Load Characteristic Influence on Power Oscillation Damping1249017/...power oscillation damping (POD)...

IN DEGREE PROJECT ENERGY AND ENVIRONMENT,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2018

Load Characteristic Influence on Power Oscillation DampingCase Study on HVDC-interconnected AC-Grids

ERIK BJÖRK

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

Abstract

The increasing share of renewables in the electric grid may have a negative impact on the systemstability to which conventional control methods are insufficient. The report focuses on the stabil-ity of low frequency inter-area modes. Along the development of renewables, power electronics aremore frequently used between the actual load and the grid. Hence the characteristics of the loadsis undergoing a change as well. This report investigates how different load characteristics influencethe damping in the grids. The investigation is performed by a simulation experiment where a modelconsisting of two AC-grids interconnected via an HVDC-link is implemented in two simulation soft-ware programs, PSCAD and Simulink. The HVDC-link has an implemented active power oscillationdamping control to improve the damping of the overall system. The report also review which soft-ware that is most suitable for the given investigation. It is shown that active power loads contributesto the damping best if they are modelled as constant impedance loads and less for constant powerloads. Reactive loads with inductive behaviour contributes to the damping best if they are modelledas constant power loads and contributes least if they are constant impedance loads. The evidencesuggest that the damping provided by the loads is uncorrelated with the damping provided by theHVDC-link. The simulation software that suited the author’s preference best was Simulink. Theseresults contribute to the understanding of how loads influence the system damping and can be usedin future studies, where it is interesting to know if the load damping is over or underestimated.

I

Sammanfattning

Den ökande andelen av förnyelsebara energikällor som integreras till elsystemet kan ha en negativp̊averkan p̊a elsystemets stabilitet vilket gör att konventionella metoder för att stabilisera systemkan visa sig vara otillräckliga. Parallellt med utvecklingen av förnyelsebara energikällor ökar andelenav kraftelektronik som kopplas mellan lasten och elnätet. Den adderade kraftelektroniken till lastenförändrar följaktligen karaktären av lasterna. Rapporten undersöker olika lastkaraktärers inverkanp̊a dämpningen av l̊agfrekventa svängningsmoder i elsystemet. Studien utförs genom ett simulerings-experiment där en modell best̊aende av tv̊a AC-nät som är sammankopplade med en HVDC-länkimplementeras i tv̊a olika simuleringsprogram, PSCAD och Simulink. HVDC-länken har en regulatorimplementerad för att dämpa de aktiva effektoscillationerna i systemet. Rapporten undersöker ävenvilket simulationsprogram som är mest lämpad för den utförda studien. Resultaten visar att lastersom konsumerar aktiv effekt gav bäst dämpning när de var modellerade som konstanta impedanslasteroch de gav sämst dämpning när de var konstanta effektlaster. Laster som konsumerade reaktiv effekt,induktiva laster, gav bäst dämpning till systemet när de var modellerade som konstanta effekt lasteroch gav sämst dämpning när de var modellerade som konstanta impedans laster. Dessutom tyderresultaten p̊a att dämpningen given fr̊an lasterna inte är korrelerade med dämningen fr̊an HVDC-länken. Simuleringsprogrammet som enligt författarens preferenser var mest lämpad för studien varSimulink. Resultaten kan bidra till en ökad först̊aelse av lasters p̊averkan p̊a elsystemets dämpningoch kan användas i framtida studier, där det är av intresse att veta om lastens dämpning är över ellerunderskattad.

II

Acknowledgements

I would like to express my great appreciation to my supervisor Joakim Björk at Department ofAutomatic Control, KTH Royal Institute of Technology, for inspiring me to the topic of the thesis,giving me guidance, encouragement and useful critique of this research work. I would also like tothank Professor Karl Henrik Johansson at Department of Automatic Control, KTH Royal Instituteof Technology, for being my examiner and his time spent reviewing my work.

Erik BjörkStockholm, 2018

III

CONTENTS

Contents

1 Introduction 1

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

1.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Simulation Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Report Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background Theory 3

2.1 Power System Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 The Electromechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 The Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.2 The Swing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Small Signal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Linearising the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.2 Load Flow Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.3 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Load Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1 Static Load Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Generator Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5.1 Classical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5.2 One Axis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5.3 Eighth-order Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Power Oscillation Damping (POD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 System Model 14

3.1 Three Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 The Perpendicular System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 The AC-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2 The Complete Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Simulations Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Implementing the AC-Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

IV

CONTENTS

3.3.2 Implementing the HVDC-link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.3 Difference Between PSCAD and Simulink Model . . . . . . . . . . . . . . . . . . . 23

4 Results 24

4.1 Three Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.1 Load Type One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.2 Load Type Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Perpendicular System - Simulink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 HVDC-link Control - PSCAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Perpendicular System - PSCAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5 POD vs. No POD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Discussion 40

6 Conclusion 42

References 43

Appendices 46

A Per Unit 46

B Derivation of the Swing Equation 46

C AC-system Properties 48

D PSCAD Data Conversion 50

E The Perpendicular System Diagrams - Simulink 51

E.1 mp = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

E.2 mp = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

E.3 mp = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

E.4 mp = 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

E.5 mp = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

F The Perpendicular System Diagrams - PSCAD 62

F.1 mp=0, mq=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

F.2 mp=1, mq=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

V

CONTENTS

F.3 mp=1, mq=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

F.4 mp=1, mq=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

F.5 mp=2, mq=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

G Increased PSS Gain - Simulink 65

G.1 mp=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

G.2 mp=0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

G.3 mp=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

G.4 mp=1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

G.5 mp=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

H No POD - Simulink 75

H.1 mp=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

H.2 mp=0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

H.3 mp=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

H.4 mp=1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

H.5 mp=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

VI

1 Introduction

1.1 Background

In order to reduce societies dependencies of fossil fuels and its emissions of greenhouse gases the EuropeanUnion has the goal to have a 100 % renewable energy system by 2050 [1]. This has led to an increasingamount of renewable energy sources being implemented into the electric power system [2]. The increasingshare of renewables in the electric grid are reducing its inertia [3] which leads to concerns about thefrequency stability in the grids [4]. It has been documented that wind and solar power has an impact onlow frequency oscillations in the power system [3,5,6]. Parallel with the development of renewables in thegrid the load characteristics in the grid are also changing. Loads with constant power characteristic isbelieved to increase in the future with the implementation of power electronic devises between the actualload and the grid [7,8]. It is known that different load characteristics has different effect on the dynamicsin the grid [9]. It was shown by [10] that changed characteristic of loads could affect the damping withup to 25 %.

Usually the powers system stabilizer (PSS) located at large generators are used in order to damp theoscillations in the grid [11]. They are often effective at damping low frequency oscillations occurringclose to the generator [12]. But the PSS are not always as effective at damping low frequency oscillationsoccurring across larger areas, so called inter-area oscillations [11, 12]. This is due to the fact that thoseoscillations are difficult to observe and control from the local measurements [12]. With the increasedpenetration of renewables there is concerns if the PSS will be sufficient enough to damp the inter-areaoscillation in the grid [4, 11, 12]. Inter-area oscillations affects the stability in the system and couldalso limit power transfer capability in the grid [10]. An additional technique can be added togetherwith the PSS in order to increase the damping in the grid. Flexible AC transmission systems (FACTS)is an additional technique that is effective at damping oscillations in the grids [13]. However, theyexperience the same bottleneck as the PSS due to the usage of local measurements they have less effect atdamping inter-area oscillations. To overcome the lack of observability of inter-area oscillations due to localmeasurements for the PSS and FACTS devices, wide-area measurement system (WAMS) can be used [14].This can improve both the controllability and observability of the PSS and FACTS devices. But the delayof the transmitted signal has to be considered because it can reduce the damping potential [12]. Anothertechnique that can be used in order to damp power oscillations in the grid is by using the controllabilityof high-voltage direct current (HVDC) transmission [15].

HVDC transmission is a commonly used technology to interconnect two asynchronous grids to improvethe load balancing [15,16]. Since the 1970s it has been investigated how the controllability of the HVDCcan be used in order to damp oscillatory modes in the grid [17–19]. By controlling the active power fromthe HVDC-terminal the speed deviation of the local generators can be reduced [20] and this improves thepower oscillation damping (POD) of the system [21].

However, the controllability of the POD-service from the HVDC-terminal can be affected if the twointerconnected asynchronous grids has oscillatory modes of similar frequencies [21]. [21] showed how thisphenomenon, that henceforth will be referred to as modal interactions, occurred between the UK andNordic Power system. It is of interest to further investigate how the modal interactions will affect thecontrollability, i.e. the potential of POD service from the HVDC-link.

With the changed dynamics in the grid affected by the increase share of renewables and constant powerloads this master thesis will focus on how different characteristics of loads influence the POD providedby a HVDC-link.

Page 1

1.2 Problem Definition

The characteristic of the load affects the dynamics of the system. The purpose of this report is to inves-tigate how the changed system dynamic of different load characteristics influence the power oscillationdamping (POD) capability from HVDC. This investigation will be conducted via simulations in two dif-ferent simulation software programs. The objective and the research question to answer is formulatedas:

How does the characteristic of a load influence the POD from HVDC?

