25 104 17.31 5971.56 81 17.19 8241.88 91 17.44 8742.51 97 17.44 904

29

5.2.2 Structure A

For structure A a similar behavior can be observed with the stability margins. In Figure22 the Nyquist plot for structure A is presented and in Table 8 key values are listed. In thesame way as with structure A∗ the gain margin increases with lower SCR and the phasemargin increases with higher SCR. The SCR has a low eect on the crossover frequencyfor this structure as well. For the case with SCR = 2.51 it can be seen that the gain ishigh at the frequency where the curve cut the real axis. This frequency corresponds toabout 20 Hz.

Figure 22: Nyquist plot for structure A with varying SCR. Higher SCR gives lower gainmargin and higher phase margin.

30

Table 7: Table with the gain margin, phase margin and crossover frequency for structureA.


Since the crossover frequencies are closely the same with dierent SCR for structure Athe speed should not change. In Figure 23 the step responses for dierent SCR can beseen and in Table 8 key values are listed. The dierences are small in terms of risetime, overshoot and settling time. Higher SCR tends to give a shorter settling time andlarger overshoot. The rise time is shorter for lower SCR. In Figure 23 the step responsewith SCR = 2.51 shows an increased oscillating behavior compared to step responseswith lower SCR. Because of these oscillations the rise time becomes lower for decreasingSCR. But the overall step response is almost unchanged when varying the SCR. Thisoscillating behavior is not a desired property of the controller and it does not improvethe step response. The frequency of the oscillations is about 20 Hz. In the Nyquist plotin Figure 22 this is indicated with a relatively large gain at 20 Hz.

31

Figure 23: Step responses for structure A with varying SCR.

Table 8: Table with the rise time, overshot and settling time for control structure A.


5.2.3 Structure B and C

For structure B the behavior is very similar as for structure A. According to the Nyquistplot in Figure 24 the gain margin decreases with increasing SCR while the phase marginincreases. According to the values in Table 9 the crossover frequency increases withincreasing SCR. I does not increase signicantly but the trend is more clear than forstructure A or A∗. For the case with SCR = 2.51 it can be seen that the gain is high atthe frequency where the curve cut the real axis. This frequency corresponds to about 20

32

Hz. The Nyquist plots for structure C is very similar or exactly as for structure B, sothe behavior of structure C is the same as for structure B.

Figure 24: Nyquist plot for structure B with varying SCR. Higher SCR gives lower gainmargin and higher phase margin.

Table 9: Table with the gain margin, phase margin and crossover frequency for structureB.


The step responses for structure B and C, with the corresponding key values, are pre-sented in Figure 25-26 and Table 10-11. The oscillating behavior and longer rise time for

33

higher SCR are the same as for structure A. The settling times are similar for both struc-tures and for all SCR:s. The rise times are also similar. The largest dierence betweenstructure B and C is the smaller overshoot for structure C. Both structure B and C havesignicantly longer rise times than structure A. But the overshoot is lower compared tostructure A. Same for structure A the oscillations in the step responses for SCR = 2.51can be related to the Nyquist plot.

Figure 25: Step responses for structure B with varying SCR.

Table 10: Table with the rise time, overshot and settling time for control structure B.


34

Figure 26: Step responses for structure C with varying SCR.

Table 11: Table with the rise time, overshot and settling time for control structure C.


5.2.4 Structure D

For structure D the Nyquist plot is shown in Figure 27 and the behavior dier from theother structures. With structure D both the gain margin and phase margin decreaseswith higher SCR. Both stability margins are aected in the same way when the SCR isincreased or decreased. In Table 12 the crossover frequency can be seen increasing withdecreasing stability. Which means the system becomes faster with decreasing stabilitylimits. The other structures did not have this behavior.

35

Figure 27: Nyquist plot for structure D with varying SCR. Higher SCR gives lower gainmargin and lower phase margin.

Table 12: Table with the gain margin, phase margin and crossover frequency for struc-ture D.


The step response in Figure 28 conrms the results from the Nyquist plot for structureD. With increasing SCR the step response becomes faster. The step also has moreoscillations with increasing SCR. In Table 13 it can be seen that with increasing SCRthe rise time and settling time decrease. But the overshoot increases. Worth to notice isthat for SCR = 2.51 the overshoot is over 25 % for structure D. This can be related tothe decreased gain margin.

