OPTIMIZATION OF MULTILINK DC TRANSMISSION FOR … · maximization) in a MTDC (Multi Terminal HVDC) transmission system and consists of eight chapters. The first chapter presents a

DEPARTAMENT OF ENERGY TECHNOLOGY

PONTOPPIDANSTRᴁDE 101

OPTIMIZATION OF MULTILINK DC TRANSMISSION FOR SUPERGRID FUTURE CONCEPTS

MASTER THESIS

Title: Optimization of Multilink DC Transmission for Supergrid Future Concepts

Semester: 9th and 10th

Semester theme: Master Thesis

Project period: 1st September 2011 – 31st May 2012

ECTS: 60

Supervisors: Remus Teodorescu, Rodrigo da Silva, Bogdan Ionut Craciun, Sanjay Chaudhary

Project group: PED/WPS 1046

_____________________________________

Tomasz Rybarski

_____________________________________

Bogdan Sfurtoc

Copies: [4]

Pages, total: [111]

Appendix: [6]

Supplements: [4 CDs]

By signing this document, each member of the group confirms that all group members have participated in

the project work, and thereby all members are collectively liable for the contents of the report. Furthermore,

all group members confirm that the report does not include plagiarism.

SYNOPSIS:

Rapid increase of renewable energy generation

sources located in remote and distant areas require

a new approach for energy transmission.

Power transportation over long distances using

traditional HVAC systems faces severe technical and

economical challenges. Thus, the concept of multi

terminal VSC based HVDC systems is introduced to

address those issues and interface remote electricity

generation units into the onshore grids, laying the

foundations for proposed future DC supergrids.

Additional converter stations sharing a common DC

link require novel control strategies for achieving

desired active power sharing between the stations.

This Master thesis provides analysis of the converter

control strategies in MTDC operation.

Furthermore, methods of engineering optimization

are introduced in order to minimize power losses in

the systems operation.

The system is modeled in PSCAD environment linked

with MATLAB for implementing the optimization

algorithm. A laboratory validation of the simulations

on a scaled down 15 kW platform is also provided.

III

Preface

The present Master Thesis was written at the Department of Energy Technology by WPS/PED Group 1046

throughout semesters 9 and 10. It is entitled ‘Optimization of Multilink DC Transmission for Supergrid Future Concepts’.

Reading Instructions The references are shown in form of numbers put in square brackets. Detailed information about literature is

presented in the References. While the format for figures and tables is (X.Y), for equations is (X-Y), where X is the number of chapter and Y is the number of equation/figure/table.

In the thesis, the chapters are consecutive numbered and the appendixes are labeled with letters. The enclosed CD-ROM contains the thesis report written in Microsoft Word, Adobe PDF format, documents used throughout the thesis, the Matlab – PSCAD MTDC optimization programs and the Matlab GUI MTDC optimization interface.

Acknowledgement The authors would like to express their gratitude to their supervisors for the support and suggestions offered

during the development of the thesis.

IV

Summary

The thesis covers the aspect of control and optimization (i.e.: power loss minimization and revenue

maximization) in a MTDC (Multi Terminal HVDC) transmission system and consists of eight chapters.

The first chapter presents a short introduction to the technical and economical challenges of the modern

power system transformation, its increased dependence on power electronics and highlights the background

behind the growth of HVDC installations worldwide. The following subchapters contain the problem

formulation, objectives and limitations of the project, followed by the outline containing the thesis structure

and contents.

The second chapter describes the state of the art in HVDC transmission. A brief introduction and overview of

the technology is presented followed by a comparison with traditional AC transmission. Principles of Voltage

Source Converter based HVDC transmission is described along with multilevel converter topologies. A

comparison between classical two terminal HVDC systems and HVDC MTDC is presented in the final

subchapter.

The third chapter focuses on modeling of the components of the MTDC system. Its main parts are described

and an equivalent model that offers an approximation of the real component’s behavior is presented.

The fourth chapter presents the control structure of the MTDC system. An overall introduction to the control

topology and objectives is presented, followed by a detailed description of the grid connected converters

control loops and synchronization methods. The wind farm converter control structure is also discussed.

Finally, the control structure in MTDC operation is presented by introducing the droop control principle.

The fifth chapter provides a thorough analysis of the three terminal MTDC system’s optimization routine.

Explanations regarding the different cost functions and constraints used are stated. Furthermore, the chapter

contains or sends reference to the calculus backing up the Matlab optimization routine results. Additionally,

the function that allows converter losses to be taken in consideration and the economical model on which the

economical optimization routine is based, are described.

Finally, in order to prove the optimization routine’s versatility, the analysis is extended to a four terminal MTDC

system.

The purpose of the sixth chapter is to build a series of scenarios that not only confirm that the system

functions properly but also prove its worthiness. The advantages of the optimization routine will be highlighted

in different situations and conclusions will be drawn.

The seventh chapter provides an experimental verification of the proposed algorithms on a laboratory

platform.

As conclusion, it can be stated that the proposed optimization algorithm has been tested and validated on

the laboratory platform and the overall project objectives were accomplished.

V

Table of Contents

1. INTRODUCTION ......................................................................................................................................................... 1

1.1 BACKGROUND .............................................................................................................................................................. 1

1.2 PROBLEM FORMULATION ................................................................................................................................................ 6

1.3 OBJECTIVES .................................................................................................................................................................. 7

1.4 LIMITATIONS ................................................................................................................................................................ 7

1.5 THESIS OUTLINE ............................................................................................................................................................ 8

2. STATE OF THE ART ..................................................................................................................................................... 9

2.1 INTRODUCTION TO HVDC ............................................................................................................................................... 9

2.1.1 Comparison of HVDC and HVAC transmission ..................................................................................................... 9

2.1.2 Applications of HVDC ......................................................................................................................................... 11

2.2 HVDC TECHNOLOGIES ................................................................................................................................................. 12

2.2.1 Voltage Source Converter HVDC ........................................................................................................................ 13

2.2.2 Converter topologies ......................................................................................................................................... 17

2.3 MTDC ADVANTAGES AND DISADVANTAGES ..................................................................................................................... 19

3. MTDC MODELING .................................................................................................................................................... 21

3.1 OVERVIEW OF THE MTDC SYSTEM ................................................................................................................................. 21

3.2 THE WIND POWER PLANT ............................................................................................................................................. 22

3.3 THE VOLTAGE SOURCE CONVERTER ................................................................................................................................ 23

3.4 THE FILTER ................................................................................................................................................................. 24

3.5 THE DC CABLE ............................................................................................................................................................ 25

3.6 THE DC CAPACITOR ..................................................................................................................................................... 26

3.7 THE GRID .................................................................................................................................................................. 27

3.8 THE TRANSFORMER...................................................................................................................................................... 28

4. THE MTDC CONTROL SYSTEM .................................................................................................................................. 29

4.1 INTRODUCTION AND OBJECTIVES OF CONTROL ................................................................................................................... 29

4.2 CONTROL OF WIND FARM AND GRID SIDE CONVERTERS ..................................................................................................... 30

4.2.1 Current Control Loop ......................................................................................................................................... 32

4.2.2 DC Voltage Control Loop .................................................................................................................................... 35

4.2.3 Active power control loop ................................................................................................................................. 38

4.2.4 AC voltage control loop...................................................................................................................................... 39

4.2.5 Phase Locked Loop (PLL) .................................................................................................................................... 41

4.3 CONTROLLING THE MTDC OPERATION ............................................................................................................................ 44

5. MTDC OPTIMIZATION ............................................................................................................................................. 47

5.1 INTRODUCTION TO THE OPTIMIZATION ALGORITHM ............................................................................................................ 47

5.2 THE MTDC SIMULATION MODEL .................................................................................................................................... 48

5.3 LOSS MINIMIZATION OF THE THREE TERMINAL MTDC SYSTEM, TAKING IN CONSIDERATION ONLY THE LINE LOSSES .......................... 49

VI

5.4 THE CONVERTER STATION’S EQUIVALENT LOSS MODEL ........................................................................................................ 54

5.5 LOSS MINIMIZATION OF THREE TERMINAL MTDC SYSTEM, TAKING IN CONSIDERATION THE LINE AND CONVERTER LOSSES ................ 61

5.6 LOSS MINIMIZATION OF THREE TERMINAL MTDC TAKING IN CONSIDERATION ECONOMICAL ASPECTS ........................................... 65

5.7 MODEL VALIDATION. EXTENDING THE ALGORITHM ON A FOUR TERMINAL MTDC SYSTEM ......................................................... 70

5.8 THE MTDC LOSS/ECONOMICAL OPTIMIZATION GUI ......................................................................................................... 75

6. STUDY CASES ........................................................................................................................................................... 77

6.1 ANALYSIS OF THE OPTIMUM POWER DISTRIBUTION IN A MTDC SYSTEM DURING 24H IN ORDER TO MAXIMIZE THE REVENUE FROM TWO

MARKETS 77

6.2 ANALYSIS OF THE INFLUENCE OF CABLE LENGTH IN DETERMINING THE OPTIMUM OPERATING VOLTAGES IN A THREE TERMINAL MTDC

SYSTEM 80

6.3 ANALYSIS OF OPTIMUM POWER SHARING IN A FOUR TERMINAL SYSTEM AFTER WIND FARM TRIP................................................. 83

7. LABORATORY WORK ............................................................................................................................................... 85

7.1 STUDY CASE 1 – POWER SHARING THROUGH SHARING FACTOR ADJUSTMENT .......................................................................... 87

7.2 STUDY CASE 2 – OPTIMIZED POWER SHARING ................................................................................................................... 93

8. CONCLUSIONS AND FUTURE WORK ........................................................................................................................ 97

8.1 CONCLUSIONS ............................................................................................................................................................ 97

8.2 FUTURE WORK ........................................................................................................................................................... 98

WORKS CITED ................................................................................................................................................................ 101

APPENDIX A – ANALYTICAL APPROACH TO THE CONVERTER AND LINE LOSS FUNCTION MINIMIZATION ...................... 105

APPENDIX B – LABORATORY SETUP PARAMETERS......................................................................................................... 109

APPENDIX C – THE MTDC OPTIMIZATION GUI ............................................................................................................... 111

VII

Nomenclature

List of abbreviations

AC Alternative Current CSC Current Source Converter DC Direct Current EMC Electromagnetic Compatibility EMI Electromagnetic Interference FACTS Flexible Alternative Current Transmission System HVAC High Voltage Alternative Current HVDC High Voltage Direct Current IGBT Insulated Gate Bipolar Transistors IGCT Insulated Gate Commutated Thyristor KCL Kirchoff Current Law KVL Kirchoff Voltage Law LCC Line Commutated Converter M2C Modular Multilevel Converter MTDC Multi Terminal Direct Current NPC Neutral Point Clamped PCC Point of Common Coupling PI Proportional Integrator PLL Phase Locked Loop PMSG Permanent Magnet Synchronous Generator PWM Pulse Width Modulation RMS Root Mean Square SQP Sequential Quadratic Programming TSO Transmission System Operator VSC Voltage Source Converter WF Wind Farm WPP Wind Power Plant WT Wind Turbine WTG Wind Turbine Generator

1

1. Introduction

This chapter presents a short introduction to the technical and economical challenges of the modern

power system transformation, its increased dependence on power electronics and highlights the

background behind the growth of HVDC installations worldwide. The following subchapters contain the

problem formulation, objectives and limitations of the project, followed by the outline containing the

thesis structure and contents.

.

1.1 Background

Electric power networks are facing a period of rapid modernization in the upcoming future as an effect

of different political, technological and sociological reasons [1]. Since more than a decade, political

change of mind has led to the liberalization of European electricity markets introducing free market rules

and competition between energy suppliers. Also, an increased interest in renewable energy sources with

many large projects planned for the near future is contributing to the transformation of the modern

power system.

According to the electricity generation projections for the future presented in Figure 1.1, the wind

power industry will encounter the most rapid growth, covering a large portion of the energy market in

the upcoming years.

Figure 1.1 - Renewable energy growth by 2020 [1]

Renewable sources of energy such as wind power, solar power (photovoltaic, concentrated, thermal),

various forms of marine energy (hydro-electric, tidal) and biomass are all alternatives to fossil fuel based

2

electricity generation. Among the main advantages of using those sources is reducing the dependence on

volatile fossil fuel markets (arab oil embargo following the Yom Kippur war of 1973 boosted oil prices and

disrupted supply), diversifying and decentralizing energy sources and reducing greenhouse gases

emissions. Future trends in the energy sector present a steady growth in the amount of power produced

from renewable sources. The European Commission’s directive on renewable energy sets a target for all

of the EU member states to reach 20% share of energy from renewable sources by 2020.

Forecasts predict a growth of installed wind capacity worldwide from current level of 197 GW up to 459

GW by the end of 2015, as presented in Figure 1.2.

Figure 1.2 - Global wind power forecast 2010-2015 [2]

A steady increase in the annual growth rate of total installed capacity can be observed during the next

five years and will average at 18.4%. [2]

The biggest contribution to such rapid development of worldwide capacity is made by Asia, primarily by

China, which became the fastest growing market in the world. China currently holds the leading position

in terms of annual growth and cumulative installed capacity and this trend is going to continue with

expected annual additions of over 20 GW by 2015. [2]

Due to advances in engineering disciplines including power electronics, materials science and thanks to

economic advantages, the scale (hub height and rotor diameter) and output power of single wind turbine

units continuously grows.

3

Figure 1.3 - Growth in size of commercial wind turbine designs [3]

The necessity of integrating growing amounts of wind power has led to installing groups of single

turbines, concentrated in locations of favorable wind conditions (either onshore or offshore) forming

wind farms.

According to Figure 1.1, the future trend in wind energy is moving generation to remote areas offshore.

Offshore wind farms present numerous advantages over onshore installations. Offshore sites have

favorable wind conditions with higher annual wind speeds and fewer fluctuations due to lack of obstacles

which affect the wind flow. Thus, larger energy potential is available at sea with possibilities to extract

maximum energy from an offshore wind farm, leading to increased utilization of the installed generation

capacity in comparison to onshore sites.

Offshore areas are not limited by space as much as in the case of onshore, allowing for bigger wind

farm installations to be built. The amount of available onshore sites which guarantee technical and

economical feasibility of building large wind farms is decreasing, thus new locations must be found

offshore. Environmental and aesthetic influence is also reduced since noise emissions and visual impact

on the landscape are no longer a subject of concern.

Reliability issues are a subject of bigger concern in offshore installations since the turbines are not as

easily accessible. Service vessels and rigs are required for access and service, which significantly boost

maintenance costs. Installation costs are also higher as offshore sites require additional foundation and

4

grid connection costs. Overall, offshore wind power presents economic challenges greater than onshore

systems.

A forecast for the amount of wind energy produced from offshore and onshore farms conducted by the

European Wind Energy Association (EWEA) is presented in Figure 1.4

Figure 1.4 - Wind power production in the EU (2000-2020) [4]

According to EWEA report, in a normal wind year, wind energy will produce 179 TWh of energy in 2010,

335 TWh in 2015 and 582 TWh by 2020. Furthermore, offshore wind energy share of total EU wind power

production will increase from 3.9% in 2008 to over 25% in 2020 [4]. Based on the presented data, one

may conclude that electricity generation from remote and distant locations which are abundant in energy

resources and delivering it over long distances to places of consumption is a practical idea for the future

power grid. This idea lies behind the concept of the supergrid – a wide area electrical transmission

network that allows the exchange of large amounts of power over great distances.

In order for those ambitious plans to become a reality, it is necessary to find a technological solution

for transferring such vast amounts of power from the point of generation to the point of consumption,

located even hundred kilometers away. The technology of choice is the High Voltage Direct Current

(HVDC) transmission system which offers numerous technical advantages as well as cost competitiveness

in comparison to standard AC networks. HVDC technology is becoming increasingly popular in long

distance bulk power transmission such as underwater power links between countries and various

locations throughout the world.

5

The main advantages of HVDC are minimization of transmission losses, lack of necessity for reactive

power compensation and easier obtainable permissions regarding the right of way, all of which are of

crucial importance for long distance transmission. Figure 1.5 presents existing HVDC links in the area of

Northern Europe.

Figure 1.5 - HVDC interconnections within Nordel Transmission System Operator [5]

6

1.2 Problem Formulation

The aforementioned HVDC is a particular case of MTDC system. The reason behind a multi terminal

VSC-HVDC transmission is basically the same as in the HVDC case, to send electrical energy from remote

power plants (e.g. offshore WPP) to grid connection points. Its main advantage over the conventional

HVDC is the freedom of energy exchange between more than two partners [6]. The transmission has to

be done as efficiently and economically as possible. The basic principle is displayed in Figure 1.6.

WPP

Offshore Onshore

VSC

VSC

VSC

Grid

Grid

DC Side AC Side

Figure 1.6 - VSC based MTDC

It can be observed that the system in question has a number of 3 terminals. It consists of two receiving

VSC converters transforming DC to AC, sharing the same DC bus. The bus is carrying energy from at least

one offshore wind power plant. The WPP is linked to the HVDC cable via another VSC with active rectifier

functions.