The research question has been divided in to the following sub goals. The goals are stated in an arbitraryorder.

1. Investigate the load characteristic influence on system damping using a linearised model.

2. Perform controlled simulations of a non-linear system model to find how the POD gets affected.

3. Verify the results from the simulations in two different software programs.

4. Compare the linearised results with the simulations result.

5. Determine which software was most suitable for the projects investigation.

1.3 Method

To investigate how the load-characteristic affects the small-signal stability a simulation experiment isconducted. This method was chosen in order to investigate the reaction and effects from a real powersystem in a controlled form. The method for the experiment is explained below.

1.3.1 Simulation Experiment

The simulation experiment can be divided in four steps. The method starts by creating a model torepresent the power system. The model sets the limitations on what parameters and behaviour that canbe observed. By determining the equations that rules the behaviour of the model the in- and out-putdata are defined. The second step is to implement the model in simulation software. In this project twosimulation software programs will be used. Using two software programs allows for comparison betweenthe results from each program. The two programs are Simulink developed by Mathworks and the secondsoftware is PSCAD developed by Manitoba HVDC Research Centre. The software programs are explainedmore in detail below in section 3.3. The third step in the process is to perform simulations of the model.The fourth step is to analyse the results after the simulations has been conducted.

1.4 Report Structure

The report is structured in the following way. First are the theoretical concepts and methods to solve theproblem presented in section 2. In section 3 the models used in the report are presented and its describedhow they are implemented in to each software. The simulations performed in this report are explainedin section 3 and the results from the simulations are shown in section 4. The results are analysed anddiscussed in section 5. Information that supports the analysis but was considered to not be essential forthe findings or claims is located in the appendices of the report.

Page 2

Steam

TurbinePm

Sha�

Tm

Te

Generator

PeRotor

� m

U

+

Figure 1: A simplified illustration of the relation between mechanical and electrical quantities.

2 Background Theory

In this section the theory required to understand the solution to the given problem is presented.

2.1 Power System Stability

This report will be concerning stability in power systems and therefore a proper clarification and definitionof the term system stability is suitable.

A system is considered to be stable when the opposing forces in a system are at a state of equilibrium. Inpower systems, stability is typically defined by two criteria, first, the ability for the system to maintain astate of equilibrium under normal operating conditions and secondly, the ability to return to an acceptablestate of equilibrium after a disturbance. When more than one generator is connected to the same system,the voltage, current and rotor speed must be in synchronism in order to have a stable system [22].

A disturbance to the system might endanger its stability by causing one or several generators to loosesynchronism with the rest of the system. The disturbance that acts on the power system can be eitherlarge or small. In the field of small signal stability the disturbances are small enough for a linearisedmodel of the system to be valid. An example of such a small disturbance can be the continuous changesof generation or load in the system. The large disturbance is concerning line faults, loss of generators orloss of loads [22].

This report will investigate small signal stability of power system and in particular stability regardinginter-area oscillations. These types of disturbances is further described in section 2.3.

2.2 The Electromechanical System

A special case of small signal stability is inter-area oscillations which are oscillations that occurs when thepower system is being subjected to a small disturbance. During inter-area oscillations, electric power isbeing transferred back and forth between groups of machines in the power system. These oscillations canbe observed both by measuring the electric power or the voltage frequency of the machines. The reasonthat these two different quantities can be used to observe the same phenomena is that the system of studyis both a mechanical system and an electrical system, hence a electromechanical system. Therefore whenmentioning power oscillations or frequency oscillations in this report both expression will be referring tothe same phenomena [22]. An illustration of this electromechanical system is shown in Figure 1, where themechanical part is illustrated by the turbine and the shaft while the electrical part is represented by thegenerator. In Figure 1 the subscript, m, denotes the mechanical properties and the subscript, e, denotesthe electrical properties. The power injected to the turbine Pm [W] is via the generator transformed intoelectrical power Pe [W] that is injected to the grid at a voltage U [V]. Note that the torque T [Nm] hasdifferent directions for the mechanical torque and the electrical torque. The angular speed of the rotor isdenoted ωm and given in [rad/s].

Page 3

2.2.1 The Dynamical System

The dynamical behaviour of a power system can be described by a set of n first order non-linear differentialand algebraic equations.

ẋ = f(x, y)

0 = g(x, y)(1)

The differential equations ẋ = f(x, y) describes the dynamic behaviour of the generator, loads or otherdynamical elements. The algebraic equations g(x, y) is based on Kirchhoff’s current law which statesthat the sum of all currents injected to a bus is zero. The states are represented by vector x and y isthe vector containing the algebraic variables [22]. The state and algebraic variables are considered to beautonomous with respect to time meaning that the dynamics are the same in the past and in the futurei.e. the system is time invariant [23].

In this project the dynamics of the generators in the system will be represented by f(x, y). The loadswill be described as algebraic equations and will be represented in g(x, y). Depending on the loadscharacteristics they will affect the equations in g(x, y).

2.2.2 The Swing Equation

To explain the dynamics in the electromechanical system the swing equation is usually used. It describesthe power oscillations that occurs in the grid during a disturbance [22]. The full derivation of the swingequation is shown in Appendix B. In this section only the vital parts from the derivation is presentedin order to get a better understanding of the swing equation. Variables with the subscript pu denotesthat its value is given in per unit. The usage of per unit and how a quantity is expressed in per unit isexplained in appendix A. The swing equation is a second order differential equation and is showed by(2).

2H

ωmsδ̈ = Tm,pu − Te,pu (2)

Where δ̈ is the acceleration of the load angle, δ. The inertia constant H given in [pu] is the normalisationof the inertia, J . The left part of the swing equation (2) equals the net-accelerating torque, Ta, that iscaused by the difference in mechanical torque, Tm, and electrical torque, Te, as seen by (3).

Ta,pu = Tm,pu − Te,pu (3)

When the mechanical and electrical torque is equal to each other the net-acceleration is zero and thesystem is stable. A perturbation from that equilibrium will cause a difference in electrical and mechanicaltorque. Thus causing the rotor to accelerate or decelerate.

The load angle, δ, is defined by

δ = ωrt− ωm,st+ δ0 (4)

Where ωr is the electrical speed of the rotor, ωm,s is the synchronous speed of the generator and δ0 isthe initial load angle. The inertia constant, H, usually falls within 3–8 s for a power system dominatedby synchronous machines [22]. The inertia constant is defined by

H =12Jω

2m,R

Sng(5)

Page 4

Where ωm,R is the rated speed of the generator given in [rad/s], Sng is the rated power of the generator[VA] and J is the inertia of the generator.

In the original swing equation (2) the damping effects of governors, damper windings and voltage regu-lators etc. are not included in the electric torque. A coefficient of damping is typically added in order torepresent these higher order dynamics. This result in

2H

ωm,sδ̈ = Tm,pu − Te,pu −Kd∆ωr,pu (6)

The swing equation can be formulated with respect to power which is usually desired due to the fact thatit is a power system that is investigated. If the stator resistance in the generator is neglected the relationbetween power and torque is

Pm = ωmTm

Pe = ωeTe(7)

In power systems where the changes of ωm and ωe is small the terms of power and torque can be usedinterchangeable. Hence when referring to stability in power system the change of torques is the same asthe change of power. This analogy is used is in section 2.3. With the torque and power relation fromequation (7), the swing equation (6) can be written with respect to the power instead

2H

ωm,sδ̈ = Pm,pu − Pe,pu −Kd∆ωr,pu (8)

In section 2.5 the swing equation (8) is formulated differently by using the relation

∆ω̇r,pu =δ̈

ωm,R(9)

and the swing equation is written as

2H∆ω̇r,pu = Pm,pu − Pe,pu −Kd∆ωr,pu (10)

2.3 Small Signal Stability

Power systems are governed by a set of non-linear equations. To analyse a non-linear system can becomplex. A common engineering approach to this problem is to linearise the system in order to applythe rich sets of tools from linear analysis. However a linearisation can only give an approximation of thelocal behaviour hence for a power system a linearisation is only valid if the system operates closes to thelinearisation point. Small signal stability studies the system close to the nominal operating point. Thedisturbances in such studies are so small that a linearisation of the system equations is acceptable [22].

The electric torque which is one of the quantities in the Swing equating can during a disturbance bedivided in to two parts the damping torque component and the synchronising torque component.

∆Te = TD∆ω + TS∆δ (11)

The damping torque component, TD∆ω, is in phase with the change of rotor speed, ∆ω. TD is thecoefficient of the damping torque. The synchronising torque component, TS∆δ, is in phase with change ofthe rotor angle ∆δ. The coefficient of the synchronising torque is TS . The generator can lose synchronism

Page 5

with the grid either by not having adequate synchronising torque or damping torque [22]. Insufficientsynchronising torque leads to an aperiodic drift of the rotor angle. The synchronising torque can beimproved by adding an active voltage regulator (AVR) to the excitations system of the generator. Lackof damping torque leads to oscillatory instability and to improve the damping torque of the systemAVR and PSS can be used [24]. Today small signal stability issues is mostly caused by lack of dampingtorque [25].