36

Figure 28: Step responses for the structure D with varying SCR.

Table 13: Table with the rise time, overshot and settling time for control structure D.


5.3 Dierent Capacitances

The eects from the reduced capacitances in the system are small. For every structurethe dierence with or without reduced capacitances can be neglected in terms of gainmargin, phase margin and step responses. In Figure 29 the Nyquist plot with structure Bfor the two cases can be seen and the stability margins are very close to being the same.The crossover frequencies are 4.02 Hz with reduced capacitances and 4.28 Hz withoutreduced capacitances. Figure 30 shows how the step response is the same with very smalldierences.

37

Figure 29: Nyquist plot with comparison of structure B without and with reduced ca-pacitances in the circuit.

Figure 30: Step responses for structure B without and with reduced capacitances in thecircuit.

38

The results for structure B are representative for all the dierent structures. There aresmall changes in stability margins and small changes in the step responses when thecapacitance is reduced with 20 %. As the step response for structure B demonstrates theimpact from 20 % reduced capacitances is very limited.

5.4 Sensitivity Analysis with Respect to Frequency Changes

From the frequency sweep on the underlying voltage source, described in Section 4.5, adependency on the frequency changes was obtained. The dependencies can be read fromFigure 31. In the gure the gain from the passive controller can be seen as a referencegain. How the converter amplify with the control system being inactive. The threestructures A,B and C can be seen damping the disturbance at low frequencies under 7Hz. For higher frequencies they amplify the disturbance. Structure A∗ has lower dampingat low frequencies but at frequencies over 10 Hz it amplify the signal less than the tunedstructure A. Structure D dampens the disturbance up to frequencies around 15 Hz beforeit starts to amplify the signal. Over 20 Hz it amplify the signal more than structure A∗

but less than the other ones.

Figure 31: The peak values of the active power output plotted against the correspondingfrequency of the disturbance. The curves for structure B are the same asfor C and they are on top of each other.

39

6 Discussion

After comparing the results between the dierent structures several interesting observa-tions can be made.

When comparing the dierent tuned structures in Section 5.1 from a stability pointof view the structures can be divided into three groups. Structure A∗ in one since ithas dierent values on the control parameters. Structure A,B and C in one group sincethey have the same type of cascade structures and their control parameters have similarvalues. The last group contains structure D which does not have the cascaded structure.

During the tuning it was discovered that structure A,B and C behave in similar ways.With minor dierences they have the same stability margins and step responses. Thisis no surprise since their structures are very similar. What seems to be the biggest dif-ference is that structure A is faster than the others in the step responses. This can beexplained with the added D-part to the GOV controller in the cascade for structure Band C. The D-part acts on the measured active power and counteracts the changes in theactive power P . This makes the controller more well damped but also slower, see Section5.1. Structure C in turn is more well damped than structure B in the step responses.The only thing separation structure B and C is the feed forward which is added, seeFigures 13 and 14. Since the power order is carried directly to the GOV controller inthe cascade in structure C it should react faster than B. Which means controller C canreact to changes faster than B. In the stability analysis structure C and B have the sameresults since the loop gain is the same. When comparing A and B the dierence is theD-part in the GOV controller in the cascade, described in Figure 12 and 13. Since theD-part slows down the GOV controller the constant K2 in the UDC CTRL controllercan be increased in B without making the middle control loop faster than the innermostloop and still be within the stability limits. Since K2 also scales up the integral part inUDC CTRL controller the phase margin decreases for structure B and C in comparisonwith A as can be seen in Figure 18. But since the D-part adds dampening to structureB and C their step responses in Figure 19 are more well damped with less overshoot.

In the step responses for structure A,B and C a small notch can be detected whichcan be related to the power balance in the system described in the theory part. When Ptransfer increases with a step the Udc decreases before the other station detects the de-crease in dc voltage and responds to the new power transfer. So before the other stationresponds the UDC CTRL controller, in the station in power control, limits the signal Pordin the cascade structures which causes a temporary drop in active power transfer.