Several advantages the MTDC system has over the two-terminal HVDC are highlighted:

- Control Flexibility – ability to share the power from the generators to the consumers

- Reliability – the MTDC system is less affected by a fault than the conventional HVDC

- Economy – the cost of a MTDC system is less than in the case of two equivalent HVDC

configuration

Several other benefits regarding the MTDC system are listed below:

- AC network interconnection over a long distance

- A large amount of power can be transmitted

- The possibility to adapt quickly to varying network conditions [6]

As mentioned above, the idea behind a MTDC system is to control the power flow between two or

more consumers who divide the same DC link. The power flow can be modified, based on certain criteria.

7

The criteria can be of economical nature, such as maximizing the profit on the entire system or it can

include ecological aspects, for example minimizing the losses in the system.

1.3 Objectives

The thesis objectives are stated below:

Build a three terminal MTDC system (1 WF terminal and 2 grid side terminals) in PSCAD using VSC

average models. The system should have power sharing capabilities.

Build an optimization routine in Matlab that can minimize the overall losses on the DC side

(converter losses included) of the MTDC system or that can maximize the profit gained by selling

the electricity on the two markets corresponding to the two grid side terminals

Create a program that links the PSCAD model to the optimization routine; the program should

run in quasi-real time

Extend the algorithm for a 4 terminal MTDC (2 wind farms and 2 grids)

Build an equivalent small scale 3 terminal MTDC system in the laboratory. Implement the

optimization algorithm and test its sturdiness.

1.4 Limitations

The limitations encountered during the development of the thesis are summarized below:

The system is simulated as an average model without the wind farm converter and its control

structure. However, a more detailed switched model including the offshore converter station has

also been built and is included on the CD.

The PSCAD simulation model of the MTDC is limited to maximum four terminals (i.e. two wind

farms and two AC grids).

The HVDC cable is modeled purely as a resistance. Effect of the temperature fluctuations on the

resistance is not considered.

Only the DC cable and converter valve losses are taken into account in the optimization function.

The laboratory setup is scaled down to 15 kW.

A programmable DC source is used in the laboratory to simulate the wind farm input.

The power is directly distributed among the grid side converters, without the use of a wind farm

converter.

8

1.5 Thesis Outline

The thesis covers the aspect of control and optimization (i.e.: power loss minimization and revenue

maximization) in a multi-terminal DC transmission system and consists of eight chapters. The structure of

the project is described in the following rows.

The first chapter presents an introduction to the topic of HVDC transmission systems. A short

background, problem formulation and objectives of the project are stated. The limitations encountered

throughout the work on the thesis are presented.

The second chapter describes the state of the art of the technology. An overview of the MTDC system is

given along with comparison to traditional high voltage AC transmission and examples of applications.

The recent development in HVDC associated with the use of VSC and its different topologies is also

presented.

The third chapter deals with the system modeling. A model of each component in the system is

presented.

Fourth chapter presents the MTDC system control structure. A brief description of the overall control

system and its objectives followed by a more detailed analysis of the control loops present in the

offshore and onshore converter stations is presented.

The fifth chapter is devoted to the optimization algorithm. The theory behind the optimization routine

is elaborated and the theoretical results are confirmed by simulations in order to validate the

optimization algorithm.

The sixth chapter presents the simulations performed on several study cases while the seventh chapter

focuses on the laboratory work.

The last chapter contains the conclusions and future work.

9

2. State of the Art

This chapter describes the state of the art in HVDC transmission. A brief introduction and overview of

the technology is presented followed by a comparison with traditional AC transmission. Principles of

Voltage Source Converter based HVDC transmission is described along with multilevel converter

topologies. A comparison between classical two terminal HVDC systems and HVDC MTDC is presented in

the final subchapter.

2.1 Introduction to HVDC

HVDC is a power transmission technology recognized as being advantageous in long distance energy

transfer applications from remote resources (hydroelectric plants, offshore wind farms) and

asynchronous grid interconnections (e.g. between two AC grids of different frequency). The first

implementation of HVDC was an undersea link between the Swedish island of Gotland and mainland in

1954. Line commutated converter stations utilizing high power thyristor valves were built for this

purpose [7].

Throughout the years, due to rapid advances in power electronics and the development of new power

semiconductor switches, there is a renewed interest in this mature technology worldwide. Deregulated

generation markets, resulting in regional differences of generation costs, lead to plans of applying the

technology in a non-traditional way. New transistor based converter designs and ongoing development

are contributing to the recent growth of HVDC transmission.

The amount of HVDC projects worldwide is increasing with new installations being built in North

America, Europe and China. HVDC will also play a key role in forming a Pan-European transmission

network for integrating large amounts of offshore wind power into the existing grid, a concept known as

the supergrid [8].

2.1.1 Comparison of HVDC and HVAC transmission

HVDC technology offers economical as well as technical benefits for long distance bulk power

transmission in comparison to traditional AC systems. Those advantages along with a brief comparison of

the two technologies are presented below.

Higher power transfer is possible over longer distances using fewer lines than in the case of an AC

transmission network. A typical HVDC system in bipolar configuration requires two core cables (one for

plus and one for minus polarity) instead of three core for the same amount of power transmitted. This

10

fact implies less right of way required for overhead lines since smaller strips of land are needed for

corridors. In long distance transmission this greatly reduces investment costs. Conductor and

infrastructure costs are also reduced due to lower utilization of copper and lighter cables (simpler towers

in case of overhead transmission). However, the high costs of the converter stations incapacitate the use

of HVDC systems for short distance power transfer [7].

For the purpose of economical analysis, the so called ”break-even” distance is introduced. The use of

HVDC transmission becomes economically justified over AC when the length of the transmission system

exceeds the break-even distance and the lower cost of the lines and right of way compensate for the

initial converter station cost.

The break-even distance and total cost in function of line length are presented in Figure 2.1

Investment Cost

Break-even

Distance

DC Line Cost

DC Terminal

Cost

AC Line Cost

AC Terminal

Cost

Distance

Total DC Cost

Total AC Cost

Figure 2.1 - Cost comparison of HVDC and HVAC systems [9]

HVDC transfer eliminates the skin effect, a specific characteristic of AC based transmission greatly

contributing to power losses in conductors [10]. Skin effect causes the alternating current density

distribution within a conductor to be largest near the surface of the conductor, decreasing towards the

center. Consequently, the current flows only through the conductor’s outer layer, leading to

underutilization of its cross section and increasing the effective resistance. Since DC transmission utilizes

the whole cross section for current flow, a smaller conductor may be used for the same amount of power

transferred. Reduced effective resistance leads to lower transmission losses.

HVDC transmission has also fewer requirements towards the line/cable insulation. Insulating AC

transmission lines is more demanding as the design has to take into consideration the peak voltage level

11

which is √2 times higher than the RMS voltage. DC operates at a constant maximum voltage allowing

better utilization of line insulation and conductor spacing. Consequently, a DC line has the capability of

transferring more power per conductor for the same insulation requirements regarding peak voltage.

Another important factor for long distance transmission is lack of generation or absorption of reactive

power in an HVDC line. A long AC line requires reactive power compensation in the form of shunt

inductors connected at regular intervals along the transmission line. This is to compensate for the lines

capacitance formed between each phase and earth, appearing in parallel with the load. This capacitance

requires additional reactive current flowing in the line to charge it and, in effect, reducing the lines power

transfer capability and generating additional losses.

HVDC may also be used for power system stability benefits. In a case of establishing an interconnection

between two asynchronous networks, the HVDC link acts as a buffer between the two systems and

prevents cascading outages propagating from one network to another. Power flow control (direction and

magnitude) allows for supporting the AC networks at both sides of the DC link.

2.1.2 Applications of HVDC

HVDC technology is used in the following fields of power transmission:

Connection of remote generation plants / long distance bulk power transmission Transmission of energy from distant generation plants (hydroelectric, offshore wind farms)

located far from the load requires the use of HVDC as an economical alternative to AC transfer.

Examples of such installations are: 800 km HVDC link from the Itaipu Hydroelectric Power Plant to

Sao Paulo in Brazil (2x 600 kV bipolar system, 3150 MW each), 1000 km link from the Three

Gorges Hydro power plant in China (total capacity of 7200 MW).

Long submarine cable transmissions HVDC provides a cost effective way of establishing long distance submarine connections; the

main reasons are savings in cable cost (due to smaller insulation requirements and using only two

core cables) and lack of capacitive charging current and reactive power compensation

requirements.

Asynchronous connection of AC power grids Connection between two asynchronous AC networks is possible using back-to-back connected

converters. Managing the two separated systems in terms of power flow control is possible with

the use of VSC based HVDC.

12

Summarizing, HVDC is an economically attractive technology for transmission of bulk power over long

distances.

2.2 HVDC Technologies

Modern HVDC systems use two basic converter technologies – the classical Line Commutated

Converters (LCC) utilizing thyristor valves and self-commutated, Voltage Source Converters (VSC HVDC)

based on controllable transistors (e.g. Insulated Gate Bipolar Transisitors – IGBT). Both topologies, the

LCC and VSC, are presented in Figure 2.2 and Figure 2.3 respectively [11].

AC 1 AC 2

Reactive power

Active power

Reactive power

Figure 2.2 – LCC based HVDC technology [11]

AC 1 AC 2

Reactive power

Active power

Reactive power

Figure 2.3 – VSC based HVDC technology [11]

The VSC based technology operates at a higher switching frequency than the line frequency and is self

commutated by a gate pulse. Pulse Width Modulation (PWM) is used for creating the desired output

voltage waveforms. Due to high frequency operation of the converter it is important to address

electromagnetic compatibility/interference (EMC/EMI) issues by installing additional filters. Switching

13

losses are also present and proportional to the PWM operation. Dealing with them is a major challenge in

high power applications of VSC HVDC.

Important advantages of HVDC transmission based on VSC is the ability to rapidly control both active

and reactive power independently of each other (reactive power control in both directions,

independently of the real power flow). Self commutation of the converter provides so called black start

capability, i.e. the ability to produce balanced three-phase voltages without relying on external electric

power transmission network.

Line commutated converters are externally commutated and require a grid that specifies the voltage

and supplies reactive power. The principle of operation is based on firing the thyristors with a specified

time delay (called the firing angle α), which introduces controllability over natural point of ignition

defined by the grid. The operating state of the converter (rectifier or inverter mode) is dependent upon

this angle. In rectifier operating mode (0 ≤ α ≤ 90 degrees) the current flow is delayed with respect to the

grid voltage by the angle α, as a consequence the converter consumes reactive power. During inverter

operation (α > 90 degrees) the presence of DC supply and an external AC source is required to provide

commutating voltage. LCC converter technology is suitable for high power applications due to higher

power handling capabilities of thyristors over IGBTs. The power range of LCC type installations is within

GW range with the largest project being the Itaipu system in Brazil, operating at 6300 MW level. VSC–

HVDC is the technology of choice for medium power levels, within the range of 300-400 MW [11].

2.2.1 Voltage Source Converter HVDC

The main advantage of VSC-HVDC over the thyristor based technology is the improved controllability

of the system allowing for independent control of active and reactive power. The reactive power can be

controlled at both terminals independent of the DC link voltage level [12].

Bidirectional power flow is possible for both the active and reactive power.

Other advantages include the following:

Dynamic support of AC voltage at the converter bus for improving voltage stability. [12]

Commutation failures due to grid disturbances (e.g. voltage drops, phase angle variation) are eliminated.

Possibility to serve isolated loads, i.e. a network where no generation source is available (VSC can operate independently to any AC source) due to self commutation and black start capability [12]

It is possible to reverse power flow without changing DC voltage polarity (advantageous in multi-terminal HVDC systems)

The VSC based HVDC transmission system topology is presented in Figure 2.4.

14

Figure 2.4 - VSC-HVDC topology [12]

The main components are two converters (a rectifier and an inverter) connected in a back to back

topology by a DC link, high frequency filters, DC link capacitors and HVDC cables.

The converters basic control strategy is high frequency PWM switching which leads to generation of

harmonics in the converter output at frequencies equal to fundamental switching frequency or its

multiples. An AC high frequency filter is required to eliminate harmonic distortion of the output

waveforms. Each converter phase leg is connected through a phase reactor (denoted as X in Figure 2.4) to

the AC system.

The common feature of all VSC based systems is the generation of a fundamental frequency AC output

voltage from a DC voltage. By controlling this voltage, both in phase and magnitude, different converter

operating modes may be obtained. The cooperation principles between a VSC and a grid are based on

controlling the phase angle δ and converter side voltage V2 by varying the PWM modulation depth.

By means of a VSC the power may be controlled to flow in either direction by setting the phase angle of

the converter AC output voltage positive or negative with respect to the AC grid voltage. Active power is

controlled by changing the phase angle of the converter AC voltage V2 with respect to the AC grid bus

voltage V1, and reactive power is controlled by changing the magnitude of the fundamental component

of the converter side voltage with respect to grid voltage. By controlling these two parameters of the

converter voltage the VSC can operate at any power factor (referred to as four quadrant operation).

Active and reactive power exchange between the converter and grid can be defined by the following

equations:

P = V1 ∙ V2 ∙sinδ

X

2-1

Q =V1

X∙ (V2 ∙ cosδ − V1)

2-2

15

Inverter operation of the VSC is analyzed below. The system from Figure 2.4 may be simplified as

follows (power flow is from the converter to the grid, thus converter voltage is represented as the

sending end, while the grid voltage – the receiving end).

Receiving endSending end

X

V2 V1

I

ΔV

Figure 2.5 - VSC and grid interconnected through phase reactor [11]

The phasor diagram of the VSC operating in inverter mode at unity power factor (injecting active power

into the grid) is presented in Figure 2.6.

ReV1

ΔVV2

δ

Im

I

Figure 2.6 - Inverter operation of VSC at unity power factor

We can easily observe that due to the proper phase shift (converter voltage leading grid voltage), the

converter acts in inverter mode and active power is generated (injected) into the grid. The rectifier

operation is presented in Figure 2.7:

16

Re

V1

ΔV

V2

δ

Im

I

Figure 2.7 - Rectifier operation of VSC at unity power factor

In case the converter voltage V2 is higher than grid voltage V1, the converter behaves as a capacitor

(voltage lags the current by 90 degrees) and reactive power flow is from the converter to grid (converter

generates reactive power). In case the converter voltage V2 is lower than grid voltage V1, the converter

behaves as an inductor (voltage leads the current by 90 degrees) and reactive power flows from the grid

to the converter (converter absorbs reactive power). Both described variants (reactive power generation

and absorption) are presented in phasor diagram form in Figure 2.8.

IV2

V1 ΔV

I

V2

V1 ΔV

Figure 2.8 - VSC reactive power control variants

As presented above, the converter can act as a rectifier or an inverter, with leading or lagging reactive

power. This allows four quadrant operation of the converter, i.e. the VSC can be controlled to operate at

any point within the limits presented in Figure 2.9. The limits are settled by the maximum current that

can flow through an IGBT, the maximum DC voltage level and the maximum DC current through a cable.

[13]

The maximum current determines (along with the actual AC RMS voltage) the maximum apparent

power the VSC can operate at. It can be seen in Figure 2.9 that this limitation draws a circle around the

17

origin of the axes. The circle can vary its radius according to the maximum MVA. The maximum DC

voltage can also set limits to the operation circle. As mentioned above, the reactive power is dependent

on the difference between the grid voltage and the voltage produced by the VSC. Due to the fact that the

VSC voltage depends on the DC link, limits are introduced on the Q axis. The limits vary along with the

variation of the grid voltage. If the latter is high, the voltage difference between the VSC and the grid will

decrease. Hence, the reactive power generation potential decreases. The last limit is represented by the

DC cable current limit. This is set by the cable’s properties.

Q inductive

P

DC Power Limit

DC Voltage Limit

Current Limit

DC Power Limit

Inverter OperationRectifier Operation

Q capacitive

Figure 2.9 - Four quadrant operation of VSC [13]

2.2.2 Converter topologies

Different VSC topologies suitable for implementation in HVDC systems have been developed

throughout the years. A brief description of those topologies is presented in this chapter.

The basic classification of a three phase VSCs can be made into two-level and multilevel topologies. The

two-level, three phase converter topology represents the simplest configuration for HVDC transmission

and is presented in Figure 2.10.

The two-level converter consists of six transistors (two per phase) with anti-parallel diodes and is able

to provide two output voltage levels VA0, VB0, VC0 (output voltage of each phase referred to supply mid-

point) equal to VDC/2 and –VDC/2.

18

VDC/2

VDC/2

oA

BC

Figure 2.10 - Two level VSC topology

Multilevel converter topologies extend the applications of the classical two-level topology to the high

power range suitable for large HVDC converters. There are numerous multi-level topologies developed,

out of which the most distinct for HVDC applications are the 3-level NPC (Neutral Point Clamped)

converter and the modular multilevel converter (M2C). The topology of a 3-level NPC converter is

presented in Figure 2.11. The phase output voltages may be switched between three different levels

(VDC/2, 0, -VDC/2).