In small signal stability the studies concerns specific modes. When these modes occur a generator or agroup of generators starts to swing against each other. The oscillations that occurs during small signalstability is characterised into the following modes [24].

• Local modes. These refer to a group of generators at a power plant swinging against the rest of thesystem. These oscillations typically have a frequency of 1-3 Hz.

• Inter-area modes. A group of generators in one area are swinging against another group of generatorsin another area. The oscillations have a frequency of less than 1 Hz.

• Torsional modes. These modes are regarding the rotation of the turbine shaft. They are causedby interaction of control equipment and governor control, exciter control etc. The frequency of theoscillations is varying between 10-50 Hz.

• Control modes. These modes are regarding oscillations of badly tuned exciters, governors, staticvar compensators (SVCs) or HVDC converters. The oscillations usually has a frequency around 3Hz.

This report is mainly concerned about inter-area oscillations/modes.

2.3.1 Linearising the System

In order to analyse the non-linear system given by (1) with linear analysis tools, such as eigenvalues anddamping, the system is linearised around a operation point (x0, y0) [22]. It results in

∆ẋ = fx∆x+ fy∆y (12)

0 = gx∆x+ gy∆y (13)

Where fx, fy, gx and gy are the Jacobian matrices

fx =

[∂f(x, y)

∂x

]x=x0,y=y0

fy =

[∂f(x, y)

∂y

]x=x0,y=y0

gx =

[∂g(x, y)

∂x

]x=x0,y=y0

gy =

[∂g(x, y)

∂y

]x=x0,y=y0

(14)

If gy is assumed to be non-singular ∆y can be solved for from equation (13).

∆y = −(gy)−1gx∆x (15)

Page 6

This is then inserted in to equation (12) which results in the linearised system from the non-linear system.

∆ẋ = (fx − fy(gy)−1gx)∆x = A∆x (16)

2.3.2 Load Flow Calculations

For the simulation software to initiate the simulations at steady state, initial power flows and bus voltagesare needed. For this purpose load flow calculations are used. At steady state the dynamic equations areequal to zero, ẋ = f(x, y) = 0. Load flow calculations are based on linear network equations. However,the boundary equations are non-linear for the load flows. Thus, an analytical solution is not tractable andthe load flow has to be solved numerically. One common numerical method to solve a set on non-linearequations is the Newton-Raphson method which will be used in this report [26].

In a power system each bus has four quantities, the voltage (U), the active power injections (P ), thereactive power injections (Q) and the angle of the voltage (θ). The buses are categorized into a threesets of buses depending on which quantity that are known for each specific bus. The buses are definedas follows

• Slack bus. At this bus the voltage magnitude and angle are known (U, θ). The powers at this busare unknown in order to absorb or emit power to account for the unknown system losses. Therehas to be at least one slack bus in the system.

• PQ bus. At this bus, the net active and reactive power (P,Q) is known. Typically loads are definedas PQ buses.

• PU bus. The net active power and voltage magnitude are known at this bus (P,U). Generatorbuses are typically defined as PU buses.

2.3.3 Eigenvalues

In this section the usage of eigenvalues in small signal stability analysis is explained. A complete derivationfor how the eigenvalues are related to system stability and its modes can be found in [22], in this sectiononly the parts that is considered for the analysis conducted in this report will be shown. The study ofeigenvalues is very useful in order to study the modes of the system.

A linear time invariant system can be written in the following form

ẋ(t) = Ax(t) +Bu(t)

y(t) = Cx(t)(17)

where x(t) is the state vector, u(t) is the control/input vector and y(t) is the output vector of the system.The matrix A is the state matrix, B is the control matrix and C is the output matrix.

Considering the autonomous system where the system response is undriven, u(t) = 0 for all t.

ẋ(t) = Ax(t) (18)

The eigenvalues, λi, of the matrix A is given by the determinant

det(A− λI) = 0 (19)

Page 7

Here I represent the identity matrix (not electric current). Each eigenvalue of matrix A is correspondingto a specific mode of the system. By studying the eigenvalues, the stability of an equilibrium can be foundfor the system. If the eigenvalue are complex it corresponds to an oscillatory mode, if the eigenvalueis real it is a non-oscillatory mode. If the real part of an eigenvalue is positive it corresponds to non-decaying mode, hence the oscillations will grow. Contrarily if the real part of the eigenvalue is negativeit corresponds to a decaying mode where the oscillation decreases. A stable system needs to have all itseigenvalues in the left half of the complex plane [22].

If the eigenvalues are complex they appear in conjugate pairs where each pair corresponds to an oscillatorymode.

λk = σk ± jωk (20)

The real component, σk, tells the damping of the k-th mode. The frequency of the oscillations is givenby ωk in [rad] which in hertz is

fk =ωk2π

(21)

The damping ratio, ζk, is given by

ζk =−σk√σ2k + ω

2k

(22)

The damping ratio gives information about how the mode propagates. If the damping ratio is positivethe oscillations are decaying and the damping ratio gives information about the rate of decay. If thedamping ratio is negative the opposites occurs, the oscillations increases. The method to retrieve thedamping ratio from the A-matrix above is suitable for linear system where an A-matrix is known. In thisreport there will be non-linear systems which can be complex to linearise hence another method to findthe damping constant can be used.

As shown in section 2.2.2, the Swing equation is a second order differential equation if written on thefollowing form

2H

ωmsδ̈ = Pm,pu − Pe,pu −Kd∆δ̇ (23)

It is also shown by [27, 28] the internal motions of the power system can be expressed by a differentialequation of the second order. In [29] it is shown that a second order system has the following poles, notethat the poles are shown in Laplace domain.

s = ω0(− ζ ± j

√1 − ζ2

)(24)

This corresponds to the eigenvalue given by (20). A solution to the second order system is given in timedomain by

x(t) = Ae−bte±jat (25)

Where a = ω0√

1 − ζ2, b = ω0ζ and A is a constant. The expression e±jat describes time varyingoscillations at a frequency of ω = ω0

√1 − ζ2, e−bt describes a time varying amplitude. If b > 0 then

the function is decaying at a rate of ω0ζ. Hence if a function of the form y1(t) = Ae−bt is fitted to the

Page 8

0 2 4 6 8 10 12 14 16 18 20

Time [s]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

[ra

d/s

]

Figure 2: System oscillations with a exponential decay curve.

positive peaks to the oscillations of the system as shown in Figure 2 then the decay constant, b, of theexponential function can be used to find the damping constant by equation (26).

ζ =b

ω0(26)

The system decay is symmetrical and therefore a function of the form y2(t) = Be−bt can be fitted to the

negative peaks of the oscillating curve in Figure 2 and the same decay constant, b, can be found. Howeverthese theories are based on linear systems and are not guaranteed to always be true for non-linear systems.

2.4 Load Model

The stability of the power system is dependent on keeping the power generation equal to the powerconsumption. The way the load is modelled is of great importance in order represent this balanceaccurately.

The power system consists of many different loads, to know the characteristic of each load and to representeach component individually in the calculations and simulations is highly impractical. This is due to thehigh computation cost and computation time necessary to perform the calculations. Consequently theload representation in power system studies is based on simplifications of the loads [22]. A load in apower system has mainly four definitions described in [30]. In this report however the definition of asystem load defined in [30] will be used. The definition is

A load is a part of the system that is not explicitly represented but is instead represented by a powerconsumption device connected to the bus [30].

There are three characteristics [30] typically used to define a load

• Constant Power Load. The power does not change with the magnitude of the bus voltage. Anexample is a constant loaded motor or power electronic devices.

Page 9

• Constant Current Load. Internal impedance is varied in order to achieve a constant current con-sumption regardless of the voltage fed to the load. The power is proportional to the voltage. Anexample is battery chargers.

• Constant Impedance Load. The power change is directly changed with the square of the voltage.For example an electrical heater has constant impedance characteristic.

However all types of loads does not fall into a specific category of constant power, current or impedanceload. Instead they can have characteristics that are a mix from each category. Which characteristicthe load has affects the dynamics in the power system [9]. To represent the loads a mathematicalrepresentation of the load is required. This mathematical representation is called load model and describesthe relationship between the bus voltage and the power flowing in to the load [30]. There exist two differenttypes of load models, static load models and dynamic load models. Dynamic load models require specificknowledge about the load characteristic and its dynamics. Static load models are suitable for simulationswhere the load is composed as a part of a system, a composite load. In this report the loads will berepresented as composite loads and hence static load models will be used [22].

2.4.1 Static Load Models

The static load model describes the active power, PL, and reactive power, QL, as functions of the busvoltage magnitude. There exist two types of models to represent static loads with, the exponential modeland the ZIP-model [22].