In structure A∗ it was discovered that the middle control loop was signicantly fasterthan the innermost and outermost loop. This makes A∗ sensitive to changes in Udc andkeep the Udc stable. This indicates that the tuning for A∗ is not optimal for P controlbut better suited for Udc control. Since UDC CTRL is faster than the other loops italso creates an overshoot in Figure 19 since the innermost loop do not have the time torespond before the second loop changes the reference to the innermost loop.

40

In structure D the cascade are removed and with that also the control with respectto Udc. So instead of getting feedback on both the voltage and power structure D onlytakes the power into account. This makes the controller faster and better damped in thestep response. The downside is that it must be assumed that Udc is kept stable by theother station.

The dierent controllers reacts in dierent ways when the SCR of the network is changed.All the controllers with cascaded structure reacts with decreasing gain margin and in-creasing phase margin with increasing SCR. But structure D has a dierent behaviorwhere the margins change in the same way. The other big dierence is that structureD generates faster responses when the SCR increases while the cascaded structures donot. So for the cascaded structures the stability margins changes with the SCR and notthe speed which means with varying SCR one or the other of the stability margins aredecreased without making the system faster. For structure D the controller gets fasterwhen both the stability margins decreases and when the controller gets slower both sta-bility margins increase. So with structure D, the loss of stability margins comes with afaster system. With the other structures the speed of the system is kept stable despitethe decrease in one of the stability margins. If aim is to keep the speed of the system thecascaded structures have the property to keep the speed. But on the other hand if theSCR of the network is changed structure D can easily change the gain of the controllerin order to compensate for this since it only has a proportional part with one parame-ter K3. If the SCR is increased then K3 can be decreased in order to compensate forthis while still keeping the speed of the system. If the SCR is reduced then K3 can beincreased to keep the speed since the stability margins improve of only the SCR is reduced.

The investigation concerning reduced capacitances showed no signicant changes to thesystem. Probably a further reduction of the capacitance in the system would be neededin order to notice any major dierence.

In the sensitivity analysis it could be seen that the cascade structures, at least the tunedones, were more sensitive than structure D to changes in the frequency of the underlyingvoltage source in the network equivalent. This means that if the network frequency isvarying, structure D would have a lower amplication of the frequency changes to theactive power output.

If only the stability margins, step responses, sensitivity to dierent SCR and sensitivityto frequency changes are taken into account structure D is the most promising solution.The downside with this solution is that it does not have any control of the dc voltage.This does not seem to be a problem in steady state operation but during faults or otherdisturbances this could be a weakness. The investigations have only been done in steadystate with relatively small changes in power output. Since only one of the two stationshas been investigated with power-synchronization control with a weak network while theother station has been in vector current control with a strong network it has been as-sumed that the station in vector current control can handle the dc voltage. The lack ofdc voltage control can also be a problem in a more realistic case where faults can occurin either of the two stations or in the dc cable. Such faults can make it impossible to

41

transfer enough power between the stations and through a dc voltage drop or increasethis can be detected. In a real case some kind of dc voltage control is needed to deal withfault cases for instance.

Another problem with structure D is the lack of an integrating part. As described in thetheory part an integrating part is needed to eliminate the static control error. Accord-ing to the description of structure D the static control error is eliminated by the extracontrol block GOV_PLL. This extra control block has not been investigated in detailin this work and could be examined further. The purpose of the block is to measurethe frequency of the network in order to eliminate the static control error. Apart fromthe analysis with varying frequency on the underlying voltage source the network has aconstant frequency in the investigation. With a more advanced network equivalent with anon-constant frequency the lack of an integrating part in structure D could be a problem.

All structures except D has a cascade structure which includes both control of the activepower P and the dc voltage Udc. With the cascade the problems with no control of thedc voltage and no integrating part disappear. Since there are two integral parts in thecascade and the second controller has the dc voltage as feedback. But as is shown in theresults the cascade makes it slower and creates other types of stability problems such asthe fact that the phase and gain margin changes in dierent directions with changingSCR. The cascade structure could be feasible if there is a need to control both P and Udcat the same time.

In the investigations the converter stations have only been in steady state. But in areal application the stations and control systems are not in steady state all the time. Ina real case there will be disturbances such as ac faults, dc faults etc. The control systemhas to be robust in order to be able to deal with these kind of disturbances and return tosteady state in a controlled way. This has not been investigated for the dierent struc-tures.