VDC/2

o

N

A B C

VDC/2

Figure 2.11 - 3-level NPC converter topology. [14]

19

The modular multilevel converter has the advantage of easily scaling the voltage output levels by

adding or removing sub-modules. Each sub-module consists of a capacitor connected across two IGBT

switches with anti-parallel diodes. By modifying the amount of sub-modules in each converter leg,

different output voltage levels may be obtained. One phase of a 17-level modular converter is presented

in Figure 2.12.

VDC/2

VDC/2

o

1

31

32

L

LA

16

2

17

Figure 2.12 - One phase of a 17-level modular converter [15]

The modular converter with 16 sub-modules per arm may produce 17 output voltage levels. The

voltage across each sub-module capacitor is equal to 1/16 VDC and the voltage stress across each switch is

limited to one capacitor voltage [15]. Multilevel converters offer advantages over classical 2-level

configurations, such as improved efficiency, lower voltage and current stresses on the transistors

allowing high power applications.

2.3 MTDC Advantages and Disadvantages

MTDC systems have been implemented in industry for a period of approximately 10 years [16]. They

allow the connection of one or more energy sources, situated in remote areas, to one or more grids.

Examples of such installations worldwide are enumerated in the following rows. It is worth mentioning

that the setups started as conventional two terminal systems [17]:

- The connection Sardinia – Corsica – Italy - The Pacific Intertie in the US

20

- The connection Hydro Quebec – New England Hydro from Canada to US

It can be seen that the MTDC is used not only for offshore wind farm applications but for other energy

sources as well (hydroenergy for example).

Although more complex than traditional two terminal HVDC system, the MTDC system proves to be

more beneficial. Several advantages of the MTDC system over the two-terminal HVDC are stated below

[17]:

- The number of IGBT based converter units is reduced, therefore a multilink HVDC is recommended over several two-terminal HVDC systems in terms of economy

- The MTDC is able to connect additional load or generation terminals with ease - A well designed energy control of the MTDC offers flexibility to the whole system which

can translate into a reduction of the electrical transmission losses, in comparison to HVDC transmission, and into an increase of the network’s availability

- Better reliability due to the fact that the energy is processed by more converters; therefore if one converter breaks, the energy transmission is redirected

21

3. MTDC Modeling

The chapter offers an insight into the MTDC system. Its main parts are described and an equivalent

model that offers an approximation of the real component’s behavior is presented. By using the

equivalent model, a good compromise between the component’s realistic behavior and simulation speed

is intended.

3.1 Overview of the MTDC System

A comprehensive picture of the MTDC system is described below. The setup consists of a single WPP

that send energy to two grids via HVDC. The main parts of that compose the system are highlighted.

Some of the setup’s sections are symmetrical therefore analogous components were not labeled.

1 2

AC Side

DC Side

8

4 5

697

3

Figure 3.1 - General electrical scheme of the MTDC

1. The Wind Power Plant 2. The Wind Farm side VSC 3. DC link capacitor bank 4. The HVDC cable 5. The Grid Side VSC 6. The AC filter 7. The step-up transformer 8. PCC (point of common coupling) 9. The electrical grid

A short description of the components and the way they were modeled will be presented in the

following rows.

22

3.2 The Wind Power Plant

A wind farm is composed out of a large number of wind turbines (tens or hundreds) situated in the

same location that produce large amounts of electrical energy. Due to the large number of wind turbines

an aggregate model of the WPP that replaces the more detailed one (in which each wind turbine

generator is modeled individually) is required. Otherwise the sheer complexity of the system will have a

big impact on the simulation speed.

The proposed WPP contains variable wind speed turbines with PMSG. The reason behind the choice is

their popularity. Due to the fact that the rated power of the wind turbines becomes higher, the blades

diameter is also increasing. Because the stresses on the blades grow proportionally, a new method of

load reduction was necessary. Pitch control offered a good solution to the problem, thus pitch variable

speed WTs became increasingly popular in the industry.

There are two methods of constructing the WPP aggregate model [18]:

- Representing the whole wind farm in a aggregate model as a power source - Modeling each generator using a simplified version

However, the second method does not give satisfying results, therefore the former is preferred. The

giant wind turbine has to be scaled to the size of the wind farm [18]. This means that its total rated

power is equal to the maximum rated power of a normal wind turbine multiplied by the total number of

WTs in the wind farm. The impedance of the wind farm cable is neglected.

The equivalent model is described in [18] and is presented in Figure 3.2.

Aggregate WT: mechanical

part & VSC rectifier

VSI

IC

S

R

Figure 3.2 – Aggregate WF model

The model contains the mechanical part of the WT and rectifier VSC equated by an ideal current

source. The switch and the damping resistor are placed in order to damp the excess power coming from

23

the WT. The excess power can be caused by a fault in the system for example. Ignoring the fault

protection system, the model can be further simplified, taking the shape of a simple ideal current source:

I.

Due to the fact that the project does not put emphasis on the WF side of the system, further

simplifications were made. The wind farm side, including the WPP aggregate model, inverter, AC cables

and filters and the MTDC rectifier are modeled as a single current source. The current source has the

equivalent power given by the equation [10]:

PWF =1

2∙ ρ · A · Cp · v3 · n · Ƞel · Ƞmec 3-1

PWF – represents the total power for a half of the wind turbine park

ρ – represents the air density

A – is the total swept area of the rotor

Cp – power coefficient

v3 – wind speed

ηel – electrical efficiency

ηmec – mechanical efficiency

n – total number of wind turbines within the farm

3.3 The Voltage Source Converter

The VSC model used in the project is a six valve converter, similar to the one illustrated in Figure 2.10. It

is both simple in structure and very widely used in the industry [19].

One of the purposes of the VSC is to generate a set of controlled AC voltages from a DC source. The

amplitude and angle are controlled via PWM. The output voltage is represented by a set of pulses with

the amplitude of Vdc

2 and duty cycle varying between 0% and 100%. The average of these pulses

constitutes a sine wave, with the maximum phase to phase voltage of 0.612·Vdc (taking in consideration

that the modulation factor is equal to unity). [19]

Due to the fact that the simulations presented in the project have to be fast and that the ripple coming

from the switching of the IGBTs is of no interest, an average model of the VSC will be used to perform the

simulations.

24

The model is based on the set of equations presented below [20]:

va = da ∙ Vdc ; vb = db ∙ Vdc ; vc = dc ∙ Vdc 3-2

IDC = ia ∙ da + ib ∙ db + ic ∙ dc 3-3

The equations are written in the abc reference frame. da, db and dc represent the duty cycles coming

from the modulation, va, vb and vc are the instantaneous AC phase to ground voltages on the, while Idc

and Vdc are the current and voltage respectively, on the DC side. Figure 3.3 - VSC Average Model

illustrates the average model of a grid side VSC modeled in PSCAD.

Figure 3.3 - VSC Average Model

The average VSC is represented by a current source on the DC side, while three voltage sources with

the voltage values coming from the modulator simulate the phase to ground voltages in the AC part. The

model also includes the converter losses, simulated by a current source in shunt with the capacitor bank.

More emphasis will be put on the losses in the coming chapters.

3.4 The Filter

The main purpose of the filter is to attenuate the high frequency harmonics at the output of the

inverter. Several low pass filter designs are available, varying in complexity and efficiency. One can

choose between an L and an LCL filter. Each type has its advantages and disadvantages.

The classical L filter, although simple in design, reaches its limitations when confronted with high power

applications. The dimensions of the inductor should be big (due to the fact that the value of the

inductance has to be high in order to cope with the high current ripple), thus having a negative effect on

system dynamics. Therefore the L filter lacks in efficiency what it gains in simplicity.

The alternative is designing a higher order LCL filter which can offer better performance than a

traditional filter keeping the cost and size down to a minimum [21]. An ideal LCL filter is presented in

Figure 3.4.

25

VSC Grid

L1 L2

C

Figure 3.4 - Ideal LCL filter

L1 is designed to reduce the ripple at switching frequency. L1 depends on the maximum current ripple

and on the DC voltage. [22]

ΔIMAX =1

8∙

VDC

4 ∙ fSW 3-4

L2 and C are designed keeping in mind that the installed reactive power on the filter should be kept to

a minimum. Moreover, the parameters should be computed such as the resonance frequency should be

far from the switching frequency and from the odd harmonics. If this is not the case, instability may

occur.

Since the project does not focus on filter design, the simulations were performed taking in

consideration a simple L filter, in order to simplify the calculations. In the laboratory work however, an

LCL filter was used.

3.5 The DC Cable

The HVDC cable offers many advantages over the classical overhead line in terms of economy and

environment. Furthermore, transmitting energy via an HVDC cable brings many technological

advantages, such as improved system stability and greater reliability. Cable designs have reached a

nominal power of 800 MW at a voltage of 500 kV, but more powerful cables (1000 MW) are within reach

due to the novel insulation materials [23].

The cables can be modeled using Pi-sections (Figure 3.5). However, since the capacitors and the

inductances do not have a noticeable influence on the DC transmission, they will be ignored. Therefore,

the HVDC cables are modeled using an equivalent series resistance. The resistance of the cable varies

according to the formula:

26

Rcable = ρ ∙l

S 3-5

ρ represents the line resistivity, l is the length of the line and S the cable’s area (without insulation). The

material used for the cables is copper, therefore ρ is equal to 16.8 nΩm, while S = 1200 mm2 [24].

Introducing the parameters into equation 3-5 and keeping in mind that the line is composed out of two

cables (for positive and negative polarity) one can compute the line’s resistance based on the formula

[25]:

Rline =2.8

100· l 3-6

𝑙 represents the cable length in kilometers.

L R

CCVin Vout

Figure 3.5 – DC cable model

3.6 The DC capacitor

The DC capacitor’s main attributes are reducing the voltage ripple on the DC cable and providing an

energy buffer between the DC and the AC sides. [26]

The challenge in designing the capacitor is making a good compromise between ripple attenuation and

system dynamics. If the capacitor is small, the dynamics of the system are better but the DC cable ripple

may not be attenuated adequately. On the other hand, if the capacitor has a high capacitance value, the

ripple on the DC side is eliminated but the system’s dynamics will suffer. The capacitor’s charging time is

introduced as a parameter for measuring the capacitor’s size. Its value is directly proportional to the

capacitor size. Keeping in mind that the energy stored in a capacitor is equal to C∙V

2 the following equation

is deduced:

27

ζ · Sconverter =1

2∙ C ∙ V 3-7

𝜁 represents the charging time, Sconverter is the active power flowing in the converter and V represents

the rated DC voltage. From equation 3-7 the capacitance can be computed. C is computed in such a way

that 𝜁is smaller than 5ms [10].

3.7 The Grid

The grid is the network through which the suppliers deliver electrical energy to the loads (consumers).

The grid is modeled using Thevenin’s equivalent circuit [10]. The circuit is presented in Figure 3.6.

Rg Lg

EgVgVpcc

PCC

Figure 3.6 - The grid model

The point of common coupling is the nearest electrical point to the load. At this point other loads can

be attached [27]. Rg and Lg represent the grid inductance (Zg).

Using Laplace, the grid’s impedance is equal to:

Zg = Rg + s ∙ Lg 3-8

Therefore VPCC is equal to:

VPCC = Zg ∙ ig + Vg 3-9

Based on the value of the grid inductance the grid can either be stiff (corresponding to a low

impedance value) or weak (in which the impedance value is high).

28

3.8 The transformer

The grid side transformers are used to step up the voltage in order to meet the grid

requirements. Another attribute is to offer galvanic isolation between the grid and the converter station.

Throughout the project the transformer is considered ideal, therefore both the no load and the load

losses are neglected.

29

4. The MTDC Control System

This chapter presents the control structure of the MTDC system. An overall introduction to the control

topology and objectives is presented, followed by a detailed description of the grid connected converters

control loops and synchronization methods. The wind farm converter control structure is also discussed.

Finally, the control structure in MTDC operation is presented by introducing the droop control principle.

4.1 Introduction and objectives of control

The main objectives that have to be accomplished by the MTDC control system may be divided into

ensuring proper power flow direction and protection of the system components in case of faults.

Unlike the conventional HVDC system interconnecting two AC grids, where bidirectional power flow

may be achieved, in the case of a HVDC transmission link in which the wind farm acts as the power

source, the power flow needs to be unidirectional (i.e. from the wind farm into the grid). This operating

mode may be easily obtained in a VSC based system by controlling the converter’s output voltage phase

angle, as presented in Chapter 2.

In the multiterminal VSC-HVDC transmission link established between an offshore wind farm and

onshore AC networks, the converter control systems have different functions. The sending converter

station has a task of keeping constant the AC voltage reference, while the receiving stations regulate the

DC voltage. Such design of the control system forces active power flow from the wind farm into the grids.

In more detail, the main task of the offshore VSC station control is maintaining a stable AC voltage of the

wind farm site, while the Grid side converters regulate the DC link voltage according to the reference

obtained from the droop controllers. By varying the DC voltage, active power flow may be controlled. As

in the case of a multiterminal system, where more than one receiving station is considered, the power

flow must be controlled in a way to achieve the desired power exchange between the two terminals

sharing the DC link. Thus, the power sharing between the onshore converter stations is the purpose of

introducing DC voltage droop control. In an ideal case, if the losses would be neglected, the HVDC system

would act as an energy buffer between the wind farm source and grids. However this is not the real life

case and power losses occur in various components of the system.

Switching and conduction losses of the converter station, as well as DC cable losses, constituting a large

amount of the overall HVDC system losses, are all considered in this thesis. An optimization method (i.e.

minimization of losses) is presented in Chapter 5.

In conclusion, the VSC HVDC control structure allows for control of active and reactive power, along

with AC and DC voltage. The general control structure is presented in Figure 4.1.

30

Figure 4.1 - General control structure of MTDC system

Since the DC link voltage has to be kept constant, the grid side converters play an important role in

maintaining stability and safe operation of the system. The DC link voltages can vary within ± 3% of the

reference voltage. Greater disturbances in the DC link voltage may cause the system to trip. A detailed

description of the grid side and wind farm converter control structure is provided in the following

subchapter.

4.2 Control of Wind Farm and Grid Side Converters

The MTDC system consists of a number of power electronics converters connected back to back and

sharing the same DC link voltage. The converters may be classified into sending (offshore) stations, acting

in rectifier mode (receiving power from the wind farm) and receiving (onshore) stations, operating in

inverter mode and feeding the power to the grid.

This thesis focuses on a 1 + 2 aggregate (i.e. 1 sending converter and 2 grid connected converters

supplying different grids). The grid connected VSC is the converter which makes the onshore connection

between the HVDC system and the AC grid. Its main purpose is active power control achieved by

regulating the DC link voltage and grid support by injecting reactive power during grid faults.

31

The wind farm VSC rectifies the AC power from the wind farm and sends it to the onshore converter

through the DC cables. The station is equipped with an AC voltage controller which controls this voltage

at the point of common coupling between the wind farm and the converter (measured on the capacitor

of the filter shown in Figure 4.2). The detailed control structure is presented in Figure 4.2.

Figure 4.2 - Detailed control structure of VSC HVDC system

The control system consists of two cascaded loops – the fast, inner current control loop (common for

both converter stations) and the outer control loops which are different for each station (in the case of

the Wind Farm VSC – an AC voltage controller, in the Grid connected VSC – DC voltage controller). The

faster inner loop has the task of controlling the AC currents, the reference for which are obtained from

the outer controllers. The reference for the active current in the grid VSC is provided by the DC voltage

controller, as for the reactive current – from the reactive power controller. In the wind farm VSC both

active and reactive current components are obtained from the AC voltage controller.

The DC voltage controller has a task of keeping the DC link voltage constant while the AC voltage

controller keeps a stable AC voltage on the wind farm side. All controllers are implemented in dq rotating

reference frame. The grid side three phase currents are transformed to d and q components by means of

the Park transformation and synchronized with the AC voltages by means of the Phase-Locked-Loop

(PLL).

The d and q voltages generated by the controller are transformed back into three-phase abc quantities

and converted into switching states for the converter in the PWM modulator. Since the outer and inner

control loops are the same for the second grid side converter, only one converter has been presented in

Figure 4.2. The design and tuning of the inner current control, as well as the outer controllers and PLL will

be described in detail in the following subchapters.

32

4.2.1 Current Control Loop

The current controller is implemented in the dq rotating reference frame and its structure is presented

in Figure 4.3. Since the control variables transformed to the dq frame are DC quantities, PI controllers are

best suited for tracking the references and removing steady state error. Decoupling network

(represented in Figure 4.3 as ωL blocks) have to be added in the controller structure since coupling terms

appear in the regulated system’s equations after the dq transformation. Grid voltage feed-forward is also

added in order to improve the controller’s performance.