The exponential load model [30] is given by

{PL = PL,0U

mp

QL = QL,0Umq (27)

Where

U =ULUL,0

(28)

UL,0,PL,0 and QL,0 are the nominal voltage and powers and UL is the current bus voltage. The exponentsmp and mq are constants that sets the characteristic of the load. The constants are almost equal to theslope dP/dU and dQ/dU when UL = UL,0. If the constants mp and mq is set to the values 0, 1, 2 thepower of the load will be characterised as a constant power, constant current or a constant impedance.However it should be noted that values in between 0, 1, 2 are also possible because a load can have acharacteristic somewhere between a constant power and constant inductance. If the load characteristicis unknown the common way is to set mp = 1 and mq = 2 [22].

The ZIP model is represented by equation

{P = P0(p1U

2 + p2U + p3)

Q = Q0(q1U2 + q2U + q3)

(29)

It is called the ZIP-model due to the fact that it is a model composed of constant impedance, Z, constantcurrent, I, and constant power, P, components. The constants p1, p2, p3, q1, q2, q3 defines the balance ofeach component [30].

In this report the exponential model will be used due to its simplicity to model and more intuitive tounderstand which characteristic of the load that influence the POD.

Page 10

A previous study was made on the western North America power system where it was found that thedamping on specific low frequency modes increased about 25 % when changing the characteristics of theconstant power loads from constant current to constant impedance [10]. The same behaviour was alsofound by [31] when tested on the Taiwan power system. It was found that the damping increased for aconstant impedance load compared to a values of mp similar to a constant current load. However thesetwo articles do not provide an explanation for the behaviour observed.

2.5 Generator Models

Inter-area oscillation are concerning oscillations between groups of generators, see section 2.3. Hence itis important to describe the dynamics of the generator properly in order to mimic the real response ofthe generator during a disturbance. In this report a group of generators in one area is modelled as oneequivalent machine. To describe the dynamics of a generator there exist several different models. But inthe following section only three models will be described, a complete derivation of all the models is givenby P.Kundur in [22].

2.5.1 Classical Model

The first model, the classical model is one of the simplest model to describe the dynamics of the gen-erator. The classical model does not take any high frequency transient behaviour of the generator into consideration hence it is more accurate when the generator is located far away from the disturbancepoint. With only two states to describe the dynamics of the generator its computations cost is low whichis useful if the system of study has a large amount of generator nodes. The dynamics of the generator isdescribed with the swing equation (10). The dynamics can be written in state form [22]

δ̇ = ω

ω̇ =1

M[Pm − Pe −Dω]

(30)

Where all the values are given in pu and

• ω = ωr − ωm,s is the angular difference.

• D represents the damping constant for the damping torque.

• M is a constant referred to as the mechanical starting time M = 2Hωms .

2.5.2 One Axis Model

In the one axis model the electrical properties of the generator are modelled using an extra state. Atransformation of the equations to dq-reference frame is made in order to simplify the description of thedynamics. The classical model assumes that the field flux Ψfd is constant. But in the one axis model Ψfdis not considered constant and thus the stator voltage is no longer constant. However the change of Ψfdis slowly. The transient field voltage E′q is proportional the field flux Ψfd [22]. The dynamic equationsof the one axis model is

δ̇ = ω

ω̇ =1

M[Pm − Pe −Dω]

Ė′q =1

Td0

(Ef −

xdx′dE′q +

xd − x′dx′d

U cos(δ − θ)) (31)

Page 11

HV

DC

Co

nn

ec�

on

Midpoint

Midpoint

System 1

System 2

PDC

� 1,1 � 1,2

� 2,2� 2,1

Figure 3: Swing board representation of the POD.

The prime symbol, ′, at the variables in (31) denotes the transient quantity of the variable [22].

2.5.3 Eighth-order Model

The last model described in this section is the eighth order model. This model ads six additional stateson the electrical properties of the generator. The additional states are Ψd,Ψfd,Ψd1,Ψq,Ψq1,Ψq2. Theextra states takes the dynamics of the excitation of the field windings in concern Ψfd and the generator’stwo damping windings, that has dynamics governed by Ψd,Ψd1,Ψq,Ψq1,Ψq2. The eight order model hasa more accurate representation of the dynamics in the generator however it has a higher computationcost compared to the classical model and the one axis model.

2.6 Power Oscillation Damping (POD)

To implement power oscillation damping (POD) control the inter-area oscillations has to be measured.There are three signals that are common to use when measuring inter-area oscillation, the rotor speed∆ω, the frequency of the busbar voltage ∆f and the line power flow fluctuations ∆P [32]. The signals∆ω and ∆P is generally not available at the HVDC-terminal stations where the POD-control will beimplemented. Some sort of communication system has to be added parallel to the transmission system inorder to provide those signals. Therefore in a real application the signal ∆f can be considered because itis available at all buses in the system. However ∆f is a weak signal compared to ∆ω and ∆P . This meansthat it is more difficult to detect the inter-area oscillations in the signal ∆f unless sensitive measurementsdevises that can withstand disturbances are used. This project is aiming at finding the most achievablePOD and will be using a simulation software where all three signals are easy to provide to all buses,therefore ∆f will not be used. ∆P needs to be phase shifted by 90◦ in order to be used which is not thecase for ∆ω [32]. Therefore ∆ω will be used as a measurement of the inter-area oscillations and as inputsignal for the POD-control. To more easily understand how active power oscillation damping works theanalogy of a swing plank described in [20] can be used. As represented in Figure 3 each system can beseen as a swing plank where the masses of the generators are balancing on a sharp edge in the middle.At equilibrium the difference in speed of the two generators in the system will be zero.

{∆ω1 = ω1,1 − ω1,2∆ω2 = ω2,1 − ω2,2

(32)

The power in the HVDC-link is described by

PDC = P0 + ∆PDC (33)

Where P0 is the nominal bulk power transfer between the two systems and ∆PDC is the power that thePOD is transferring. If there is a difference in speed between the two generators an unbalance occurs

Page 12

and the swing plank starts to tilt. To restore the balance the HVDC link can remove energy from thefirst generator in order to bring it down from the air. A negative proportional feedback controller i.e.P-controller regulating the term ∆PDC is used in this project to control the POD. The controller isdescribed by the following equation.

∆PDC = −K1∆ω1 +K2∆ω2, K1,K2 > 0 (34)

The signs in front of the constants K1,K2 is taking into account the HVDC-link’s predefined powerdirection. An example of the control is if ω1,1 becomes larger than ω1,2 in system 1 then generator 1 willfiguratively rise to the air and generator 2 touch the ground. Power has to be drawn from System 1 inorder to drag generator back to equilibrium hence the negative sign in front of K1 and positive sign infront of K2.

Page 13

G1

P , QL L

R + jX1 R + jX2

P , Qe,1 e,1 U

U

IInduc�ve Capaci�ve

Figure 5: Voltage current behaviour of a SVC described in [22].

It is known that active power transfer is mainly governed by the angular differences between two busesand that the reactive power transfer is governed by the voltage difference at those buses [22]. The voltagebehaviour of a reactive load can be exemplified by studying the voltage-current behaviour of a static varcompensator (SVC) seen in Figure 5 and is described in [22]. If a inductive load is assumed then whenthe reactive load increases the voltage at the bus decreases.

To understand how the loads characteristic interacts with the system and how they contribute to thedamping the following interpretation can be made. First some quantities of interest are stated in order toenhance the understanding. The angular difference between the generator bus and load bus is θ1−θ3 = δ.The power transfer between these buses is denoted, P13, Q13. The following reasoning assumes line losses.

If the active load increases the angular difference, δ, will increase and hence the power transfer, P13,will increase. A small note, this is what initiates inter-area oscillations, the increased power demand willdecrease the speed of the generator until it is able to speed up and the oscillations has started. As P13increases, the losses across the line increases and hence the voltage at bus 3, U3, decreases. The decreasedvoltage, U3, will decrease the load power for load constants, mp > 0. Hence this will decrease the powerdemand and the angular difference and this helps to damp the inter-area oscillations.

Now studying the behaviour of a reactive load and how it reacts during a disturbance. When the activeload increases it will as shown previously increase the angular difference. The power transfer will thereforeincrease and the losses along the line as well and hence the voltage, U3, will decrease. Inspecting (36)the reactive load will decrease for all mq > 0. When the reactive load decreases, the voltage at bus 3will increase hence it will counteract the damping that the active load contributes. However now it isassumed that the recitative load consumes reactive power as an inductance. If the reactive load wouldbe an capacitance the opposite would happen. Then for mq > 0 it will not counteract the damping fromthe active load it will instead assist the damping from the active load.

To conclude this reasoning, it is believed that the active load gives less damping as mp = 0 and thedamping will increase as mp increases. If it is assumed that the reactive load is inductive then worstdamping is achieved as mq = 2 and the damping increases as mq approaches 0. This hypothesis is testedin section 4. Parameter for the Three bus system us given in table 1 and 2. The parameters was chosenarbitrary after typical standard values given in [22]. It will be interesting to see if the same behaviourcan be seen in the Perpendicular system as is described in next section.