In structure A∗ the power-synchronization control and vector current control partly makeuse of the same controllers and control loops with identical control parameter values.Since the two control methods, described in Section 2.3, are somewhat dierent the useof identical control parameter values are probably not ideal. Instead the involved con-trollers could be tuned with two dierent sets of parameters values for the two dierentcontrol methods. The use of same control parameter values could be why structure A∗

has inner loops that are slower than the outer loops described in the results. If the con-trollers could be assigned dierent control parameters depending on control method thiscould be avoided. For structure A∗ the power-synchronization control and vector currentcontrol are closely coupled together both in structure and control parameter values.

7 Conclusions and Future Work

Based on the Nyquist plots and step responses after the tuning and the comparison be-tween the structures, the conclusion is that structure D is the most promising solution. It

42

has the fastest response with comparable stability margins. It is not sensitive to dierentnetwork strengths and can easily be adapted to work for networks with dierent SCR.As long as the other station is able to transfer enough power through the HVDC linkand as long as no faults occur and the HVDC is in steady state structure D is the mostpromising. Structure D however needs to be investigated further in order to determine ifit can handle fault cases, not steady state operation with fast changes and how the con-troller responds to a realistic ac network with varying frequency. Another suggestion tofuture work is to look at GOV_PLL to see how it can be used to improve the performance.

Another conclusion is that structure A,B and C in general performs better than struc-ture A∗ in the step responses. As mentioned in the discussion in Section 6 the vectorcurrent control and power-synchronization control are closely connected in A∗. Based onthis, regardless of what control structure to use, the parameter values of the controllersshould be separated between vector current control and power-synchronization control.

To further evaluate the structures additional simulations with bigger and more realis-tic ac networks is needed. Simulations of dierent fault cases are also needed in orderto determine how the control structures handles situations which not can be consideredto be steady state. Another suggestion for future studies is to investigate if the HVDCcan handle a case with weak ac networks at both ends of the HVDC. As it is now onlyone station is in power-synchronization control, the station in active power control, whenthe HVDC is connected to two weak ac networks. The other station is in dc voltagecontrol which uses vector current control. It would be interesting to let both stations usepower-synchronization control and investigate how the dierent control structures couldhandle that case. So the station in dc voltage control also uses power-synchronizationcontrol.

43

8 References

[1] Nahid-Al-Masood, N. Modi, and R. Yan, Low inertia power systems: Frequencyresponse challenges and a possible solution, in 2016 Australasian Universities PowerEngineering Conference (AUPEC), pp. 16, Sept 2016.

[2] E. Ørum, M. Kuivaniemi, M. Laasonen, A. I. Bruseth, E. A. Jansson, A. Danell,K. Elkington, and N. Modig, Future system inertia, ENTSOE, Brussels, Tech.Rep, 2015.

[3] L. Zhang, Modeling and control of VSC-HVDC links connected to weak AC systems.PhD thesis, KTH, 2010.

[4] B. M. Weedy, B. J. Cory, N. Jenkins, J. B. Ekanayake, and G. Strbac, Electric powersystems. John Wiley & Sons, 2012.

[5] L. Zhang and L. Dofnas, A novel method to mitigate commutation failures in HVDCsystems, in Power System Technology, 2002. Proceedings. PowerCon 2002. Inter-national Conference on, vol. 1, pp. 5156, IEEE, 2002.

[6] G. Asplund, K. Eriksson, K. Svensson, et al., DC transmission based on voltagesource converters, in CIGRE SC14 Colloquium, South Africa, pp. 17, 1997.

[7] IEEE Guide for Planning DC Links Terminating at AC Locations Having LowShort-Circuit Capacities, IEEE Std 1204-1997, pp. 1216, Jan 1997.

[8] P. Kundur, N. J. Balu, and M. G. Lauby, Power system stability and control, vol. 7.McGraw-hill New York, 1994.

[9] R. Ottersten and J. Svensson, Vector current controlled voltage source converter-deadbeat control and saturation strategies, IEEE Transactions on Power Electron-ics, vol. 17, pp. 279285, Mar 2002.

[10] J.-W. Choi and S.-K. Sul, Fast current controller in three-phase AC/DC boostconverter using d-q axis crosscoupling, IEEE Transactions on Power Electronics,vol. 13, pp. 179185, Jan 1998.