Figure 4.3 - Current controller structure in dq frame

The block diagram for the d-axis of the inner current control loop is presented in Figure 4.4. The

decoupling network and grid voltage feed-forward were considered as disturbances in the control system

and omitted during the tuning process. Since the control loop for the q-axis has identical dynamics, it was

enough to perform tuning for the d-axis and apply the same controller parameters for the q-axis.

Figure 4.4 - Block diagram of current control loop

33

The control loop structure consists of the following main components:

Current controller implemented in the form of a PI (proportional – integral) controller represented by the transfer function:

Gctrl = KP +Ki

s 4-1

For which the values of the proportional and integral gain parameters (Kp and Ki,

respectively) have to be found in the process of tuning.

Plant, representing the filter inductance through which the current is being passed. A simple L filter is considered represented by a first order transfer function in the form of:

Gplant =1

s ∙ L + R

4-2

L is the filter inductance and R – the parasitic resistance. For further simplifications, considering the fact

that the parasitic resistance is of negligible value, the final plant equation may be considered in the form:

Gplant =1

s ∙ L

4-3

Further simplifications in the control structure are introduced by eliminating the delay blocks since

their effect is insignificant for the system. A simplified structure of the current control loop is presented

in Figure 4.5.

Figure 4.5 - Simplified block diagram of current control loop

34

A properly tuned controller should fulfill design requirements in terms of settling time and damping,

i.e. it should offer a fast response to satisfy performance requirements and an overshoot within a certain

limit for safety reasons. Using the root locus tuning method, the design requirements for the controller

are defined by the bandwith (associated with the settling time) and damping (related to the amount of

overshoot).

The design requirements for the controller are the following:

Controller bandwith of:

ωn =2 ∙ π ∙ fs

10

4-4

sf is the switching frequency in [Hz]

Damping factor of: ζ = 0.707

For those requirements the values of Kp = 0.38 and Ki = 691 are found. The controller response is

analyzed in the SISOtool package from MATLAB. The root locus and Bode diagrams for d-axis current loop

are presented in Figure 4.6.

102

103

104

105

-90

-45

0

Frequency (rad/s)

Phase (

deg)

-30

-20

-10

0

10

Bode Editor for Closed Loop 1 (CL1)

Magnitu

de (

dB

)

101

102

103

104

105

-180

-150

-120

-90P.M.: 65.4 deg

Freq: 3.16e+003 rad/s

Frequency (rad/s)

Phase (

deg)

-40

-20

0

20

40

60

80

G.M.: -Inf dB

Freq: 0 rad/s

Stable loop

Open-Loop Bode Editor for Open Loop 1 (OL1)

Magnitu

de (

dB

)

-3000 -2500 -2000 -1500 -1000 -500 0 500-1500

-1000

-500

0

500

1000

15000.20.380.560.70.810.89

0.95

0.988

0.20.380.560.70.810.89

0.95

0.988

5001e+0031.5e+0032e+0032.5e+0033e+003

Root Locus Editor for Open Loop 1 (OL1)

Real Axis

Imag A

xis

Figure 4.6 - Root locus and Bode diagrams for current control loop

35

Time (seconds)

Am

plit

ude

0 1 2 3 4 5 6

x 10-3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

System: Closed Loop r to y

I/O: r to y

Peak amplitude: 1.13

Overshoot (%): 12.9

At time (seconds): 0.00134


I/O: r to y

Settling Time (seconds): 0.00367

0 1 2 3 4 5 6

X10-3

0.2

0.4

0.6

0.8

1

1.2

1.4

Am

plit

ud

e

Figure 4.7 - Step response of the current loop

From the root locus it can be observed that the complex poles are in the left side of the s-plane

indicating the system is stable. Their location with respect to the origin guarantees that the requirements

in terms of damping and bandwidth are fulfilled. The step response is presented in Figure 4.7, where it is

clearly visible the controller has a fast response and an overshoot within the design limits.

4.2.2 DC Voltage Control Loop

The DC voltage control is implemented in the Grid connected converters and is providing active

current reference to the inner current controller. Its purpose is regulating the DC link voltage to the

reference value (obtained from the droop controllers). By regulating this DC link voltage, it is ensured

that the active power is being transferred from the DC link into the AC grid. The block diagram of the DC

voltage control loop, with delay blocks disregarded, is presented in Figure 4.8.

Figure 4.8 - Block diagram of DC voltage control loop

36

The DC voltage is controlled across the capacitor of the DC link, thus the plant transfer function is given

by equation 4-5.

GDC =1

s ∙ C

4-5

C is the capacitance in [μF]

The ( )i sG block represents the closed loop transfer function of the inner current loop.

The design requirements set on the outer controllers are identical in terms of damping, however the

controller bandwith should be at least 10 times smaller than the inner loop bandwith, since the outer

loops should have a slower response. Thus, the following requirements are set on the DC voltage

controller:

Controller bandwith of:

ωn =2 ∙ π ∙ fs

100 4-6

Damping factor of: ζ = 0.707

The following figures present the design plots and step response of the loop.

101

102

103

104

105

106

107

108

-180

-135

-90

-45

0

Frequency (rad/s)

Phase (

deg)

-150

-100

-50

0

50

Bode Editor for Closed Loop 1 (CL1)

Magnitu

de (

dB

)

101

102

103

104

105

106

107

108

-180

-150

-120

-90P.M.: 60 deg

Freq: 314 rad/s

Frequency (rad/s)

Phase (

deg)

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

G.M.: -Inf dB

Freq: 0 rad/s

Stable loop

Open-Loop Bode Editor for Open Loop 1 (OL1)

Magnitu

de (

dB

)

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

x 106

-5

0

5x 10

4

0.9850.9970.9990.99911

1

1

0.9850.9970.9990.99911

1

1

5e+0051e+0061.5e+0062e+0062.5e+0063e+006

Root Locus Editor for Open Loop 1 (OL1)

Real Axis

Imag A

xis

Figure 4.9 - Root locus and Bode diagrams for DC voltage control loop

37

Time (seconds)

Am

plit

ude

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450

0.2

0.4

0.6

0.8

1

1.2

1.4System: Closed Loop r to y

I/O: r to y


Overshoot (%): 24.4

At time (seconds): 0.0104


I/O: r to y


0 0.005 0.01 0.015 0.02 0.03

0.2

0.4

0.6

0.8

1

1.2

1.4

Am

plit

ud

e

0.025 0.035 0.04 0.045

Figure 4.10 - Step response of DC voltage loop

As it can be observed, the settling time of 0.03 [s] is 10 times larger than that of the inner loop and thus

meets the requirements of designing the outer loops slower than the inner loops to maintain system

stability. However, the controller was later retuned to obtain an even bigger settling time for stability

purposes during the simulation run.

Moreover, an overshoot of above 20% is unacceptable considering the voltage levels that are present in

HVDC transmission systems (+/- 300 kV). In order to improve the system’s stability and limit the

overvoltage, a pre-filter block has to be included in the control architecture as presented in Figure 4.8

[28].

The pre-filter transfer function is expressed in the form of a low-pass filter and is of the form:

Gpf =1

τ ∙ s + 1

4-7

Where τ =1

ωZero PI

is the time constant of the filter and ωZeroPI is the frequency of the PI controller’s

zero (in rad/s). Thus, an additional pole is introduced into the system in order to cancel out the effect of

the controller’s zero and, as consequence, reduce the overshoot to an acceptable value. The effect of

introducing the pre-filter into the system is presented in the step response in Figure 4.11.

38

Time (seconds)

Am

plitu

de

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

0.2

0.4

0.6

0.8

1

1.2

1.4


I/O: r to y


Overshoot (%): 8.85

At time (seconds): 0.0178 System: Closed Loop r to y

I/O: r to y


0.4

0.6

0.8

1

1.2

1.4

Am

plit

ud

e

0.2

0 0.005 0.01 0.015 0.02 0.030.025 0.035 0.04

Figure 4.11 - Step response of compensated DC voltage loop

4.2.3 Active power control loop

The active power controller is implemented in the outer control structure of the sending converter only

in the case of a HVDC link established between two AC grids, where an active power reference may be

chosen at any instance. In the case of interconnecting a grid with a wind farm as source, active power

control loop cannot be used since it is impossible to set the power reference due to the fluctuating

nature of this power source. An AC voltage controller is used instead, its description is provided in

Chapter 4.2.3.

The instantaneous power theory defines the instantaneous active and reactive power equations with

respect to the instantaneous currents and voltages, in dq reference frame, as:

P = ud · id + uq · iq 4-8

Q = uq · id − ud · iq 4-9

Represented in matrix form as:

PQ =

ud

uq uq

−ud ∙

id

iq 4-10

39

From equations 4-8 and 4-9 the reference for the active and reactive currents may be obtained:

id

iq =

1

ud2 + uq

2 · ud

uq uq

−ud ·

Pref

Qref 4-11

Where Pref and Qref are the reference values for the active and reactive power. The reference for active

current from equation 4-11 is thus:

id =Pref ∙ ud + Qref ∙ uq

ud2 + uq

2 4-12

The block diagram of the active power controller is presented in Figure 4.12.

Figure 4.12 - Block diagram of active power control loop

4.2.4 AC voltage control loop

In the case of an HVDC connection between a wind farm and a grid where the wind farm acts as the

power source, an AC voltage control loop is implemented in the wind farm side converter [29].

The AC voltage control is realized by regulating the voltage drop over the filter capacitor Cf of the wind

farm converter [30]. The controller is implemented in dq reference frame and presented in Figure 4.13.

Vabc are the 3 phase voltages measured across the AC side filter, later transformed into dq and

compared with the reference. D-axis voltage is responsible for giving the active current reference Id_ref,

while the q-axis voltage generates the reactive current reference Iq_ref. A decoupling feed-forward

compensation is used to eliminate the coupling between Vd and Vq voltages.

40

Figure 4.13 - AC voltage controller structure in dq reference frame

The d and q axis loops for the controller have the same dynamics, thus tuning is performed only for

the d-axis and the controller parameters are adapted for the q-axis. The block diagram of the controller is

presented in Figure 4.14.

Figure 4.14 - Block diagram of AC voltage control loop

The controller had to meet the same design requirements as imposed on the outer control loops, i.e. a

10 times slower response than that of the inner loop.

The step response of the AC voltage control loop is presented in Figure 4.15.

41

Figure 4.15 - Step response of AC voltage loop

4.2.5 Phase Locked Loop (PLL)

The phase locked loop (PLL) is an important part of the control structure of grid-connected converters.

The phase angle of the grid voltage vector is a necessary to perform the synchronization of the converter

with the grid at the point of common coupling (PCC). This phase angle also plays an important role in

transforming the feedback variables to the dq reference frame, in which the control system is designed.

The PLL keeps an output signal synchronized with a reference input signal both in frequency and phase

[31]. The phase angle is detected by synchronizing the PLL rotating reference frame and the grid voltage

vector. The purpose of using the PLL is to synchronize in phase the converter output current with the grid

voltage to obtain unity power factor operation. The structure of the PLL in the dq reference frame is

presented in Figure 4.16.

Time (seconds)

Am

plit

ude

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

0.2

0.4

0.6

0.8

1

1.2

1.4


I/O: r to y


Overshoot (%): 8.8

At time (seconds): 0.0177 System: Closed Loop r to y

I/O: r to y


42

Figure 4.16 – Structure of the PLL [32]

The inputs for the PLL are the three-phase grid voltages, transformed into the synchronous rotating

reference frame using Park transformation. The output is the tracked phase angle.

Setting the quadrature axis reference voltage (Vg_q) to zero results in lock in of the PLL output on the

phase angle of the grid voltage vector. This quadrature component represents the phase difference

between the grid and converter voltage. A PI regulator is used to control this variable and its output is the

grid frequency. After integrating the grid frequency, the angle of the grid voltage vector is obtained,

which is fed back to the abc – dq transformation block. The feed-forward term of the grid angular

frequency (2πf) is introduced to improve the tracking performance of the PLL.

The plant is defined by a pure integrator, thus the transfer function of the PLL presented in equation 4-13

is of similar form as the standard second order transfer function shown in equation 4-14 [28].

H s =kp ∙ s +

kp

Ti

s2 + kp ∙ s +kp

Ti

4-13

G s =2 · ζ · ωn · s + ωn

2

s2 + 2 · ζ · ωn · s + ωn2 4-14

By comparing the two transfer functions, the controller gains may be obtained. The design

requirements imposed on the controller are:

Damping factor ζ of: 0.707

Settling time Ts of: 0.04 [s]

The parameters of the PI controller of the PLL can be set as a function of the settling time as follows

[28]:

Kp =9.2

Ts 4-15

43

Ti =Ts · ζ

2

2.3 4-16

The obtained dynamic response to a 1 Hz step in frequency is presented in Figure 4.17

Figure 4.17 – Step response of PLL for 1 Hz frequency boost

The obtained step response fulfills the desired requirements. The step is applied at t = 0.3 s and settles at

0.34 s with an overshoot of 20%.

The obtained grid phase angle is presented in Figure 4.18.

Figure 4.18 – Phase angle of the grid voltage

0.29 0.3 0.31 0.32 0.33 0.34 0.3549.6

49.8

50

50.2

50.4

50.6

50.8

51

51.2

51.4

51.6

Grid frequency

PLL frequency

0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.19 0.195 0.2 0.205

1

2

3

4

5

6

Time [sec]

Theta

[ra

d]

44

4.3 Controlling the MTDC operation

The control of an HVDC system in multiterminal configuration is much more demanding than in case of a

point to point system. In a point to point system only one sending station and one receiving station is

present. Increasing the number of connections to create a multiterminal system means that more than

two converters share the DC bus and the system becomes more complicated. The challenge is the control

of the DC voltages and the power sharing between the grid connected converter stations.

A simplified steady state DC equivalent circuit of the MTDC system is presented in Figure 4.19.

WF VSC

GS VSC1

GS VSC2

I2

I3

V2

V3

I1

V1

R2

R3

Figure 4.19 – DC equivalent circuit of the MTDC system [33]

The power is injected into the DC link by the wind farm VSC and shared between two receiving grid side

converters (GS VSC1 and GS VSC2) according to equations 4-17 and 4-18.

P2 = V2 ∙ I2 4-17

P3 = V3 ∙ I3 4-18

The voltages on the receiving stations may be defined as:

V2 = V1 − R2 ∙ I2 4-19

V3 = V1 − R3 ∙ I3 4-20

By varying the DC link voltages and currents it is possible to share the power injected by the sending converter station.

45

Thus, in MTDC operation a droop controller is introduced to change the DC voltage reference in each receiving station according to a defined droop. The droop’s main purpose is to offer a mathematical correlation between the reference voltage, the measured current and the DC link voltage on the terminal.

Figure 4.20 presents the structure of the MTDC under droop control where the wind farm is represented by a current source injecting current into the DC link and grid side converters are considered as ideal voltage sources.

V2, P2 V3, P3V1x x

I1, P1

R2, I2 R3, I3

Vref

α2 α3

Vref

Figure 4.20 - The Voltage Droop Control Principle

The main elements of droop control are the virtual resistances (denoted as α2 and α3 in Figure 4.20)

and the sharing factor. The virtual resistances are used in the DC link for setting the DC voltage reference

for each converter station. The sharing factor parameter, denoted as k, determines the distribution of

current among the receiving stations, thus has a direct effect on the power sharing.

The sharing factor is defined as the ratio of DC link currents:

𝐤 =𝐈𝟐𝐈𝟑

4-21

Power sharing between the onshore converter stations is achieved by finding voltages V2 and V3

(equations 4-19 and 4-20) for a specified sharing factor, after which the virtual resistances are found from

4-22 and 4-23.

The use of virtual resistance in the DC link allows setting different references for the receiving

converter stations DC voltages, as presented in Figure 4.21.

α2 =Vref − V2

I2 4-22

α3 =Vref − V3

I3 4-23

46

Figure 4.21 - Grid side converters DC voltage reference generation

Adding virtual resistances to the control structure of the grid side converter stations produces a voltage

drop across this resistance which, in turn, leads to generating different DC voltage references at the

terminals of the converter stations.

The next chapter describes the use of optimization methods to finding the optimal values for the droop

coefficients.

47

5. MTDC optimization

The main part of the chapter highlights a thorough analysis of the three terminal MTDC system’s

optimization routine. Explanations regarding the different cost functions and constraints used are stated.

Furthermore, the chapter contains or sends reference to the calculus backing up the Matlab optimization

routine results. Additionally, the function that allows converter losses to be taken in consideration and the

economical model on which the economical optimization routine is based, are described.

Finally, in order to prove the optimization routine’s versatility, the analysis is extended to a four

terminal MTDC system.

5.1 Introduction to the optimization algorithm

The optimization routine is written in Matlab and is part of the Matlab MTDC Optimization Algorithm. It

is based on the integrated fmincon function. Basically, the routine acquires information about the

present state of the system (nodal voltages, line currents and, during the economical analysis, energy

prices), introduces them into the fmincon function gathers the optimized state vector containing the new

currents and voltages and computes the optimization droops.