3.2 The Perpendicular System

In previous studies where the potential of damping inter-area oscillations in two AC-systems by POD viaan HVDC-link has been investigated, a set-up like the one shown in Figure 6 has been used [16, 20]. Inthis report a similar set-up will be used for the model implemented in the two simulation programs. Inthe previous studies the AC-system was represented by a two machine model which allowed for multiplesimplifications in order to perform an analytical study of the damping potential. However in this reportanalytical studies of the system will not be conducted instead a numerical study will be conducted of

Page 15

Table 1: Generator Data for the Three Bus System

Description Symbol Value Unit

Inertia Constant H 4 MWs/MVADamping Constant D 0 pu

d-axis Synchronous Reactance Xd 1 pud-axis Transient Reactance X ′d 0.15 pu

Terminal reactance Xt 0.1 pud-axis Open Circuit Transient Time Constant T ′do 6 s

Rated Active Power Pm 1 pu

Table 2: Network Data for the Three Bus System

Description Symbol Value Unit

Voltage Magnitude, Bus 1 U1 1 puVoltage Magnitude, Bus 2 U2 1 pu

Bus angle, Bus 2 θ2 0 degreesLine reactance, from Bus 1 to Bus 2 X1 0.3 puLine resistance, from Bus 1 to Bus 2 R1 0.01 puLine reactance, from Bus 2 to Bus 3 X2 0.5 puLine resistance, from Bus 2 to Bus 3 R2 0.02 pu

Nominal Active Load PL,0 0.7 puNominal Reactive Load QL,0 0.7 pu

the system which allows for a more complex dynamical system. The AC-systems will be represented ofa benchmark system for small signal stability studies called the two area four generator-system and isshorten to 2A4G-system [34]. The model was first used in [35, 36] and properly explained in [22] and isdefined as a benchmark system in [34]. The 2A4G-model was chosen due its similarity to the systems usedin the previous studies [16,20] and because it had a small amount of buses compared to other benchmarksystems. A small amount of buses has a smaller computation cost compared to larger system with morebuses and to ensure that the simulations runs as fast as possible its desired to keep the computationcost low. The HVDC-link is using voltage source converter (VSC) technology. The VSC-terminals hasindependent active- and reactive-power control, this controllability allows to implement control featuressuch as POD [15, 16, 21]. A more detailed explanation of the AC-systems and the HVDC-link and howthey were implemented in the simulation programs is given below.

3.2.1 The AC-system

The 2A4G-system is shown in Figure 7 where it can be seen that bus 6 is dividing the system into twoequal areas hence the name. When a load change occurs the inter-area oscillations will be governed bygenerator 1 and 2 swinging against generator 3 and 4. The model parameters are from the benchmarksystem [34], the power base is Sbase = 100 MVA, the voltage base for the generator is UGen, base = 20 kV.The lines are rated for 230 kV hence the lines the voltage base is Ubase = 230 kV. All data for the systemis shown in tables in Appendix C. The parameters of the lines in pu-values are

R = 0.0001[pu/km] X = 0.001[pu/km] bc = 0.00175[pu/km]

Bus and line data are shown in table 12 and 13. The lines are modelled using the pi-model. The nominalfrequency of the system is 60 Hz.

The generators step up transformer is ∆/∆-connected and they are rated for 900 MVA with a positiveleakage reactance of 0.15 pu. The magnetization losses and winding resistance are neglected as shown intable 16.

Page 16

AC

DC

DC

AC

AC

System 1

AC

System 2

PDC,1 PDC,2

Figure 6: Simplified representation of the Perpendicular system.

Gen 4

AC

DC

Ge

n 3

Ge

n 2

Gen 1Bus 2 Bus 1Bus 4 Bus 3 Bus 5 Bus 6 Bus 7

T4

T3 T2

T1L5

L7C5 C7

25 km 10 km 110 km 110 km 10 km 25 km�/�

�

/��

/�

�/�

Figure 7: The two area four generator system.

The capacitor banks are directly connected at bus 5 and 7, hence rated for 230 kV. Their specified valuesat 1 pu is shown in table 15.

At bus 5 and 7 there are loads directly connected to the 230 kV buses. The loads specified data areshown in table 17. The characteristic of the loads will be varied through different solutions but are asdefault set as a pure current source for the active power and as an impedance for the reactive power(mp = 1,mq = 2).

The generators have exciter and PSS controls. The exciter is a IEEE standard static exciter suitable forpower system stability studies called DC1A [37]. The parameters for the exciter is given by [34] and isshown in table 19. Some parameters is however altered from the IEEE standard exciter in order to fit themodel. Some parameters do even alter between the two software programs. Read about the parameterchoice below. The PSS that is used is created by P.Kundur and is described on pages (814-815) in [22].To implement this PSS the IEEE standard PSS called PSS1A [37] can be used P.Kundurs version isachieved by choosing the constants in the PSS accordingly. The parameters for the PSS is from [34] andis shown in table 20.

3.2.2 The Complete Model

A HVDC-link is added to the original benchmark system. The HVDC-links inverter is connected at bus 5in both of the two AC-systems. The HVDC-link along with the inverter station is assumed to be losslessand have a nominal power transfer of 0 MW thus it will primarily not affect the system, just when thePOD control finds it necessary.

With the AC systems defined as mentioned above in section 3.2.1 a complete image of the system can becreated from the simplified system shown in Figure 6. The complete system is shown in Figure 8. Notethat the length is between bus 5 and 7 is different for system 2 compared to system 1. Changing theimpedance of the lines will change the frequency of the inter-area modes. As was found by [21] modalinteractions occurred when the modes had similar frequencies and for the system to be controllable weneed the inter-area modes to be different [16]. The values for the line properties in system 2 is shown intable 14.

Page 17

Gen 4

AC

DC

Ge

n 3

Ge

n 2


T4

T3 T2

T1L5

L7C5 C7

25 km 10 km 110 km 110 km 10 km 25 km�/�

�

/��

/�

�/�

Gen 4

DC

AC

Ge

n 3

Ge

n 2


T4

T3 T2

T1L5

L7C5 C7

25 km 10 km 76.4 km 76.4 km 10 km 25 km�/�

�

/��

/�

�/�

System 2

System 1

Figure 8: The Perpendicular system.

3.3 Simulations Software

To see how the behaviour of the system reacts to different characteristics of loads the dynamical equationshas to be solved for every instant of time. The equations can be complex to solve analytically due itsnon-linear characteristics. Therefore numerical simulations are performed. The system in Figure 8 isbuilt in the simulations software Simulink and PSCAD.

Simulink is a software program developed by Mathworks and is a tool to simulate dynamic systems. Theadvantage of using Simulink is that each element in Simulink is well documented hence it is easy to builda model with known elements. This is important because the power system is very complex and henceit is desired to have known equations in order to explain the behaviour of the system with the specifiedequations.

PSCAD is software that is well known and used in both academia and industry [38]. The downside withPSCAD is that its components are not as well documented compared to Simulink due to confidentialityreasons. The equations that rules the behaviour of the component is therefore often unknown to the user.

The following section is structured in the same manner as the system was implemented in to eachsoftware program. First is the AC-systems created and after that the HVDC-link that interconnectsthe AC-systems is created. Under each sub section it is described how this was done for each softwareprogram. The nominal frequency for the simulation model was 60 Hz. This was due to the fact that bothSimulink and PSCAD had it as their standard frequency.

3.3.1 Implementing the AC-Grid

The two area four generator system that is shown in Figure 7 is implemented differently for each softwareprogram.

Page 18

(a) A illustration of a 25 km pi-line in PSCAD.

(b) A illustration of the line connection between bus4 and 3 in PSCAD. Each line segment represents api-line.

Figure 9: A illustration of line models in PSCAD.

Simulink

Simulink has the 2A4G-system available in the MATLAB library by typing ”power PSS” into the com-mand window [39]. The loads in the model were changed to the load block called Three-Phase DynamicLoad [40] in order to change the characteristics in the same way as in equation (27) between each simu-lation. Note that the time constants should be set to zero in order to have a static load model.

The generators in Simulink is modelled with the eighth order state model where six states describes theelectrical dynamics and two states describes the mechanical states with the swing equation (30) [41]. ThePSS gain was set to 5 for system 1 and 4 for system 2. This was because the PSS was to strong withthe original gain values. By reducing its gain value its impact was reduced and the inter-area oscillationswere easier to observe.

PSCAD

A PSCAD model of 2A4G-system was created for this thesis. The 2A4G-system has the line propertiesgiven in pu [34] but in PSCAD the line quantities has to be given in their base unit. To transform thepu-values given in table 13 to the base values see Appendix D. Each line segment is modelled as a pi-lineas seen in Figure 9a. The pi-line is then implemented into a box where each line segment represents api-line. The connection between bus 4 and bus 3 is shown in Figure 9b.

The capacitor banks are modelled as a capacitance connected to ground. The capacitance value in Faradis the reactive power given at nominal voltage. The conversion from capacitor banks power value to Faradis shown in Appendix D.

In PSCAD, the dynamic equations for the synchronous machine are not fully stated. This is one of thedownsides with PSCAD, the models and blocks are not fully documented due to confidentially reasons.But the model has been proven to be reliable and accurate [38].