[11] T. Glad and L. Ljung, Reglerteknik: grundläggande teori, vol. 2. Studentlitteratur,2006.

[12] K. Åström and T. Hägglund, Advanced PID Control. ISA - The Instrumentation,Systems and Automation Society, 2006.

44

Appendices

A Nyquist plots for all control loops

In this appendix the stability analysis is presented in detail for all loops in each controlstructure. For each control loop a Nyquist plot is shown and for each structure key valuesare listed in a table.

A.1 Structure A∗

In Table A.1 the gain margin, phase margin and cross over frequency for each loop instructure A∗ are listed. The corresponding Nyquist plots are presented in Figure A.1-A.3.The gain margin for the UDC CTRL is close to 1 which is low. Not optimal from a activepower point of view.

Table A.1: Table with the gain margin, phase margin and crossover frequency for eachcontrol loop in structure A∗.

Control loop Gain margin Phase margin [] Crossover frequency [Hz]GOV 7.84 62.57 1.90UDC CTRL 1.10 36.43 17.92P CTRL 5.89 76.27 2.23

Figure A.1: Nyquist plot with structure A∗ for the control loop with controller GOV.

45

Figure A.2: Nyquist plot with structure A∗ for the control loop with controller UDCCTRL.

Figure A.3: Nyquist plot with structure A∗ for the control loop with controller P CTRL.

46

A.2 Structure A

In Table A.2 the gain margin, phase margin and cross over frequency for each loop instructure A are listed. The corresponding Nyquist plots are presented in Figure A.4-A.6.

Table A.2: Table with the gain margin, phase margin and crossover frequency for eachcontrol loop in structure A.


Figure A.4: Nyquist plot with structure A for the control loop with controller GOV.

47

Figure A.5: Nyquist plot with structure A for the control loop with controller UDCCTRL.

Figure A.6: Nyquist plot with structure A for the control loop with controller P CTRL.

48

A.3 Structure B and C

In Table A.3 the gain margin, phase margin and cross over frequency for each loop instructure B and C are listed. The corresponding Nyquist plots are presented in FigureA.7-A.9. As discussed in Section 5 the Nyquist plots for structure B is equal to the onesfor structure C.

Table A.3: Table with the gain margin, phase margin and crossover frequency for eachcontrol loop in structure B and C.


Figure A.7: Nyquist plot with structure B and C for the control loop with controllerGOV.

49

Figure A.8: Nyquist plot with structure B and C for the control loop with controllerUDC CTRL.

Figure A.9: Nyquist plot with structure B and C for the control loop with controllerP CTRL.

50

A.4 Structure D

In Table A.4 the gain margin, phase margin and cross over frequency for each loop instructure D are listed. The corresponding Nyquist plots are presented in Figure A.10.

Table A.4: Table with the gain margin, phase margin and crossover frequency for eachcontrol loop in structure D.

Control loop Gain margin Phase margin [] Crossover frequency [Hz]GOV 2.48 77.07 5.95

Figure A.10: Nyquist plot with structure D for the control loop with controller GOV.

B Qualitatively the Eect of Dierent Parameter Val-

ues

This appendix contains gures that describe which eects dierent parameters in thecontrol structures have on the Nyquist plots. The eects are demonstrated in FigureB.1-B.9. The gures only demonstrates how the Nyquist curve changes with an increas-ing or decreasing value on one parameter. The values listed in the Figures are valuesscaled with a constant, it is not the actual values on the parameters. To describe this,Nyquist plots from structure B have been used but each parameter have the same eecton the other structures. The values on each parameter in control structure B during thesimulations where Figure B.1-B.9 were obtained. It is the behavior which is interesting.

51

The hexagon on each curve corresponds to the same frequency in all the gures. Thehexagon demonstrates how the cross over frequency should change with dierent param-eters. Which parameters that are included in each structure is explained in Section 3. Toshortly summarize the behavior of the constants. The constants K1, K2 and K3 aectsthe gain margin and the crossover frequency. The constants K2,D and K3,D aects thegain margin, the cross over frequency and in some extent the phase margin. The timeconstants T1, T2, T2,D and T3,D moves the curve almost like a clock pointer, see FigureB.2 where the curve turns from the solid yellow line to the dashed blue line when T1 ischanged. For a more illustrated explanation see Figure B.1-B.9.