The aforementioned function has a cost function that will be minimized taking in consideration a set of

constraints. Several cost functions will be used, depending on whether the converter losses are taken in

consideration and whether the economical parameters are taken into account. Basically, the routine will

try to minimize the line and converter losses, will try to maximize the profits on the whole circuit, or will

take in consideration both aspects.

The constraints are based on Kirchoff’s laws and are nonlinear. Thus, an optimization algorithm that

minimizes a cost function taking in consideration nonlinear constraints is required. The SQP (Sequential

Quadric Programming) algorithm was chosen. The algorithm comes as a standard solving approach for

Matlab’s fmincon function.

Furthermore, certain limitations have to be taken in consideration, otherwise the routine will display

unfeasible solutions that will lead to system instability.

The limitations for the state variables are the following:

- The voltages are set to vary between 291 and 309 kW (0.97 – 1.03 p.u.); if the voltages exit this

range, the system is at risk of becoming unstable. Therefore the current’s maximum value for P =

1 p.u. is equal to 1.03 p.u. at Vmin=0.97 p.u.

- The powers vary between 0 and 1 p.u.

48

The initial state variables which represent the starting vector x0 for the algorithm are the values of the

system’s parameters (V1, I1, I2, I3) from the previous time step. At the beginning of the run x0 is allocated

several values within the imposed limits (the closer the values are from the optimum point, the faster the

algorithm reaches the solution, therefore a good starting point is an advantage). It is worth mentioning

that the SQP algorithm does not need a feasible initial solution in order to function properly.

Taking into consideration the cost function, limitations, constraints, starting vector and algorithm

described above, the optimization routine is initialized by invoking the fmincon function.

After the results are obtained, a realistic mode of setting the V2 and V3 (obtained from the V1 and the

currents) as voltage references for the voltage controllers is required. Therefore, the voltage droop

control method is adopted. The droop control strategy is presented in detail in Chapter 4.

5.2 The MTDC simulation model

The MTDC simulation model’s parameters are listed below:

- rated power: 400 MW

- rated voltage: ±150 kV

The simulations are performed with the wind power plant sending the rated power to the grids, unless

otherwise mentioned (P = 1 p.u.). Even though the two grid side converters are sharing the power from

the wind farm, both should be able to withstand its rated power in order to be able to transmit all the

power to one grid. Thus the GSVSC ratings are 400 MVA each.

The cable resistances are: R2 = 1.8 Ω and R3 = 3.6 Ω. These correspond to cable lengths of 64.3 km and

128.6 km respectively [25], [34]. Taking in consideration that the WPP is situated offshore and that the

cables are submersed, the values are feasible. The three terminal MTDC model is presented in the figure

below.

AC

AC

DCOffshore Onshore

DC cable

resistanceDC cable

resistance

WPP

Grid

VSC

VSC

VSC

AC

FilterTransformer

Figure 5.1 - Three terminal MTDC

49

In the last part of the chapter, the four terminal model is going to be tested. In this situation two

WPPs situated at a close distance to each other (R = 0.5 Ω, corresponding to a distance of approximately

20 km) are sending the power to the two grids. The rated power of the first WPP is 300MW (0.75 p.u.)

while the second WPP’s power is 100MW (0.25 p.u.). This value is taken in order to be able to keep the

VSC ratings the same. Both power plants will send their rated powers to the grids. The model is

presented in the figure below.

AC

AC

DC

DC cable

resistance

Grid 1

R1

R2

R3

Grid 2

WPP 2

WPP 1 VSC

VSC

VSC

VSC

AC

FilterTransformer

Figure 5.2 - Four terminal MTDC

5.3 Loss minimization of the three terminal MTDC system,

taking in consideration only the line losses

The optimization function’s objective is to adjust the power flow on the offshore terminals in such a

way that the system will have its overall losses minimized. In the first instance only the cable losses are

taken in consideration. The simplified MTDC system is illustrated in Figure 5.3.

V2, P2 V3, P3V1x x

I1, P1

R2 R3

I2 I3

Figure 5.3 - Simplified MTDC system’s representation

50

The minimization problem can be solved in one of several ways. One approach is to find the line

currents for which the sum of the line losses is minimum. The terminal voltages will be computed from

the aforementioned currents. Therefore, the cost function is represented by the following equations:

f = Plost line 2+ Plost line 3

5-1

f(I2, I3) = R2 · I22 + R3 · I3

2 5-2

The cost function is to be minimized based on the following constraints:

I1 = I2 + I3 5-3

I1 =P1

V1

5-4

It can be observed that the system has a number of 2 degrees of freedom. It can be observed that the

minimization problem is not linear. In order to simplify the calculus, the presumption that V1 reaches the

maximum allowable value is made. Since the line losses are quadratically dependent on the currents

flowing through the terminals, I1 should reach the smallest value possible. Consequently, since P1 is a

preset value (corresponding to the power coming from the wind farm side converter), V1 will rise

inversely proportional with I1’s decrease until it reaches the maximum allowable limit (set to 1.03 p.u.).

Having found I1, I3 is replaced in equation 5-2 with I1-I2, obtained from 5-3.

The simplified cost function, depending only on I2 is:

f I2 = R2 · I22 + R3 · I1 − I2

2 5-5

Deriving df (I2)

dI2 the stationary point I2 =

I1 ∙R3

(R2+R3) is obtained. Therefore:

I2 =I1 ∙ R3

(R2 + R3)

5-6

I3 =I1 ∙ R2

(R2 + R3)

5-7

The voltages are further computed as:

V2/3 = V1 −I1 ∙ R2 · R3

(R2 + R3)

5-8

The droops are computed using equations 4-22 and 4-23. Taking into consideration the simulation

model, V2=V3=1.024 p.u. and I2=2·I3=0.647 p.u.

51

Note: The algorithm implemented in Matlab does not use the simplification mentioned above.

Further, a graphical representation of the minimization problem is presented. Equation 5-2 can be

written as:

I22

yR2

+I3

2

yR3

= 1, y − min 5-9

The equation represents an ellipse with the origin in the center of the coordinate system. The major

and minor axes are depending on the dimensions of R2 and R3. Taking in consideration the constraints

the graphical optimization problem is reflected in Figure 5.4.

Figure 5.4 - Graphical interpretation of the MTDC loss minimization problem (line losses considered)

For a better visualization I2 and I3 are represented in kA while the cost function outputs MW. From

the graphic I2 = 0.85 kA = 0.64 p.u., I3 = 0.42 kA = 0.32 p.u. while Plosses = 2 MW = 0.005 p.u. The graphical

representation confirms the theoretical results.

It can be seen that the optimum point is located in the place where the constraints line is tangent to

the ellipse with the smallest radii described by the cost function.

The constraints line depends on the value I1 which, in turn, depends on V1. But in order for I1 = I2+I3 to

be tangent to the ellipse with the smallest radii, I1 has to have the smallest value possible. This value is

52

constrained by equation 5-4. Therefore, in this case V1 goes to the maximum allowable value. The arrows

in the graph indicate that the constraints line can change its position according to the variation of V1.

The remaining part of the subchapter is intended to confirm the theoretical results by running the

Matlab optimization routine. The DC voltages, currents and transmitted powers are plotted in Figure 5.5.

In Figure 5.6 the voltage profiles are represented by excel columns. Terminal 1 (corresponding to the

wind farm) is represented by the red color, terminal 2 (corresponding to the line with the smaller line

losses) is represented by the blue color and terminal 3 is represented by cyan. The convention will be

kept throughout the project, unless otherwise specified.

Figure 5.5 - Optimization Routine Taking in Consideration Only the Line Losses

Terminal 1 – red, terminal 2 – blue, terminal 3 - cyan

2 3 4 5 6 7 8 9 10 11 121.02

1.025

1.03

1.035

Time [s]

Vdc [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

Time [s]

Pdc [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

Time [s]

Idc [

p.u

.]

53

V1 V2 V3 I1 I2 I3 P2 P3 Plosses

1.03 1.0248 1.0248 0.971 0.647 0.323 0.663 0.331 0.005

Table 5.1 - Optimization Routine Results

Figure 5.6 - Voltage histogram for the optimization routine with line losses

As anticipated, V1 goes to the maximum value of 1.03 p.u., while the sharing factor I2/I3 becomes 2. The

system’s voltages, currents and powers can be seen in Table 5.1. The losses on the circuit are equal to

0.005 p.u. The value is expected to be the smallest in comparison to the losses obtained with manual

droop control.

Figure 5.7 displays the power losses during various sharing factor values. It can be seen that the sharing

factor obtained using the optimization routine gives the smallest losses for the MTDC system.

Predictably, the highest losses are registered when trying to send more power to the terminal with a

higher line resistance (I2/I3 < 1).

1,018

1,02

1,022

1,024

1,026

1,028

1,03

1,032

1sec 2sec 3sec 4sec 5sec 6sec 7sec 8sec 9sec 10sec

V1

V2

V3

54

Figure 5.7 - Ploss curve

5.4 The converter station’s equivalent loss model

An important economic consideration is regarding the power lost in the system while transmitting

the energy from the wind farm to the grids. Minimizing the losses in the system has a direct influence on

the profit maximization of the system.

The total loss of a converter station represents approximately 1.6% of the whole transferred power

at rated power [35]. Figure 5.8 describes the proportion of the losses in a converter station.

Figure 5.8 - Losses in a converter station

0 5 10 15 201

1.5

2

2.5

3

3.5

4x 10

-3

Sharing Factor (I2/I3)

Plo

ss [

p.u

.]

Valve losses70%

Transformer losses13%

Reactor losses8%

Other devices9%

Losses in a converter station

55

A large portion of the HVDC connection losses are due to the valve losses within a converter station.

Therefore, these cannot be ignored when building a MTDC loss minimization algorithm. The MTDC in

question has voltage source converters composed of IGBTs with diodes. The IGBT is of modular design. A

large number of IGBT and diode sub-modules are connected in parallel in order to increase the maximum

power the valve can withstand [36]. Therefore, the losses in a valve are calculated for a single IGBT or

diode sub-module before multiplying them by the total number of sub-modules. Since the number of

valves in a 2 level, 3 phase converter is 6 and since the currents and voltages in all IGBTs are identical

(although phase shifted), it is, therefore, sufficient to consider the losses in only one converter valve.

These will be multiplied by the total number of valves (six), in order to obtain the total losses in the

converter.

Taking in consideration the simplifications mentioned above, a loss function that reflects the power

loss changes in a converter according to the variation of the current and voltage will be determined in the

following rows. The IGBT and diode losses are divided into three categories: conduction losses, switching

losses and blocking losses [37]. Normally the blocking losses are too small to be taken in consideration,

therefore:

Plosses = PSW + Pcond 5-10

The IGBT conduction losses can be calculated using the approximation:

uCE = uCE 0 + rc · ic 5-11

The anti-parallel diodes have an analog conduction voltage drop given by equation 5-12.

uD = uD0 + rD · iD 5-12

uce0 and ud0 represent the semiconductor’s on state zero-current collector-emitter voltage, rc and rd are

the semiconductor’s on slope resistance and ic/id are the currents flowing through the IGBT and diode,

respectively.

The average values of the IGBT losses are equal to:

Pcon dIGBT=

1

TSW∙ uCE (t) ∙ iC(t) ∙ dt

TSW

0

5-13

By replacing the voltage uCE in 5-12 with the value from 5-11 one can obtain:

Pcon dIGBT=

1

TSW∙ uCE 0 · iC(t) + rC · iC

2 (t) ∙ dt

TSW

0

5-14

By taking in consideration the modulation function, however, in the case of carrier based sinusoidal

PWM and by integrating 5-14, equation 5-15 is deduced [36], [38].

56

Pcond IGBT=

uCE 0 ∙ IACMAX

2 ∙ π∙ 1 +

ma ∙ π

4∙ cosρ +

rC ∙ IACMAX

2

2 ∙ π∙ π

4+

ma ∙ 2

3∙ cosρ

5-15

It can be observed that the conduction losses are linearly and quadratically dependent on the current.

Similarly, the losses equation for the diode is obtained:

Pcond diode=

uD0 ∙ IACMAX

2 ∙ π∙ 1 −

ma ∙ π

4∙ cosρ +

rD ∙ IACMAX

2

2 ∙ π∙ π

4−

ma ∙ 2

3∙ cosρ

5-16

ma represents the modulation index of the converter, 𝜌 represents the displacement angle between

the converter voltage and the load current (𝜌 =1 for Q=0) and IAC_MAX represents the maximum AC phase

voltage.

uce0, rc, ud0 and rd can be found in the IGBT datasheets. In the present case [36]:

- UCE0 = 1.14 V; UD0 = 1.05 V

- rC = 1.2m Ω; rD = 0.65m Ω

The diode and IGBT conduction losses are added, multiplied by the number of the IGBTs in one valve

and multiplied by the number of valves in a two-level converter in order to obtain the total losses of a

converter.

Pcond = 6 · NIGBT · (Pcond IGBT+ Pcond diode

) 5-17

The number of IGBTs in a valve for a converter with Srated = 400MW and Vrated =±150kV is equal to

approximately 300 [36].

Taking in consideration all the aforementioned values, one will obtain Pcond in function of IAC_max and ma.

Pcond = IAC MAX2 · 0.29 + 0.1 · ma + IAC MAX

· (0.612 + 0.028 · ma)

5-18

The switching losses of an IGBT depend on its switching on and switching off energies [37]:

EON IGBT= uCE (t) ∙

TON

0

iC(t) ∙ dt

5-19

EOF FIGBT= uCE (t) ∙

TOFF

0

iC(t) ∙ dt

5-20

Eon_IGBT represents the total energy lost in an IGBT during turn-on. In practice, it is measured from the

moment the collector current begins to rise to the moment when uce almost zero. Eoff_IGBT is the total

energy lost during turn-off. TOFF starts from the point where the collector-emitter voltage starts to rise

until the point where the collector current falls to 0 [39].

57

Figure 5.9 shows a basic representation of the switching states of an IGBT [40]. The slopes are

linearized and the integrals are computed in order to obtain Eon and Eoff.

Figure 5.9 - Turn ON/OFF Representation – IGBT

The IGBT datasheets contain the EON, EOFF values for the reference values of voltage and current. For

values different than the references the following formulas apply:

EONIGB T= EON IGBT ref

·V

Vref∙

I

Iref

5-21

V and I are the voltage and current values, respectively, for which one wants to determine Eon.

Regarding the diode, the turn-off losses are small enough to be ignored, while the turn-on losses are

calculated similarly to the IGBT turn-on losses.

Therefore:

EONdiode= EON diode ref

·V

Vref∙

I

Iref

5-22

The total switching losses are equal to the sum of the turn-on and turn-off energies on both the IGBT

and diode. The values are multiplied by the number of sub-modules in an IGBT and further multiplied by

the total number of valves in the converter. Equation 5-23 gives the total switching losses in a three

phase two level converter [41].

PSW = 6 · NIGBT ·fSW

π· (EON IGB Tref

+ EOFF IGBT ref+ EOFF diode ref

) ·VDC

Vref∙

IAC MAX

Iref

5-23

NIGBT represents the number of IGBTs in a valve, fSW represents the switching frequency, equal to 1150

Hz; Vref is the reference voltage equal to the blocking voltage of one IGBT; Iref is the conduction current

58

after commutation, Iref = 1.333kA. The energies are can be found in the semiconductor’s datasheet. It can

be observed that the losses depend on both current and voltage.

Taking in consideration the values mentioned above, one can deduct the switching losses according to

the VDC and IAC_max.

PSW = 0.005 · VDC · IAC MAX 5-24

It was considered that the voltage drop on one IGBT is equal to the DC voltage divided by the number

of series connected IGBTs in a valve (Figure 5.10).

V_IGBT

Vdc/2

Vdc/2

Figure 5.10 – Voltage Drop ON an IGBT

The total losses are computed using equation 5-10.

Plosses = 0.0054 · VDC · IAC MAX+ 0.612 + 0.028 · ma · IAC MAX

+· IAC MAX2 · (0.27 + 0.1

· ma)

5-25

In order to have an optimization algorithm that uses only DC parameters as inputs, a relationship

between IAC_max and IDC has to be found. Starting from the power conservation equation the following

relation can be written:

PDC = PACAVERAGE+ Plosses 5-26

Since Plosses is expected to be small in comparison to PDC and PAC_AVERAGE (1-2%), it can be ignored when

computing IAC_max(IDC).

Expanding equation 5-26 one will obtain:

VDC · IDC = 3 · ma · 3

2 · 2· VDC ·

IAC MAX

2 5-27

59

IAC MAX=

4

3 · ma· IDC 5-28

The amplitude modulation factor ma is defined as the ratio between the peak of the sinusoidal control

signal and the amplitude of the triangular signal. In the case of a two level three phase converter, with

PWM modulation, which operates in the linear region (ma ≤ 1) the following relation can be found

between the AC side voltage, DC voltages and the amplitude modulation factor [19]:

ma =VLL RMS · 2 · 2

3 · VDC

5-29

Considering the VLL_RMS constant at 170 kV, ma will be equal to:

ma =278

VDC 5-30

By replacing IAC_max with IDC, equation 5-25 becomes:

Plo sses = 0.005 · VDC ·4

3 · ma· IDC + 0.612 + 0.028 · ma ·

4

3 · ma· IDC +

16

9 · ma2 · IDC

2

· (0.27 + 0.1 · ma)

5-31

By introducing equation 5-30 in equation 5-31, Plosses can be expressed in terms of VDC, IDC and PDC.