In PSCAD the generator is consisting of the synchronous machine block, Torsional shaft model, Exciter,PSS, Turbine model and governor. The data from table 18 was inserted in the synchronous machine block.The exciter data from table 19 was implemented in the exciter block which had the chosen dynamicsof DC1A however the exciter gain was reduced from KA = 200 to a value of 46. This was due to thefact that the system was not stable with an exciter gain of KA = 200. The PSS data from table 20 wasinserted in the PSS block with PS1A-dynamics.

With the given settings of the systems parameters it was difficult to implement a steam turbine that wasstable enough to run the simulations required. Instead a hydro turbine along with a hydro governor wasimplemented which had a more stable performance with the given settings. The disadvantage of a hydroturbine is because it has a non-minimum phase behaviour which sets limitations to the model. This ismore explicit explained in [22]. The hydro turbine was given by the standard model Hydro Tur 1 fromPSCAD along with the standard governor Hydro Gov 1. The torsional shaft model, modelled four movingmasses at 900 MVA similar to Simulink.

To have different types of turbines in each simulation program, steam turbines in Simulink and hydroturbines in PSCAD, is off curse not ideal when it is desired to compare the results from each program.

Page 19

(a) The generator model in PSCAD. (b) The generator model block in PSCAD.

Figure 10: Illustration of the generator model in PSCAD.

Figure 11: An illustration of the 2A4G-system in PSCAD.

However, as the goal of the study is to investigate how load characteristic influence the damping in thepower system, and real power systems can have both hydro and steam turbines in them, it can be ofinterest to see if the same inter-area mode behaviour can be observed independent of the type of turbineused.

The loads are implemented in the block called Fixed Load. This block allows for changing the characteristicof the load in the same way as in equation (27) and in the Simulink block Three-phase Dynamic Block.The transformers are implemented in the block 3 Phase 2 Winding Transformer. Where the data fromtable 16 is inserted. By connecting all components as specified by Figure 7 the PSCAD model is createdand can be seen in Figure 11.

3.3.2 Implementing the HVDC-link

The AC-grids is now connected using a HVDC-link. A simple model of a HVDC-link can be viewed inFigure 12. The model consists of two VSC-terminals and a DC-link. The VSC terminals has two degreesof freedom which enables them to control the active P and reactive power Q individually. The active

Page 20

AC

DC

DC

ACUAC,1 UAC,2

++

P Q1 1

P Q2 2

+

UDC C

Figure 12: A illustration of the HVDC-link.

AC

DCU

AC

+

P Q

+

DCU C

Figure 13: A illustration of a STATCOM.

power is correlated to the DC-voltage UDC and the reactive power is correlated with the AC-voltageUAC . Therefore the VSC-terminal can control either the active power or the DC-voltage and the reactivepower or the AC-voltage [42]. In order to not conflict with the exciter control in each system bothterminals of the HVDC-link will control the reactive power and not the AC-voltage. The reactive powerwill have a reference of 0 MVAr. The terminal connected to system 1 will control the voltage, UDC , inthe HVDC-link. The terminal connected to system 2 will control the active power P2 injected to system2. So far everything is equal for both Simulink and PSCAD. The difference of the two programs is howHVDC-link is modelled. The voltage in the HVDC-link is set to 640 kV.

Simulink

It is assumed that the HVDC-link has no transmission line losses which has been assumed in previousarticles [16, 43]. This allows for the voltage in the DC-link to be written as function of the current fromeach terminal.

dUDCdt

=1

C(i1 − i2) (37)

If the terminals are of VSC-technology then they can operate in such a speed that the DC voltage can beassumed to be constant and hence it can be assumed that P2 = −P1 [16]. Due to this assumption it isnot necessary to model the dynamics in the HVDC-line only the dynamics at the terminals is modelled.To only model the dynamics of the terminals a static var compensator (STATCOM) can be used. ASTACOM is a device that consist of a capacitor in parallel with a VSC connected to the AC grid [44]. Ageneral visualization of a STACOM is shown in Figure 13.

To create a model of the HVDC-link two STATCOMs are connected at each AC-system. Since the HVDCdynamics are outside the bandwidth of interest it was decided to use the Simulink model Statcom (PhasorModel). This model does not have switching actions as would have been for a more detailed model ofthe STATCOM. The phasor model just takes the reference values and converts it to the actual values ofAC-voltage and currents [45]. It can be seen how it is implemented in the Simulink model in Figure 14.

The power direction in the DC-link is set as going from system 2 to system 1. The bulk power transfer isset to P0 = 0 MW in order to not affect the initial steady states for the benchmarked AC-systems. Thecontrol for the POD was described in section 2.6 where the following equation was described

∆PDC = −K1∆ω1 +K2∆ω2, K1 = K2 > 0 (38)

Page 21

multiplied capacitans with 5Trip

Vref

Pref

Vabc_conv_SH

Iabc_SH

P

A

B

C

VSC, P-reg

100 MVASTATCOM1

Vdc

Vabc_Preg_SH

Iabc_Preg_SH

Vabc_Vreg

Iabc_Vreg

Vdc

DC Voltage

Computation

Trip

Vref

Vabc_Vreg

Iabc_Vreg

P

A

B

C

VSC, V-reg

100 MVASTATCOM2

1

1

Pref

1

A

2

B

3

C

4

a5

b6

c

1

P1

2

P2

-1

Vdc (pu)

Figure 14: How the STATCOMs are connected in Simulink.

The rotor speed deviation ∆ω1 and ∆ω2 will be defined differently now when there is more than twogenerators in the system. The inter-area oscillations will be generator 1 and 2, referred to as area 1,swinging against generator 3 and 4, area 2. Hence the rotor speed deviation used for the control of PODwill be the weighted frequency of the generators in area 1. This is also referred to as the frequency tothe centre of inertia (COI) [46] which is defined in (39).

ωCOI,1 =M1ω1 +M2ω2M1 +M2

ωCOI,2 =M3ω3 +M4ω4M3 +M4

(39)

However as shown in section 2.5, M = 2H/ωm,s and hence the inertia constant can be used and due to thefact that the inertia constant is the same for generators in respective area, see table 18, a simplificationis made.

∆ω1 =H1ω1,1 +H2ω2,1

H1 +H2− H3ω3,1 +H4ω4,1

H3 +H4=ω1,1 + ω2,1

2− ω3,1 + ω4,1

2

∆ω2 =H1ω1,2 +H2ω2,2

H1 +H2− H3ω3,2 +H4ω4,2

H3 +H4=ω1,2 + ω2,2

2− ω3,2 + ω4,2

2

(40)

The HVDC feedback gains K1,K2 were set to

∆PDC = −0.133∆ω1 + 0.1444∆ω2 (41)

The gains were set in order to optimize the damping for a linearised version of the system. Constantimpedance characteristic of the loads were used in the system when it was linearised. Hence using theseconstants in the non-linear system they are an approximation of the optimal gains.

PSCAD

PSCAD has on their website finished models of a HVDC-link that has terminals with modular multi-level converter (MMC) technique [47]. MMC is as the name suggesting composed of several layers ofconverters. The advantage of this configuration is less losses in the conversion, a filter to get rid ofharmonics is nearly eliminated and it is easier to handle faults. This type of converter valves are beingincreasingly used in the industry [48]. The PSCAD-model is created to study how a fault in the HVDC-link affects its performance. Therefore it has detailed models of the dynamics in the cables and terminalsof the HVDC-link. Line losses is considered here compared to the model in Simulink. In the file thereis multiple different topologies of HVDC-links. Symmetrical monopole configuration which is used bythe majority of today’s VSC-based HVDC systems [49] is chosen for our model. As mentioned before adisadvantage with PSCAD is the lack of documentation of its components and models. It is the same for

Page 22

Figure 15: The HVDC-link model interconnecting the two AC-systems in PSCAD.

this model, hence it unknown for the author how the MMC are modulated and what equations that rulesthe model. But the modulation of MMC terminals is not the topic of this thesis project and the authoris able to control the terminals input reference and when tested later shown in section 4 the controlimplemented in the model is able to track the given reference with sufficient results. The complete modelfor the system is shown in Figure 15.

The POD-control is the same as the one used in Simulink. The same feedback gains (41) is implementedin the PSCAD model although the model differs slightly from the Simulink model. The gains will give asufficient approximation of the optimal gains.

3.3.3 Difference Between PSCAD and Simulink Model

To conclude the differences between the model in the two software programs. PSCAD is using hydroturbines and governors instead of steam turbines that is used in Simulink. The hydro turbines has a bitslower dynamic responds compared to the steam turbines and can set limitations to the system. PSCADhas a much richer model of the HVDC-link compared to Simulink. The HVDC-link in PSCAD takes theswitching actions in the valves in consideration and losses along the DC transmission line.