Figure B.1: Nyquist plot with structure B for the control loop with controller P CTRLthat illustrates how K1 aects the stability margins. The hexagon marksthe same frequency for all curves.

52

Figure B.2: Nyquist plot with structure B for the control loop with controller P CTRLthat illustrates how T1 aects the stability margins. The hexagon marksthe same frequency for all curves.

Figure B.3: Nyquist plot with structure B for the control loop with controller UDCCTRL that illustrates how K2 aects the stability margins. The hexagonmarks the same frequency for all curves.

53

Figure B.4: Nyquist plot with structure B for the control loop with controller UDCCTRL that illustrates how T2 aects the stability margins. The hexagonmarks the same frequency for all curves.

Figure B.5: Nyquist plot with structure B for the control loop with controller UDCCTRL that illustrates how K2,D aects the stability margins. The hexagonmarks the same frequency for all curves.

54

Figure B.6: Nyquist plot with structure B for the control loop with controller UDCCTRL that illustrates how T2,D aects the stability margins. The hexagonmarks the same frequency for all curves.

Figure B.7: Nyquist plot with structure B for the control loop with controller GOVthat illustrates how K3 aects the stability margins. The hexagon marksthe same frequency for all curves.

55

Figure B.8: Nyquist plot with structure B for the control loop with controller GOVthat illustrates how K3,D aects the stability margins. The hexagon marksthe same frequency for all curves.

Figure B.9: Nyquist plot with structure B for the control loop with controller GOVthat illustrates how T3,D aects the stability margins. The hexagon marksthe same frequency for all curves.

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C Abc-αβ-dq Transformation

This appendix describes the Abc-αβ-dq transformation which is used and briey explainedin Section 2.3.3. A three phase system with 120 phase shift between the phases can beexpressed as in Equation C.1.

uv,a = Uv sin(ωt)uv,b = Uv sin(ωt− 2π

3)

uv,c = Uv sin(ωt− 4π3

)(C.1)

where Uv is the amplitude of the respective uv and ω (rad/s) is the angular frequency ofuv. The relation between angular frequency ω and frequency f is ω = 2πf . The indicesa, b and c are for the three phases. This three phase ac quantity can be transformed to atwo phase dc quantity via αβ-transformation (also known as the Clarke transformation)and dq -transformation. This is described in Figure C.1 and C.2. In Figure C.1 a voltagevector u is placed in a αβ-frame. This is a xed vector in time but not in the αβ-frame.The vectors a, b and c are the three phases of the ac voltages. If the voltages ua,b,c arealigned in the αβ-frame as in Figure C.1 the xed vector u will rotate with the angularfrequency of ua,b,c. All the three phases can be expressed in terms of u and the angleθ. In the same way both uα and uβ can be expressed in terms of u and θ. So the threephases a, b and c can be expressed in terms of uα and uβ. Since u is a xed vector θ willchange with time and how u is represented by uα and uβ will vary with time.In otherwords u is not xed in the αβ-frame.

α

β

a

c

b

u

uα

uβ

θ(t)

Figure C.1: Three-phase variables to a rotating vector in the αβ-frame.

In Figure C.2 the vector u is expressed in a dq-frame in terms of ud and uq. The dq-frameis a coordinate system which rotates with the angular frequency of the ac network. Withthe dq-frame the ac voltage u can be expressed in terms of ud and uq which are constantdc voltages in the dq-frame which is an advantage when controlling of the voltages. On

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constant dc voltages PID-controllers can be applied in order to attenuate steady stateerrors.

α

β

d

q

ud

uq

u

uα

uβ

θdq(t)

Figure C.2: From a stationary αβ-frame to synchronous dq-frame

A requirement for the dq-frame to be useful is that θdq is estimated in a correct way. Ifthe frequency of the system always was xed to 50 Hz this would not be a problem. Butsince the frequency of the ac system varies around 50 Hz an estimation of θdq has to bemade. This is done using a Phase Locked Loop (PLL). The PLLs purpose is to keep thedq-frame aligned with the ac network frequency.

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control strategies for vsc-hvdc links in weak ac...

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