Plosses = 6.2 · 10−6 · PDC2 + 2.6 · 10−5 · PDC · VDC + 6.4 · 10−4 · PDC · IDC + 2.9 · 10−3 · PDC

+ 0.037 · IDC 5-32

Plosses = a · PDC2 + b · PDC · VDC + c · PDC · IDC + d · PDC + e · IDC 5-33

a = 6.2 · 10−6 [

1

W]

b = 2.6 · 10−5 [1

V]

c = 6.4 ∙ 10−4 1

A

d = 2.9 ∙ 10−3 e = 0.037 [V]

5-34

Presuming the voltage being kept constant at 1 p.u. (300 kV), Plosses becomes dependent both

quadratically and linearly on the DC current:

Plosses = 0.75 · IDC2 + 3.25 · IDC 5-35

60

The weight corresponding to the linear term is bigger, therefore the linearity of Plosses(IDC) is

pronounced. At nominal values of power and voltage (PDC = 1 p.u. = 400 MW, VDC = 1 p.u. = 300 kV) the

losses are equal to 5.66 MW, approximately 1.41% of the nominal power, comparable to the results form

[36].

Data regarding the losses (expressed in percentage of nominal power) at different power inputs and

voltages is presented in the table below:

PDC=0.125p.u. PDC=0.25p.u. PDC=0.375p.u PDC=0.5p.u. PDC=0.75p.u. PDC=1 p.u.

VDC=0.97p.u. 0.14% 0.29% 0.44% 0.61% 0.98% 1.40%

VDC=1 p.u. 0.14% 0.29% 0.45% 0.62% 0.99% 1.42%

VDC=1.03p.u. 0.14% 0.30% 0.46% 0.64% 1.02% 1.44%

Table 5.2 – Converter Losses at Different Power and Voltage Inputs

It can be seen that the losses grow with the increasing voltage.

Figure 5.11 – Converter Losses at Different Power and Voltage Inputs

In contrast, the line losses vary quadratically with the current, hence decrease with the voltage growth.

Moreover, they vary linearly with the total cable resistance, thus are proportional to the cable length. In

an HVDC system the cable lengths influence decisively the ratio between the converter and cable losses.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Power Input [p.u.]

Pow

er

Losses [

%]

Vdc = 1 p.u.

Vdc = 1.03 p.u.

Vdc = 0.97 p.u.

61

If the interconnecting line is short, the cable losses can be neglected. Consequently, the optimum way to

transmit energy would be decreasing the voltage as much as possible, in order to keep the converter

losses low. On the other hand, if the lines are long, the minimum total losses will be reached if the

voltage reaches the maximum value. For medium lines, though, an equilibrium between the DC current

and voltage must be reached in order to obtain maximum transmitted power. The statement will be

further analyzed in subchapter 5.5. A comparison between the converter losses and the line losses is

displayed in Figure 5.12.

Figure 5.12 – Converter Losses vs. Line Losses at V=1 p.u.

5.5 Loss minimization of three terminal MTDC system, taking

in consideration the line and converter losses

In order to have an optimization routine that reflects the power distribution in the real MTDC system,

the converter station losses have to be taken in consideration. This can be computed by adding the

converter losses from all three converters to the cable losses into the minimization function 5-36.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

Power Input [p.u.]

Pow

er

Losses [

%]

Converter Losses

Line Losses, R=1 ohm





62

Plosses = Plosses converter 1+ Plosses converter 2

+ Plosses converter 3+ Plost line 2

+ Plost line 3 5-36

f Ii , Vi , Pi = R2 ∙ I22 + R3 ∙ I3

2 + (a · Pi2 + b · Pi · Vi + c · Pi · Ii + d · Pi + e · Ii)

i

; i = 1,2,3 5-37

The minimization function has the following equality constraints:

I1 = I2 + I3 5-38

Ii =Pi

Vi; i = 1,2,3 5-39

V2/3 = V1 − I2/3 · R2/3 5-40

Due to the fact that the offshore converter has a big impact on the overall losses of the system,

especially when using short cables in the design, the cost function above is computed taking in

consideration the wind farm converter losses. In order to have a minimization function depending only

on the state variables I1,2,3 and V1, equation 5-37 will be further modified. Equation 5-41 is obtained.

f(I1 , I2 , I3 , V1) = R2 ∙ I22 + R3 ∙ I3

2 + a ∙ P12 + I2 ∙ V1 − R2 ∙ I2

2+ I3 ∙ V1 − R3 ∙ I3

2 + b

∙ P1 ∙ V1 + I2 ∙ V1 − R2 ∙ V2 2 + I3 ∙ V3 − R3 ∙ V3

2 + c

· I1 ∙ P1 + I22 ∙ V1 − R2 ∙ I2 + I3

2 ∙ V1 − R3 ∙ I3 + d

∙ P1 + I2 ∙ V1 − R2 ∙ I2 + I3 ∙ V1 − R3 ∙ I3 + e ∙ Ii

i=1,2,3

5-41

Since the optimization function is quite complex, a direct analytical approach to solve the problem was

too cumbersome. A method that implied fewer computations was chosen instead. The method can be

found in the Appendix A. It focuses on the linearization of the converter losses before being inserted in

the cost function. The table below presents the analytical results at nominal power.

V1 V2 V3 I1 I2 I3 P2 P3

0.986 0.98 0.98 1.01 0.69 0.32 0.67 0.31

Table 5.3 – Analytical Results (Line+Converter losses)

The optimization routine results when taking in consideration the simulation model, are presented

below. The mismatch is due to the linearization in the analytical model.

V1 V2 V3 I1 I2 I3 P2 P3 Plosses

0.97 0.965 0.964 1.03 0.651 0.38 0.628 0.366 0.0319

Table 5.4 – Optimization Routine Results (Line+Converter losses)

63

Figure 5.13 - Optimization Routine Results (Line+Converter losses)

Terminal 1 – red, terminal 2 – blue, terminal 3 - cyan

Taking in consideration the simulation model and observing Figure 5.12, the line losses are much lower

than the converter station losses, hence in order to have maximum power transmitted overall,

minimizing the converter station losses becomes a priority. Consequently, V1 is expected to reach the

lower limit of 0.97. The resistances still have an impact on the power sharing between the grid side

converters, most current flowing through the line with lower resistance.

2 3 4 5 6 7 8 9 10 11 120.96

0.98

1

1.02

1.04

Time [s]

Voltage [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

Time [s]

Pow

er

[p.u

.]

2 3 4 5 6 7 8 9 10 11 120

0.5

1

1.5

Time [s]

Curr

ent

[p.u

.]

64

Figure 5.14 - Voltage profiles for optimization routine with line and converter losses

Figure 5.14 offers a better overview of the voltage levels in the system. It can be observed that V1 decreases (in order to cope with the onshore voltages’ decreasing values) from the initial reference to the minimum limit thus keeping the system losses down to minimum.

Figure 5.15 - Ploss curve comparison

0,964

0,966

0,968

0,97

0,972

0,974

0,976

0,978

0,98

0,982

0,984


V1

V2

V3

0 1 pf2 pf1 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045


Pow

er

Losses [

p.u

.]

converter&line losses

line losses

65

Figure 5.15 shows the power losses during various sharing factor values. It can be seen that the sharing

factor obtained using the optimization routine gives the smallest losses for the MTDC system. In

comparison to the curve illustrating the optimization taking in consideration only the line losses, it can be

seen that the optimum power sharing factor has shifted from 2 to 1.72. Naturally, due to the additional

converter losses, the curve translated up on the Ploss axis (pf 1 represents the optimum sharing factor for

the MTDC system with line losses, while pf2 represents the optimum sharing factor for the MTDC system

with both line and converter losses). The big difference on the Y-axis denotes that the converter losses

have a higher weight in the calculation of the total system losses than the cable resistances. The

difference will be reduced with the increase of R2 and R3. This aspect will be further debated in the study

cases.

5.6 Loss minimization of three terminal MTDC taking in

consideration economical aspects

The energy markets deal with the trade of electrical energy. Since the liberalization of the energy

market, the producers try to sell the energy at the best possible price while the suppliers buy the

electricity from the producers and deliver it to the consumers. The prices are decided by balancing the

supply and demand of energy on the market and can change according to numerous factors. They can

vary depending on the time of the day, the day of the week, season, weather etc. [42], [43]

Having a wind power plant connected via an MTDC system to a minimum of two markets can constitute

a big advantage, since the provider can choose what market he should play in. Taking in consideration

the forecast produced power for the next day (or hour) and the prices prediction for the same period of

time, one can decide where and what quantity of power to send in order to have maximum profit. The

bids will be sent to the TSOs.

The following approach in the MTDC optimization routine analyzes the prices for the next 24h from the

two markets the provider will play in (Denmark and Germany in this case) and decides what power

quantities to send to each market. The decision is based on a cost function that minimizes the difference

between the maximum profit possible and the actual profit on the two branches.

f = Pwindfarm · max c2 , c3 − c2 · Pterminal 2+ c3 · Pterminal 3

5-42

c2 and c3 represent the energy prices per MWh on the two markets at a certain hour. Therefore the

cost function changes dynamically, depending on the time of the day, in order to obtain the maximum

profit.

f I1 , I2 , I3 , V1 = Pwindfarm · max c2 , c3 − c2 · I2 · V1 − I2 · R2 − c3 · I3 · V1 − I3 · R3 5-43

The constraints remain the same.

66

I1 = I2 + I3 5-44

I1 =P1

V1

5-45

By making the presumption that, in order to achieve maximum profit, V1 has to remain at its peak

value, I1 is computed from 5-45.

I3 is replaced by (I1-I2) in equation 5-43. Therefore one obtains:

f I2 = I22 · c3 · R3 + c2 · R2 + I2 · V1 · c3 − c2 − 2 · c3 · I1 · R3 − c3 · I1 · V1 + TP 5-46

TP – represents the maximum profit possible at the power produced by the wind farm.

By putting the condition df (I2)

dI2= 0, the currents and voltages corresponding to the maximum profit are

obtained.

I2 =2 · c3 · R3 · I1 − V1 · (c3 − c2)

2 · (c3 · R3 + c2 · R2)

5-47

I3 =V1 · c3 − c2 + 2 · c2 · R2 · I1

2 · (c3 · R3 + c2 · R2)

5-48

V2 = V1 −2 · c3 · R2 · R3 · I1 − V1 · R2 · (c3 − c2)

2 · (c3 · R3 + c2 · R2)

5-49

V3 = V1 −V1 · R3 · c3 − c2 + 2 · c2 · R2 · R3 · I1

2 · (c3 · R3 + c2 · R2)

5-50

Consequently, three cases can be distinguished based on the economical factors c2 and c3:

1) The price per MW/h on both markets is equal (c2=c3). In this case the optimum sharing factor

will coincide with the optimum sharing factor from chapter. Since there are no economical advantages on

neither of the markets, the maximum profit is obtained by minimizing the losses on the system. Figure

5.16 and Table 5.5 come to confirm the theoretical results.

67

Figure 5.16 - Optimization routine taking in consideration the economical factors (c2=c3)

V1 [p.u] V2 [p.u] V3 [p.u] I1 [p.u] I2 [p.u] I3 [p.u] P2 [p.u] P3 [p.u] Revenue

(Euros/h)

1.03 1.0248 1.0248 0.971 0.6473 0.3236 0.6633 0.3317 12337.65

Table 5.5 – Optimization routine results (c2 = c3)

2) The price per MW/h on the market corresponding to the transmission line with smaller losses is

higher (e.g.: c2>c3; R2<R3). Taking a look at equation 5-47, all the power will go through terminal 2 if

I2≥I1. Therefore:

2 3 4 5 6 7 8 9 10 11 121.02

1.025

1.03

1.035

Time [s]

Vdc [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

Time [s]

Pdc [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

Time [s]

Idc [

p.u

.]

68

I2 ≥V1 ∙ (c2 − c3)

2 ∙ c2 ∙ R2

5-51

c3

c2≤ 1 −

2 · R2 · I2

V1≤ 1 −

2 · R2 · I1

V1

5-52

For the simulation model taken in consideration in this chapter, the ratio c3/c2 has to be smaller or

equal to 0.985 in order for all the power from the wind farm to be transmitted on terminal 2.

In other words, in almost all cases when c2>c3 and R2<R3, the power goes on the terminal with smaller

losses. The theoretical results are confirmed in Table 5.6 and Figure 5.17. The following energy prices are

considered: c2 = 30.2 Euros/MWh; c3 = 29.74 Euros/MWh.

V1 [p.u] V2 [p.u] V3 [p.u] I1 [p.u] I2 [p.u] I3 [p.u] P2 [p.u] P3 [p.u] Revenue

(Euros/h)

1.03 1.0222 1.03 0.971 0.971 0 0.9925 0 11988.75

Table 5.6 – Optimization routine results (c2>c3)

Figure 5.17 - Optimization routine taking in consideration the economical factors (c2>c3)

2 3 4 5 6 7 8 9 10 11 121.02

1.025

1.03

1.035

Time [s]

Vdc [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120

0.5

1

Time [s]

Pdc [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120

0.5

1

Time [s]

Idc [

p.u

.]

69

3) The price per MW/h on the market corresponding to the transmission line with bigger losses is

higher (e.g.: c2<c3; R2<R3).

In this case, the power is shared between the terminals according to equations 5-47 and 5-48. Taking

c3 = 31.2 Euros/MWh and c2 = 31 Euros/MWh, the currents on the two terminals are: I2 = 0.681 kA =

0.511 p.u., I3 = 0.613 kA = 0.46 p.u. The voltages are: V2 = 1.023 p.u. and V3 = 1.026 p.u.

The simulation results come to confirm the theoretical values. The simulation values can be seen in

both Figure 5.18 and Table 5.7.

V1 [p.u] V2 [p.u] V3 [p.u] I1 [p.u] I2 [p.u] I3 [p.u] P2 [p.u] P3 [p.u] Profit

(Euros/h)

1.03 1.026 1.0226 0.971 0.511 0.46 0.524 0.4705 12369.76

Table 5.7– Optimization routine results (c3>c2)

Figure 5.18 - Optimization routine taking in consideration the economical factors (c3>c2)

2 3 4 5 6 7 8 9 10 11 121.02

1.025

1.03

1.035

Time [s]

Vdc [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120.4

0.6

0.8

1

Time [s]

Pdc [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120.4

0.6

0.8

1

Time [s]

Idc [

p.u

.]

70

Figure 5.19 displays the hourly profit for the system at different sharing factors. It can be seen that the

sharing factor obtained using the optimization routine gives the highest profit for two established prices.

Figure 5.19 - Profit comparison

It can be also be observed that the sharing factor corresponding to the biggest profit shifts to the left (I3

gets higher) when c3/c2 increases. The curve also translates up or down on the graph, according to the

price values. It should be taken into consideration that the efficiency of the optimization routine will

grow when the difference in price between the two markets grows. This will further be developed in the

simulations chapter.

5.7 Model validation. Extending the algorithm on a four

terminal MTDC system

In order to prove the robustness and versatility of the optimization algorithm, the routine was

extended on a four terminal model. The model is shown in Figure 5.2.

In order to keep the converter size to 400 MVA, the WPP sizes are reduced. The length between the

two WPP is considered to be small, thus the resistance between wind farms is smaller than either of the

other cable’s resistances. The simulation is performed with WPP1 supplying 300 MW (0.75 p.u.) and

WPP2 supplying 100MW (0.25 p.u.). The cables corresponding to the grid side terminals are kept to the

0 1 1.1 2 3 4 4.2 5 6 7 8 9 101.222

1.224

1.226

1.228

1.23

1.232

1.234

1.236

1.238

1.24

1.242x 10

4


Revenue [

Euro

s/h

]

c2 = c3 = 32 Euros/MWh

c2 = 32 Euros/MWh; c3 = 32.2 Euros/MWh

c2 = 32.2 Euros/MWh; c3 = 32 Euros/MWh

c2 = c3 = 32.2 Euros/MWh

71

same values. The DC cable resistance connecting the two wind farms is considered to be equal to 0.5 Ω.

Therefore the wind power plants are considered to be near each other.

The optimization on this setup is significantly more complex than in the 3 terminal MTDC. For this

reason only the theoretical calculus for the MTDC line loss optimization are going to be displayed. In the

case of the model with converter losses and in the economical analysis case, the cost functions and

constraints will be listed and the simulation results will be presented.

The simplified MTDC system is represented in Figure 5.20.