Page 23

4 Results

The results from the Three bus system and the Perpendicular system is presented in this section. Specificmodel parameters and how the models were implemented is described in section 3. The nominal frequencyfor the simulations is set to 60 Hz due to the fact that the simulations software had it as a standard. Asdescribed in section 2.4 the equation to alter the characteristic of the loads is

{PL = PL,0U

mp

QL = QL,0Umq (42)

This will be the equation to focus on in the following section. The constants mp and mq will be altered foreach simulation in order to see how the damping in the system gets affected by the different characteristics.

4.1 Three Bus System

The Three bus model described in 3.1 is linearised with the method explained in section 2.3.1. Analgebraic expression of the matrix A is not shown in the report due to the complicated expression thatis created. Hence there is not possible to make any analytical conclusions of how the different loadcharacteristics affect the damping in the system. Therefore numerical methods were used to solve thismodel.

With the given data from section 3.1 load flows are performed. This results in the following initial valuesfor the simulation.

x0 =[δ0,1 ω1,0 E

′q0,1

]=[44.6529◦ 0 1.2545

](43)

y0 =[θ1 θ3 U1 U3

]=[33.1577◦ 11.0185◦ 1 0.7717

](44)

To test how the damping changes as the characteristic of the load is changed five values is chosen for theactive and reactive loads each.

mp =[0 0.5 1 1.5 2

]mq =

[0 0.5 1 1.5 2

] (45)The values of 0, 1 and 2 reflect the constant power, current and impedance characteristic. To observethe trend of damping between these three types of characteristic the values of 0.5 and 1.5 is chosen aswell. Each value of mp is tested with all values of mq hence there will be 25 simulations in total.

Two different types of reactive loads will be tested, load type one and load type two. The first reactiveload is modelled as a consumer of reactive power with QL = 0.7 MVAr. This is to test the behaviour ofan inductive load as argued for in section 3.1. This load should give best damping as a constant powerload and worst damping as a constant impedance load. In the same section it was also argued for thatthe opposite will occur if the reactive load is a producer of reactive power, a capacitance. Hence thesecond load type tested is then the same system with the same values however the load has a reactiveload of QL = −0.7 MVAr.

4.1.1 Load Type One

For load type one the reactive loads are consuming reactive power. The system damping, ζ, is shown intable 3 where is can be seen how it changes with different load characteristics. The damping constant,

Page 24

ζ, along with the mode specific frequency, f0, is found from the eigenvalues of the A matrix as shown insection 2.3.3. The nominal frequency is set to 60 Hz.

Table 3: Resulting damping coefficients for loads using load type one. Each value of mp is divided in toeach sub-table.

(a) mp = 0

mp mq ζ f0 [Hz]

0 0 0.0322 0.75650 0.5 0.0213 0.79900 1 0.0159 0.82630 1. 0.0128 0.84540 2 0.0109 0.8595

(b) mp = 0.5

mp mq ζ f0 [Hz]

0.5 0 0.0401 0.71110.5 0.5 0.0256 0.76210.5 1 0.0187 0.79530.5 1.5 0.0149 0.81860.5 2 0.0124 0.8359

(c) mp = 1

mp mq ζ f0 [Hz]

1 0 0.0490 0.66841 0.5 0.0302 0.72711 1 0.0217 0.76561 1.5 0.0169 0.79291 2 0.0140 0.8132

(d) mp = 1.5

mp mq ζ f0 [Hz]

1.5 0 0.0592 0.62811.5 0.5 0.0353 0.69371.5 1 0.0248 0.73721.5 1.5 0.0191 0.76811.5 2 0.0156 0.7912

(e) mp = 2

mp mq ζ f0 [Hz]

2 0 0.0712 0.58972 0.5 0.0409 0.66172 1 0.0282 0.70982 1.5 0.0214 0.74412 2 0.0173 0.7698

The change of the damping constant from table 3 is visualised in Figure 16. Each step of active loadcharacteristic mp is shown in each sub figure. The lines from each sub figure in Figure 16 is showntogether in Figure 17.

Page 25

0 0.5 1 1.5 2

mq

0.0109

0.0128

0.0159

0.0213

0.0322

Dam

pin

g C

oeffic

ient,

mp = 0

(a) mp = 0

0 0.5 1 1.5 2

mq

0.0124

0.0149

0.0187

0.0256

0.0401

Dam

pin

g C

oeffic

ient,

mp = 0.5

(b) mp = 0.5

0 0.5 1 1.5 2

mq

0.014

0.0169

0.0217

0.0302

0.049

Dam

pin

g C

oeffic

ient,

mp = 1

(c) mp = 1

0 0.5 1 1.5 2

mq

0.0156

0.0191

0.0248

0.0353

0.0592D

am

pin

g C

oeffic

ient,

mp = 1.5

(d) mp = 1.5

0 0.5 1 1.5 2

mq

0.0173

0.0214

0.0282

0.0409

0.0712

Dam

pin

g C

oeffic

ient,

mp = 2

(e) mp = 2

Figure 16: The change of damping can be seen as mq is altered when mp is constant. The data fromtable 3 is used. The data points is marked with a ring in each figure and the line is drawn between eachdata point to show the trend.

Page 26

0 0.5 1 1.5 2

mq

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Da

mp

ing

Co

eff

icie

nt,

Change of Damping

Figure 17: The change of damping in the Three bus system for load type one. The characteristic constantmq is changed for each line of mp.

It can be seen from table 3 and Figure 17 that the damping increases as mp increases from 0 to 2. Forevery value of mp it can be seen that best damping is achieved when mq is 0 and that the damping getsworse when mq increases in value.

4.1.2 Load Type Two

Load type two has reactive loads that produces reactive power, as capacitors. With change of load thepower flow calculations has to be repeated. The new ones results in the following initial values.

x0 =[δ0,1 ω1,0 E

′q0,1

]=[38.2192◦ 0 0.9852

](46)

y0 =[θ1 θ3 U1 U3

]=[23.5187◦ 7.4276◦ 1 1.0892

](47)

The results of how the system damping changes for each characteristic of the loads when the loads producereactive power is shown in table 4. The damping constant, ζ, along with the mode specific frequency, f0,is found from the eigenvalues given by the A matrix as shown in section 2.3.3. The nominal frequency isset to 60 Hz.

Page 27

Table 4: Resulting damping coefficients for loads using load type two. Each value of mp is divided in toeach sub-table.

(a) mp = 0

mp mq ζ f0 [Hz]

0 0 0.0088 1.01640 0.5 0.0093 1.01240 1 0.0099 1.00770 1.5 0.0107 1.00210 2 0.0116 0.9954

(b) mp = 0.5

mp mq ζ f0 [Hz]

0.5 0 0.0094 1.00550.5 0.5 0.0100 1.00060.5 1 0.0107 0.99500.5 1.5 0.0116 0.98840.5 2 0.0127 0.9805

(c) mp = 1

mp mq ζ f0 [Hz]

1 0 0.0101 0.99471 0.5 0.0108 0.98911 1 0.0116 0.98261 1.5 0.0126 0.97491 2 0.0139 0.9659

(d) mp = 1.5

mp mq ζ f0 [Hz]

1.5 0 0.0108 0.98411.5 0.5 0.0115 0.97771.5 1 0.0124 0.97041.5 1.5 0.0135 0.96181.5 2 0.0150 0.9516

(e) mp = 2

mp mq ζ f0 [Hz]

2 0 0.0114 0.97372 0.5 0.0123 0.96662 1 0.0133 0.95852 1.5 0.0145 0.94902 2 0.0161 0.9377

The change of the damping constant from table 4 is visualised in Figure 18. Each step of active loadcharacteristic mp is shown in each sub figure. The lines from each sub figure in Figure 18 is shown inFigure 19.

Page 28

0 0.5 1 1.5 2

mq

0.0088

0.0093

0.0099

0.0107

0.0116

Dam

pin

g C

oeffic

ient,

mp = 0

(a) mp = 0

0 0.5 1 1.5 2

mq

0.0094

0.01

0.0107

0.0116

0.0127

Dam

pin

g C

oeffic

ient,

mp = 0.5

(b) mp = 0.5

0 0.5 1 1.5 2

mq

0.0101

0.0108

0.0116

0.0126

0.0139

Dam

pin

g C

oeffic

ient,

mp = 1

(c) mp = 1

0 0.5 1 1.5 2

mq

0.0108

0.0115

0.0124

0.0135

0.015D

am

pin

g C

oeffic

ient,

mp = 1.5

(d) mp = 1.5

0 0.5 1 1.5 2

mq

0.0114

0.0123

0.0133

0.0145

0.0161

Dam

pin

g C

oeffic

ient,

mp = 2

(e) mp = 2

Figure 18: The change of damping can be seen as mq is changed when mp is constant. The data fromtable 4 is used. The data points is marked with a ring in each figure and the line is drawn between eachdata point to show the trend.

Page 29

0 0.5 1 1.5 2

mq

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

0.016

0.017

Da

mp

ing

Co

eff

icie

nt,

Change of Damping

Figure 19: The change of damping in the Three bus system for load type one. The characteristic constantmq is changed for each value of mp.