V4, P4

V3, P3x x

I1, P1

R2

R1

I5

x

I2, P2

R3

I3

I4x

V1

V2

Figure 5.20 - Simplified four terminal MTDC system representation

Starting from equations 5-1 and the cost function for the four terminal MTDC is derived.

f = Plost line 1+ Plost line 2

+ Plost line 3 5-53

f I3 , I4 , I5 = R1 · I32 + R2 · I4

2 + R3 · I52 5-54

The following constraints are taken in consideration:

I1 = I3 + I5 5-55

I2 = I4 − I5 5-56

V2 = V1 − I5 · R3 5-57

P1 = V1 · I1 5-58

P2 = V2 · I2 5-59

In order to simplify the calculations, some initial presumptions are applied. In order to have minimal

losses on the line, the voltages V1 and V2 have to be as big as possible. Therefore, either V1 or V2 will

reach the maximum limit of 1.03 p.u. (depending on I5’s flow in the circuit). The other voltage will be

72

computed with respect to equation 5-57. The power flow through cable 3 is expected to go from the WPP

with the higher power production to the WPP with the lower output power. Of course, the direction also

depends on the sharing factor between the two grid converters and the cable resistances (when no

converter losses or economical aspects are taken in consideration). If a large amount of current needs to

be sent on the terminal corresponding to converter number 3, than the power flow on the third cable will

be reversed. Presuming I5 flowing in the direction indicated in Figure 5.20, V1 is considered to reach 1.03

p.u. I1 will be calculated by dividing the power generated by the first wind farm to the voltage V1.

By applying the Lagrange multipliers on equation 5-54, the following formula is obtained:

f I3 , I4 , I5 = R1 · I32 + R2 · I4

2 + R3 · I52 + λ1 · I3 + I5 − I1 + λ2 · I4 − I5 − I2 5-60

Deriving f in function of I3, I4, I5, λ1 and λ2 and replacing I2 with P2

V1−I5·R3 one reaches the system

presented below:

2 · I5 · R3 + λ1 − λ2 = 02 · I3 · R1 + λ1 = 02 · I4 · R2 + λ2 = 0

I1 = I3 + I5

P2 = I4 − I5 · (V1 − I5 · R3)

5-61

I5 is obtained from 5-61.

I5

=V1 · Ri

31 − R1 · R3 · I1 − (V1 · Ri

31 − R1 · R3 · I1)2 − 4 · (P1 · R1 − P2 · R2) · R3 · Ri

31

2 · R3 · Ri31

5-62

Furthermore, V2, V3 and V4 can be computed based on I5. V2 is calculated from 5-57, I2 is equal to P2

V2

while the converter voltages are given by equations 5-63 and 5-64.

V3 = V1 − (I1 − I5) · R1 5-63

V4 = V2 − (I2 + I5) · R2 5-64

By introducing the simulation model values in the above equations the following results are obtained:

- V1 = 309 kV = 1.03 p.u.; V2 = 308.95 kV = 1.0298 p.u.; V3 = 307.43 kV = 1.0247 p.u.;

V4 = 307.43 KV = 1.0247 p.u.

- I5 = 0.0988 kA = 0.0741 p.u.; I1 = 0.97 kA = 0.728 p.u.; I2 = 0.32 kA = 0.24 p.u.; I3 = 0.87 kA

= 0.65 p.u.; I4 = 0.42 kA = 0.315 p.u.

It can be observed that the line losses are not expected to increase by much, in comparison to the 3

terminal MTDC. This is due to the small value of the resistance between the two WPP.

73

The behavior of the line loss optimization routine is displayed in Figure 5.21, the results are illustrated

in Table 5.8. Their values are expected to match the theoretical results.

Figure 5.21 - Line Loss Optimization Routine

WPP 1 – red, WPP 2 – blue, line 1 – cyan, line 2 – green, line 3 - yellow

V1

[p.u]

V2

[p.u]

V3

[p.u]

V4

[p.u.]

I1

[p.u.]

I2

[p.u.]

I3

[p.u.]

I4

[p.u.]

I5

[p.u.]

P3

[p.u]

P4

[p.u.]

1.03 1.0298 1.0248 1.0248 0.7282 0.2428 0.6542 0.3167 0.0739 0.6704 0.3245

Table 5.8– Line Loss Optimization on 4 Terminal MTDC

2 3 4 5 6 7 8 9 10 11 121.02

1.025

1.03

1.035

Time [s]

Vdc [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

Time [s]

Pdc [

p.u

.]

2 3 4 5 6 7 8 9 10 11 120

0.2

0.4

0.6

0.8

Time [s]

Idc [

p.u

.]

74

It can be observed that the optimized sharing factor has increased (in comparison to the 3 terminal

MTDC optimization) from a value of 2 to 2.06. Moreover, the grid-side converter DC voltages V3 and V4

remain equal. As anticipated, the total line losses (Plosses=0.00672) are higher than in the 3 terminal case.

The small resistance between the two WPP permits the power to flow without big losses in the circuit,

therefore the power transmitted to the grids is comparable to the 3 terminal case. The voltage histogram

is presented in the figure below. It can be seen how V1 and V2 get close to the limit of 1.03 p.u., with the

grid side converter voltages slowly reaching the preset value of 1.0248.

Figure 5.22 – 4 Terminal MTDC Voltage Histogram

In the last part of the subchapter, the 4 terminal MTDC models with converter losses and economical

aspects will be highlighted. Due to the fact that the calculations are too complex to solve via analytical

methods, no theoretical results will be presented. The cost functions and constraints will be presented

for each case. In order to simulate these complex MTDC loss optimization issues, a graphical interface has

been created.

Hence, when taking in consideration the converter losses, equation 5-54 becomes:

Plosses = Plosses converter 1+ Plosses converter 2

+ Plo sses converter 3+ Plost line 2

+ Plost line 3 5-65

f Ii , Vi , Pi = R2 ∙ I22 + R3 ∙ I3

2 + R1 · I12 + a · Pi

2 + b · Pi · Vi + c · Pi · Ii + d · Pi + e · Ii

i

;

i = 1,2,3,4

5-66

a, b, c, d, e are the parameters computed in Chapter 5.4

The constraints remain the same as in the line loss optimization part.

1,016

1,018

1,02

1,022

1,024

1,026

1,028

1,03

1,032


V1

V2

V3

V4

75

In order to make the function dependent on the state variable vector given by: x = [I1, I2, I3, I4, I5, V1,

V2]T, the following replacements are made: V3 = V1 − I3 · R1; V4 = V2 − I4 · R2; Pi = Vi ∙ Ii; i = 1,2,3,4

In the case of optimization taking in consideration the economical aspects, the economical cost

function is equal to:

f I1 , I2 , I3 , I4 , I5 , V2, V3 = (PWPP 1 + PWPP 2 ) · max c3, c4 − c3 · I3 · V1 − I3 · R1 −

−c4 · I4 · V2 − I4 · R2 5-67

c3 and c4 are the energy costs corresponding to the two markets. Like in the 3 terminal case, the cost

function represents the difference between the maximum available revenue (obtained when sending all

the power coming from the wind farm, with no losses taken in consideration, to the terminal with a

higher energy price) and the actual revenue (depending on the amount of power transmitted and the

energy cost) at each grid side terminal.

The constraints remain the same as in the previous 4 terminal MTDC cases.

5.8 The MTDC Loss/Economical Optimization GUI

Due to the infinite number of scenarios that can be analyzed, a graphical interface was built in order to

allow the user to further explore the behavior of the 3 and 4 terminal MTDC systems. The GUI is

presented in Appendix C. The MTDC models behind the interface are designed in PSCAD. Therefore, not

only does the GUI optimize the MTDC circuit according to the user’s demands, but it also communicates

with the PSCAD models in quasireal time.

Among the features, it is worth mentioning the possibility to choose between a 3 or 4 terminal system

and to select the optimization type. The user can minimize the losses in the circuit (line or converter+line)

or can maximize the profit of the system with regard to the energy prices for the two markets and the

electrical losses. He also has the possibility to turn the optimization feature off and manually select a

sharing factor (I2

I3) the system should follow, in order to compare and contrast the results to the

optimization cases.

The results can be observed on the three displays (DC voltage, DC current and DC power). Moreover,

the textboxes keeping track of the outputs offer an insight on the state of the system (e.g.: the textboxes

display the energy prices, total revenue, instantaneous losses in the system, the optimization routine

voltages, currents and terminal powers that the MTDC should achieve etc.). Most of the results are

displayed in a per unit system.

The economical part of the interface lets the user insert the prices in a text file. The file will be read and

interpreted by the optimization routine. Regarding the wind farm powers, they also can be read from text

files. This feature allows the user to insert power steps as inputs and analyze the system behavior.

76

Moreover, observations taking in consideration real wind profiles can be made by inserting a text file

with the wind speeds at different moments. Therefore the MTDC system’s ability to cope with real

conditions can be explored.

Among other parameters that can be changed one can find the line resistances and the simulation

time.

77

6. Study cases

The purpose of the chapter is to build a series of scenarios that not only confirm that the system

functions properly but also prove its worthiness. The advantages of the optimization routine will be

highlighted in different situations and conclusions will be drawn.

The first scenario puts emphasis on the optimization routine including the economical aspects. The

second scenario offers an insight into the optimization routine with the converter losses and analyzes the

system’s behavior with the increase of line length.

The third scenario focuses on the four terminal MTDC and analyzes the system behavior during a trip at

one of the wind farms.

6.1 Analysis of the optimum power distribution in a MTDC

system during 24h in order to maximize the revenue from two

markets

With the power markets becoming competitive, the generator companies can send the generated

power to power distributors, directly to the consumers or sell it in an energy pool [43]. As mentioned in

subchapter 5.6, a wind farm connected to a MTDC system may offer the provider the possibility to

distribute the energy on more than one market. The decision where to send the power can be made

based on obtaining the biggest profit. The price for a MWh of energy varies hourly in a single day,

therefore a dynamic profit optimization should be implemented in order to get the maximum revenue.

The supplier has to forecast the energy generation and the prices at different hours during the day.

Afterwards the optimization routine is performed and the bids can be made. Moreover, on optimization

routine can be performed with the excess energy generated by the wind farm. This can be sold on the

energy spot market according to the results.

In order to prove the worthiness of the economical optimization routine, a simulation performed with

real energy prices is going to be carried out over a period of 4 minutes. 10 seconds in the simulation are

associated with one hour in real-time, therefore, the hourly energy prices will change once every 10

seconds. The total revenue is going to be computed in the end and will be compared to the revenue

obtained when running the optimization routine in line loss minimization mode.

The prices for the energy are shown in Euros/MWh and are taken from the European Energy Exchange

and European Power Exchange websites. The following table contains information from the 9th of May,

2012 regarding the energy prices in Denmark and Germany.

78

Hourly Interval (h)

Price per MWh Denmark (Euros/MWh)

Price per MWh Germany (Euros/MWh)

00 – 01 36.1 40.99

01 – 02 33.60 36.00

02 – 03 31.20 33.13

03 – 04 28.60 30.53

04 – 05 31.10 33.09

05 – 06 36.21 35.55

06 – 07 47.20 47.21

07 – 08 54.90 54.93

08 – 09 56.00 56.03

09 – 10 61.10 61.18

10 – 11 59.03 59.04

11 – 12 60.90 60.96

12 – 13 57.70 57.73

13 – 14 55.10 55.46

14 – 15 53.20 53.53

15 – 16 50.10 50.28

16 – 17 48.00 49.25

17 – 18 51.40 52.10

18 – 19 52.00 52.10

19 – 20 52.90 53.39

20 – 21 55.10 55.39

21 – 22 57.40 57.00

22 – 23 54.90 52.90

23 – 00 44.20 44.20

Table 6.1– Energy Prices 09/05/2012

The simulation routine results are shown in Figure 6.1. The total revenue for 24h is equal to 469950

Euros.

It can be seen that the revenue prone optimization function adjusts the sharing distribution according

to the prices. If the price difference between the two markets is high relative to the cable resistances

(according to equations 5-46 and 5-47) during a moment of the day, all the energy will flow towards the

region with the higher price. Otherwise, the optimization function will compute an optimal sharing factor,

corresponding to the biggest profit that can be obtained. In order to have a comparison, a simulation in

which the power was sent to the market with a bigger energy selling price was performed. The profit

obtained was 467791 Euros/day, 2159 Euros smaller than the optimized result. This translates into a

yearly revenue increase of approximately 800000 Euros.

79

Figure 6.1 – 24h Economical Optimization Routine Results

In conclusion, it was proven that the economical optimization routine is capable of turning a

considerable profit during an extended period of time. In order for the routine to be worthwhile, the

market prices have to be comparable. Otherwise, the system will receive all the wind farm power to the

terminal with higher price (action that can be performed without an optimization algorithm).

0 0 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241

1.025

1.05

Time of Day [h]

Voltage [

p.u

.]

0 0 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

0.2

0.4

0.6

0.8

1

Time of Day [h]

Pow

er

[p.u

.]

0 0 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

0.2

0.4

0.6

0.8

1

Time of Day [h]

Curr

ent

[p.u

.]

80

6.2 Analysis of the influence of cable length in determining

the optimum operating voltages in a three terminal MTDC

system

In Chapter 5.5 it was proven that for short lines the optimum voltage references are kept as low as

possible, in order to have maximum power transmission. This is due to the fact that the converter losses,

which are greater than the line losses, have the tendency to decrease when the DC voltage reduces. For

longer cables, however, the line current starts to play a key role in determining the optimum operation

point (since energy lost in a cable is quadratically dependent on the DC current) a balance between the

optimum currents and voltages has to be achieved.

The power level that is produced by the wind farm is also an essential factor when determining the

optimum operation voltages and currents. Therefore the study case will also include optimum operation

analysis at different power steps.

Table 6.2 offers an overview of the optimum voltages and currents when varying the length of a cable.

The power input is kept constant.

P1 [p.u.]

Cable2 [km]

Cable3 [km]

V1 [p.u.]

V2 [p.u.]

V3 [p.u.]

I1 [p.u.]

I2 [p.u.]

I3 [p.u.]

Plosses [%]

1 64.3 128.6 0.97 0.964 0.963 1.03 0.651 0.38 3.19

1 82.1 128.6 0.97 0.965 0.963 1.03 0.607 0.424 3.28

1 89.3 128.6 0.975 0.97 0.968 1.026 0.588 0.438 3.31

1 100 128.6 0.994 0.99 0.987 1.005 0.554 0.452 3.35

1 117.9 128.6 1.022 1.018 1.015 0.979 0.507 0.472 3.41

1 132.14 128.6 1.03 1.023 1.022 0.971 0.48 0.491 3.46

Table 6.2– Optimum operation point when varying the 2nd cable’s length

It can be observed that, with the increase of the line length, the cable losses have a higher weight in

determining the optimum operation point of the MTDC. For short lines the voltage will go to the

minimum value in order to keep the converter station losses as low as possible. On the other hand, when

the cable lengths grow, the optimization function will gradually put more emphasis on rising the voltage

levels (in order to keep the line losses low). When the line lengths are high enough, the WF voltage level

will reach the maximum allowable level of 1.03 p.u.

The combined line length for which the voltage starts to rise from the lowest allowed value is equal to

214 km. The total line length for which the optimum operation point voltage (V1) reaches 1.03 p.u. is

equal to 263 km. The rate of increase in the optimum voltage values can be seen in Figure 6.2.

81

Figure 6.2 – Optimum Voltages Variation in Function of Cable Length

Moreover, Table 6.2 shows how the currents vary according to the line resistances. As the second

cable’s resistance increases, the current intensity drops. As predicted, when R2 is bigger than R3, I3 will

become larger than I2.

Further, the optimum operation point of the system having increasing power steps as input is analyzed.

The reason behind this analysis is that when the WPP produces less than the nominal power, the current

on the DC buses drops. Thus, the energy dissipated on the cables diminishes. The voltage, on the other

hand, stays within the normal operation limits (V1 = 1 p.u. ± 3%). Consequently, the converter stations’

influence over the total losses of the system increases. As a result, the loss optimization algorithm will try

to keep the voltages bound to the lower limit for small power inputs. The voltage levels are expected to

rise as the WPP power approaches the nominal power.

The loss optimization routine results at different input power levels are displayed in Table 6.3.

Furthermore, the system’s response to the power steps can be observed in Figure 6.3. The loss

optimization results match the expectations, while the MTDC system’s control successfully tracks the

references set by the algorithm.

0,92

0,94

0,96

0,98

1

1,02

1,04

100 km 193 km 211 km 218 km 229 km 247 km 261 km 268 km

Vo

ltag

e [p

.u]

Total Line Length [km]

V1 [p.u.]

V2 [p.u.]

V3 [p.u.]

82

P1 [p.u.]

Cable2 [km]

Cable3 [km]

V1 [p.u.]

V2 [p.u.]

V3 [p.u.]

I1 [p.u.]

I2 [p.u.]

I3 [p.u.]