For load type two it can be seen in table 4 that the same behaviour of mp can be seen as for load typeone. The damping increases as mp increases from 0 to 2. For the value of mq it can be seen that bestdamping is achieved when the reactive load has the characteristic of a constant impedance mq = 2. Thedamping reduces as mq reduces and approaches a constant power load mq = 0. As seen in Figure 19 thetotal system damping is increasing as the value of mq increases for all values of mp.

4.2 Perpendicular System - Simulink

The Perpendicular system described in section 3.3.1 is simulated for 20 s. A disturbance is initiated insystem 1 at t = 1 s. The disturbance is a load increase at bus 5 in system 1 with 180 MW. The systemwas simulated with different values on the exponents mp and mq. The values were

mp =[0 0.5 1 1.5 2

]mq =

[0 0.5 1 1.5 2

] (48)25 simulations were made in order to test all the values of mp against all the values of mq. The solversettings in Simulink are similar to the built in Simulink system (power PSS). The ode23tb solver withvariable time step was used. The ode23b is faster for stiff systems compared to other solvers for example”ode15” [50]. The time step is variable for each simulation and is set by Simulink itself.

The method used for linearised system to find the damping coefficient, ζ, in previous section cannot beused in this section due to the fact that the simulated system is non-linear. Hence an A-matrix will notbe available. However the eigenvalues can be described with an exponential function as shown in section2.3.3. The quantity of interest that were used to measure the oscillations was the relative frequency, ∆ω1,in system 1 (40). The relative frequency, ∆ω2, in system 2 and the power transfer in the HVDC-link,PDC , could have been used as well but ∆ω1 was the signal where the oscillations were most visible.

To fit an exponential curve to the measured signal the peaks of the oscillation curve was found. Withthe known peaks the built in application in Matlab called Curve Fitting was used to fit an exponentialcurve to the peak values. The original signal and fitted curves are shown in Appendix E. The frequencyof the oscillations was found by measuring the period between each peak value and then take the averagefrequency for the oscillation. It was also tested to use Fourier analysis of the frequency but it wasfound that the dominant frequency from that analysis always overestimated the actual frequency of theoscillation that can be seen in the figures in Appendix E. The damping constant was extracted from the

Page 30

decay constant of the fitted curve and below in table 5 follows the values of the damping constant alongwith its mode frequency, f0, as the characteristic of the loads are varied.

Table 5: Resulting damping coefficients for active power loads. Each value of mp is divided in to eachsub-table.

(a) mp = 0

mp mq ζ f0 [Hz]

0 0 0.0402 0.68600 0.5 0.0398 0.68570 1 0.0391 0.68440 1.5 0.0390 0.68510 2 0.0386 0.6845

(b) mp = 0.5

mp mq ζ f0 [Hz]

0.5 0 0.0447 0.68340.5 0.5 0.0443 0.68350.5 1 0.0440 0.68270.5 1.5 0.0438 0.68260.5 2 0.0435 0.6823

(c) mp = 1

mp mq ζ f0 [Hz]

1 0 0.0481 0.67981 0.5 0.0479 0.68031 1 0.0477 0.67951 1.5 0.0474 0.67981 2 0.0472 0.6794

(d) mp = 1.5

mp mq ζ f0 [Hz]

1.5 0 0.0510 0.67631.5 0.5 0.0509 0.67621.5 1 0.0507 0.67631.5 1.5 0.0505 0.67521.5 2 0.0503 0.6758

(e) mp = 2

mp mq ζ f0 [Hz]

2 0 0.0536 0.67262 0.5 0.0534 0.67212 1 0.0532 0.67202 1.5 0.0531 0.67142 2 0.0529 0.6713

The values of the damping consonant in table 5 is visualised in Figure 20. Where each sub figure showshow the damping in the system varies for one value of mp compared with all values of mq. All the linesof mp in Figure 20 is showed together in Figure 21.

Page 31

0 0.5 1 1.5 2

mq

0.0386

0.039

0.0391

0.0398

0.0402

Dam

pin

g C

oeffic

ient,

mp = 0

(a) mp = 0

0 0.5 1 1.5 2

mq

0.0435

0.0438

0.044

0.0443

0.0447

Dam

pin

g C

oeffic

ient,

mp = 0.5

(b) mp = 0.5

0 0.5 1 1.5 2

mq

0.0472

0.0474

0.0477

0.0479

0.0481

Dam

pin

g C

oeffic

ient,

mp = 1

(c) mp = 1

0 0.5 1 1.5 2

mq

0.0503

0.0505

0.0507

0.0509

0.051

Dam

pin

g C

oeffic

ient,

mp = 1.5

(d) mp = 1.5

0 0.5 1 1.5 2

mq

0.0529

0.0531

0.0532

0.0534

0.0536

Dam

pin

g C

oeffic

ient,

mp = 2

(e) mp = 2

Figure 20: The change of damping can be seen as mq is changed when mp is constant. The data fromtable 5 is used. The data points is marked with a ring in each figure and the line is drawn between eachdata point to show the trend.

Page 32

0 0.5 1 1.5 2

mq

0.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052

0.054

Dam

pin

g C

oeffic

ient,

Change of Damping

Figure 21: The change of damping in the Perpendicular system. The characteristic constant mq ischanged for each value of mp.

Page 33

2 4 6 8 10 12 14

Time [s]

-10

-8

-6

-4

-2

0

2

4

6

8

10

P [M

W]

Power Transfer in HVDC-link

Figure 22: The power transfer in the HVDC-link when following a sinusoidal reference with an amplitudeof 10 MW.

4.3 HVDC-link Control - PSCAD

The HVDC-link in the PSCAD model has as mentioned in section 3.3.2 unknown dynamical properties.Hence it is required to test how it can follow a reference to ensure that the controls is fast enough to makethe assumption P2 = −P1. The AC system is not represented by the 2A4G-system. The AC-systems ateach side of the HVDC-link is instead represented by three phase voltage sources. This will make thesimulation run faster and it is just interesting to see the speed of the HVDC-link and not to study thedynamics for inter-area oscillations in this test instance.

The test will be the following. The active power reference for terminal connecting to system 2 needs tofollow a sinusoidal reference at 0.7 Hz. The frequency was chosen because as can be seen from previoussection 4.2 the inter-area oscillation is about 0.7 Hz. The magnitude of the sinusoidal is for one test 10MW and for another 200 MW. 10 MW is chosen because it is similar to the largest peaks of the PODthat was observed in the Simulink set-up. However 10 MW power transfer is not what this HVDC-linkis rated for hence there will be a lots of noise in the measured signal. When the amplitude is 200 MWthen the noise is reduced due to the fact it is a power level closer to its rated operating point at 900 MWand there will not be needed to apply a filter to the signal. First the system is having a reference of 0MW in order for the system to start-up. After 4 s the reference is changed to the sinusoidal reference.

The first simulation was with a reference amplitude of 10 MW. The reference, P2,ref , is for the powerinjected to the HVDC-link from system 2, P2. The power ejected at the opposite end of the HVDC-link,to system 1 is P1. The result is shown in Figure 22.

The same test was now performed with a reference amplitude of 200 MW. The results is shown in Figure23.

It can be seen from both Figure 22 and 23 that P2 is not able to follow the reference, it is measured tohave a static error of about 25-30 % in both figures. The signal power, P2, lags behind the reference,P2,ref . It can also be seen in the figures that the relation P2 = −P1.

Page 34

2 4 6 8 10 12 14

Time [s]

-200

-150

-100

-50

0

50

100

150

200

P [M

W]


Figure 23: he power transfer in the HVDC-link when following a sinusoidal reference with an amplitudeof 200 MW.

The static error is believed to be due to a badly tuned gain in the controller. A third simulations isperformed where the same sinusoidal reference signal with an amplitude of 10 MW is amplified with again of 1.3 in order to see if this will improve the reference tracking of 10 MW. The results is seen inFigure 24.

4.4 Perpendicular System - PSCAD

It is desired to compare the results from the simulations performed in Simulink with the ones performedin PSCAD. Therefore the same test is applied to the PSCAD model as to the Simulink model in section4.2. A disturbance is initiated in system 1. The disturbance is a load increase at bus 5 in system 1 with180 MW. The time step in PSCAD simulations is fixed and set to Ts = 100µs. Due to the fact that thesimulations in PSCAD was very time consuming only five simulations of different load characteristics wastested. This was to verify the results from Simulink. The five different characteristics is showed in table6.

Table 6: Load Characteristics Tested in PSCAD.

mp mq

0 01 01 11 22 2

The HVDC-link in PSCAD requires power in order to keep the voltage level in the HVDC-link. Thereforewill the HVDC-terminal that controls the voltage level in the HVDC-link draw power from the AC-systemit is connected to which in this chase is system 1. It was measured that the HVDC-link draws 7 MWof active power and 1.5 MVAr of reactive power. Thus the load flows for the 2A4G-system is no longer

Page 35

2 4 6 8 10 12 14

Time [s]

-15

-10

-5

0

5

10

15

P [M

W]


Figure 24: The power transfer in the HVDC-link while following a reference where the reference gain isamplified 130 % to improve the reference tracking of 10 MW.

valid for system 1 and new ones were performed which resulted in table 7.

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