Plosses [%]

0.7 132.14 128.6 0.97 0.964 0.964 0.962 0.476 0.487 2.14

0.75 132.14 128.6 0.97 0.964 0.964 0.773 0.382 0.391 2.35

0.8 132.14 128.6 0.9701 0.963 0.963 0.824 0.408 0.417 2.56

0.85 132.14 128.6 0.985 0.978 0.978 0.862 0.426 0.436 2.77

0.9 132.14 128.6 1 0.997 0.997 0.896 0.443 0.453 3

0.95 132.14 128.6 1.022 1.015 1.015 0.929 0.459 0.47 3.22

1 132.14 128.6 1.03 1.023 1.022 0.971 0.48 0.491 3.46

Table 6.3– Optimum operation point when varying the power input

Figure 6.3 – MTDC System’s Response to the Increasing Power Steps

10 20 30 40 50 60 700.95

1

1.05

Time [s]

Voltage [

p.u

.]

10 20 30 40 50 60 700.2

0.4

0.6

0.8

1

Time [s]

Pow

er

[p.u

.]

10 20 30 40 50 60 700.2

0.4

0.6

0.8

1

Time [s]

Curr

ent

[p.u

.]

83

6.3 Analysis of optimum power sharing in a four terminal

system after Wind Farm trip

The simulation for this study case was performed on a four terminal system to present the optimized

power sharing between the receiving stations under the condition of disconnecting one wind farm (WPP

2, from Figure 5.2. The optimization algorithm considered is the minimization of line losses.

For this study case, the system operates in the beginning with WPP 1 generating 0.75 p.u. and WPP 2 –

0.25 p.u. of power (respectively 300 MW and 100 MW). The step is applied at t = 12 [s] after which WPP 2

is disconnected (i.e. power drops to 0 p.u.), while WPP 1 keeps generating at the level of 0.75 p.u.

The effect of the trip on the system voltages, current and power distribution are presented in Figure

6.4.

Figure 6.4 – Line loss optimization for Study Case 3

2 4 6 8 10 12 14 16 18 20 220.85

0.9

0.95

1

1.05

Time [s]

Voltage [

p.u

.]

2 4 6 8 10 12 14 16 18 20 220

0.2

0.4

0.6

0.8

Time [s]

Pow

er

[p.u

.]

2 4 6 8 10 12 14 16 18 20 22

0.2

0.4

0.6

0.8

1

Time [s]

Curr

ent

[p.u

.]

84

The disconnection of WPP 2 results in reducing the amount of active power injected into the DC link,

since WPP 1 becomes the only source of active power under such conditions. This leads to disturbances

in the DC link voltage, since the amount of energy needed to keep the voltages constant and within

specified limits also drops. The Grid Side Converters, operating in voltage control mode, manage to

maintain the system stable and a constant DC link voltage is restored (as presented in Figure 6.4).

Before the disconnection of WPP 2, both wind farms generate to their own grids with little current (0.15

p.u.) flowing from WPP 1 to Grid 2 through resistance R3. The majority (0.85 p.u.) of power from WPP 1

is sent to Grid 1 since resistance R1 is significantly smaller than the combined resistances of R2 and R3.

Grid 2 receives all of the power from WPP 2 with an additional 0.15 p.u. from WPP 1. This can be

explained by the optimization algorithm working to minimize the I2R losses and evacuating some of the

WPP 1 current into Grid 2.

After disconnecting WPP 2, Grid 2 gets entirely supplied from WPP 1, resulting in an increase of current

flowing through the interconnecting branch (yellow waveform in Figure 6.4) and decrease of DC voltage

V2 on the terminal of WPP 2.

85

7. Laboratory work

The presented optimization algorithm has been implemented in a scaled down laboratory platform for

verification. The schematic of the laboratory set-up is presented in Figure 7.1 - Schematic of laboratory

set-up.

DC POWER

SOURCE

dSpace

Control

System

PWM

PWM

PC

Controldesk +

Optimization

Serial

communication

GRID 1

GRID 2

HVDC

CABLE

VSC 1

VSC 2

FILTER

FILTER

VDC2*, P2*

VDC3*, P3*

Optimized V2*, V3*

P1, V1,I1,I2,I3

IDC2

VDC2

IDC3

VDC3

Iabc, Vabc

Iabc, Vabc

R2

R3

I1 I2

I3

V2

V3

V1

Figure 7.1 - Schematic of laboratory set-up

The DC power source represents the Wind Farm together with the offshore converter station and

injects current into the DC link, which is shared between the two converters according to reference set by

the droop controllers. A simplified model of the HVDC cable is considered with only the resistive

component taken into account. Danfoss VLT FC-302 converters are used as grid side converters,

represented by VSC 1 and VSC 2 in Figure 7.1. Control for the converters is implemented in the dSpace

system connected to a PC running MATLAB/Simulink and ControlDesk graphical user interface.

Furthermore, the optimization algorithm is executed offline from the PC. A serial communication link is

established between the PC and dSpace interface board for the purpose of receiving measurement data

from the system and sending the optimized voltage references for the converter control system. A

detailed technical specification of the system is presented in Appendix B.

86

The voltage references for the controllers may be obtained from two different sources – either directly

from the optimization algorithm, or indirectly by setting a desired sharing factor k.

In the case of adjusting the sharing factor, the final voltages are obtained from the following steps,

presented in equations from 7-1 to 7-8. Firstly the sharing factor parameter is defined by the ratio of DC

link currents and the KCL for the circuit.

2

3

Ik =

I 7-1

1 2 3I = I + I 7-2

After which the DC link currents I2 and I3 are found in function of the input current I1:

12

kII =

(k+1) 7-3

13

II =

(k+1) 7-4

The voltages at the receiving stations are defined as by equations 7-5 and 7-6.

2 1 2 2V = V -R I 7-5

3 1 3 3V = V -R I

7-6

Which, after substituting the current equations from 7-3 and 7-4 turn to the final form:

1 22 1

kI RV = V -

(k+1) 7-7

1 33 1

I RV = V -

(k+1) 7-8

The optimization algorithm communicates with dSpace via a RS-232 connection. The algorithm samples

data regarding the currents, the voltages and the WPP’s power, performs the same loss/economical

optimization routine used in the simulations and outputs the voltage references for the two converters.

The boundaries and constant parameters were adjusted in order to meet the parameters and demands

of the laboratory setup. In order to exclude the erroneous information (due to the noise that can appear

on the RS-232 line), the routine samples an array of data for each of the input variables and applies basic

filtering procedures that exclude the faulty values and performs an average on the rest. The same

procedure applies for an output signal. An array of droop values will be sent for each of the two

converters only to be sorted and averaged by the dSpace model. Because the RS-232 connection

supports only the transmission of uint8 variables, a data class conversion is performed before and after

sending or receiving the information. Uint8 variables are written on 8 bits, therefore their range varies

from 0 to 28-1. This means 256 levels of information for each variable.

87

As mentioned before, prior to the transmission each variable is converted into a corresponding

information level. After this process, the accuracy of the transmitted information will suffer, depending

on the resolution of the conversion. The resolution can be improved by either establishing a narrower

range between the minimum and the maximum values of the variable to be transmitted or by splitting

the variable that needs to be sent into several parts, each corresponding to a uint8 variable. After the

transmission, the information can be rebuilt.

The following subchapters contain a presentation of the study cases performed during laboratory work.

The purpose of the study cases performed on the laboratory platform was testing the power sharing

between the two converters under the conditions of the optimization algorithm turned off and turned

on. In the case of optimization being deactivated, the different reference voltages for the converter

stations were obtained by adjusting the sharing factor.

7.1 Study case 1 – Power sharing through sharing factor

adjustment

The objective of this study case is to observe the changes in power sharing and current distribution

for different values of the sharing factor parameter with the optimization algorithm turned off. The tests

were performed for a value of k equal to 1, 2, 3, 0.5 and 0.33. Laboratory results are validated on a

simulation model and both results are presented.

Results for k = 1

Figure 7.2 – Current sharing for k=1

88

Figure 7.3 – DC voltages for k=1

k = 1 Measured Values Simulation Values

Currents I1 = 5 A; I2 = 2.45 A; I3 = 2.55 A; I1 = 5 A; I2 = 2.5 A; I3 = 2.5 A;

Voltages V2 = 615 V; V3 = 614.5 V; V2 = 614.8 V; V3 = 613.7 V;

Table 7.1 - Laboratory and simulation results for k=1

Results for this value of k present equal sharing among both converter stations. The injected current of

5A from the DC source is equally shared among VSC 1 and VSC 2, with an average of 2.5 A flowing

through each branch (Figure 7.2).

Results for k = 2

89





Voltages V2 = 613 V; V3 = 615.5 V; V2 = 613 V; V3 = 615.8 V;


Setting the sharing factor to a value of 2 results in obtaining the ratio of currents: I2:I3 = 2:1 (Figure 7.4).

90

Due to a high current I2 as well as line resistance R2, the voltage drop on this resistance is greater than on

R3, resulting in a lower voltage V2 on the terminals of converter station VSC 1 (Figure 7.5), according to

equation 7-5.

Results for k=3



For a value of k=3, the I2:I3 currents ratio grows to 3:1, as a result having an even greater effect on the

voltage differences between the converter stations (Figure 7.7).

91



Voltages V2 = 612.5 V; V3 = 617 V; V2 = 612.1 V; V3 = 616.9 V;


Results for k=0.5

Figure 7.8 – Current sharing for k=0.5

Figure 7.9 – DC voltages for k=0.5

92

k = 0.5 Measured Values Simulation Values


Voltages V2 = 617 V; V3 = 612 V; V2 = 616.5 V; V3 = 611.7 V;

Table 7.4 - Laboratory and simulation results for k=0.5

For a sharing factor of 0.5, more current will flow through VSC2. The measured DC voltages on the

converter stations also shift – due to a higher voltage drop on R3, voltage V3 is now below V2.

Results for k=0.33

Figure 7.10 – Current sharing for k=0.33

Figure 7.11 – DC voltages for k=0.33

93

k = 0.33 Measured Values Simulation Values



Table 7.5 - Laboratory and simulation results for k=0.33

Sharing factor of 0.33 sets the I2:I3 ratio to 1:3, as presented in Figure 7.10.

The study case presented a validation of the laboratory model, showing that correct sharing is achievable

by varying the sharing factor parameter. The distribution of currents and voltages in the converter

stations followed the desired sharing factor and results were as expected. The DC voltages of the

converter stations kept within the limits of +/- 3% at all times.

7.2 Study case 2 – Optimized power sharing

This study case was performed with the optimization algorithm turned on. The design goals of the

optimization was minimizing the line losses during the first run and minimizing the line plus converter

losses during the second run. The resistances of the lines have been modified for obtaining a clearer

picture of the sharing. R2 in this case is set to 2.1 Ω and R3 to 5 Ω. For this study case, a serial link

connection was established between the PC and dSpace interface card. The PC received on the input

measured quantities of the MTDC system (P1, V1, I1, I2, I3 ) and sent the result of the optimization in the

form of reference voltages V2 and V3.

Results for line losses optimization

Figure 7.12 – Optimized current sharing for line losses

94

Figure 7.13 – Optimized DC voltages for line losses

Line Optim. Measured Values Simulation Values



Table 7.6 - Laboratory and simulation results for line loss optimization

By analyzing the sharing results under the condition of line loss optimization turned on, the conclusions

are the following:

- Higher current is passed through the line with lower resistance R2 corresponding to VSC1.

- However, not all of the current flows through the line with lower resistance. A significant amount

of current is also passed through R3. This can be explained by the fact that the main contribution

to the line losses comes from the current, i.e. it influences the line losses quadratically , according

to the formula I2 R. Thus, increasing the current flow through branch R2 does not satisfy the

conditions for optimization, although the equivalent resistance is smaller.

95

Results for line and converter losses optimization

Figure 7.14 – Optimized current sharing for line and converter losses

Figure 7.15 – Optimized DC voltages for line and converter losses

96

Conv .Optim. Measured Values Simulation Values


Voltages V2 = 622.9 V; V3 = 622.9 V; V2 = 622.5 V; V3 = 622.5 V;

Table 7.7 - Laboratory and simulation results for line and converter losses optimization

Due to the fact that the resistances are big, the impact the converter losses have in the system is very

small (the fact that there are only two converters in the system also contributes to this). Therefore, the

voltage profiles are expected to remain the same as in the line loss minimization case. A small difference

in voltage is expected, as a consequence of the voltage drop on the converter’s switches. Moreover, the

sharing factor decreases by a very small percent.

97

8. Conclusions and future work

The chapter presents the conclusions drawn subsequent to the simulation analyses and laboratory work

performed throughout the project. In the last part of the chapter, several possible future work topics are

listed.

8.1 Conclusions

The project was aimed at successfully building and testing an optimization routine that either

minimizes the power losses or maximizes the revenue gained from selling the electrical energy for a

MTDC system. Two MTDC systems were built and simulated in PSCAD, a 3 terminal system with one

offshore and two onshore converters and a 4 terminal system with two WPPs and two grid side VSCs. In

both cases, the grid converters’ outer loop operates in DC voltage control mode. The reference voltage is

set by the optimization algorithm via droop control. The optimization algorithm was built in Matlab,

therefore a program that establishes a communication link between the latter and PSCAD had to be

elaborated.

System loss minimization and revenue maximization were the main priorities when building the MTDC.

Consequently, more than one optimization function was designed and used (constraints and limits were

added to the optimization where needed). Each simulation is accompanied by an analytical analysis that

computes and confirms the optimization results.

Firstly the system’s line losses were analyzed. The optimization algorithm had the task to minimize the

cable losses in the circuit. It was demonstrated (both analytically and via simulation) that, for the 3

terminal MTDC system, the currents on the two receiving branches have a ratio inversely proportional to

the ratio of the equivalent resistances of the cables. Moreover, the WPP DC side voltage reaches

maximum allowable value, in order to keep the line losses as low as possible, with the grid terminal

voltages having equal values.

Secondly, the converter losses were added to the system losses equation. A novel converter loss

function was build for this purpose. The function is sensitive to both DC voltage and current changes. It

was observed that the function is more susceptible to voltage changes than to current changes at values

close to the nominal voltage. This translates into smaller converter loss values at voltages equal to the

lower voltage limit. The simulations attest that for small cable lengths, the voltages obtained with the

optimization algorithm tend to go to the lower limit. Analytical analysis also confirms the results. Further

studies performed in Chapter 6 studied the variation of the terminal voltages with the cable length of the

MTDC system. Predictably, the influence of the cable losses in the objective function gradually grows with

the cable length. Eventually, a balance between keeping the voltages as low as possible in order to keep

the converter losses low and increasing the voltages as much as possible in order to have cable losses as

small as possible must be achieved. In the case of very long cables, the smallest overall losses are

98

achieved by making the voltages go to the upper limit. The receiving terminals’ current ratio is also

affected by the converter losses. The sharing factor loses its exclusive dependency on the cables ratio due

to the current dependent losses in the converter. Therefore, the sharing factor decreases or increases

(depending on whether the line losses sharing factor is higher or lower than 1) towards unity.

Further, an economical optimization was performed on the MTDC system. Based on the presumption

that the two grid terminals correspond to two separate energy markets with comparable energy prices

and that the energy producer can unobstructedly send any amount of energy to the market of his choice,

the best voltage profiles, corresponding to the maximum revenue of the system, are chosen. It was

proven that the losses in the circuit play a significant role in choosing the amount of power that has to be

sent to each terminal and that the revenue maximization algorithm is more efficient than sending all the

available power to the market that has a higher energy price. It should be noted that for big price

differences or for similar losses on the grid terminals the profit maximization algorithm will send the

power to the terminal with the highest price.

In the case of the 4 terminal MTDC, the analyses (both analytical and simulations) presented in the

project took in consideration only the line losses. For the other cases, a graphical interface was created to

further study the behavior of such systems. A study case regarding the system’s behavior in case of a trip

at one of the wind farms is presented in Chapter 6. The results confirm that the MTDC system can

continue to function properly given the conditions. Its behavior matches a 3 terminal system, with the

cable resistance linking the two WPPs adding to the line corresponding to the terminal directly connected

to the wind farm that tripped.

The laboratory work involved building a downscaled model of the MTDC system and testing the

optimization algorithm in real life. Several problems regarding the physical setup (e.g.: voltage offsets in

the DC measurements, high system sensitivity due to small line resistances, the DC source’s limited

power) were encountered and dealt with before collecting the data displayed in Chapter 7. The manual

sharing, line and converter & line optimization algorithms were successfully tested. In conclusion, the

laboratory results match the simulations. However, further tests must be performed and power analyzers

should be included in the circuit to attest the fact that the smallest DC losses are achieved when running

the system with the optimization algorithm on.

8.2 Future Work

A lot of improvements can be brought to a project of such scale. In terms of simulation analysis, the

PSCAD models can be given a more detailed approach to the HVDC components modeling (e.g.: more

realistic power plant behavior, real cable models, LCL filters, switching converter model etc). Moreover, a

more thorough economical analysis must be performed and realistic constraints (both economical and

technical) must be taken in consideration when performing the economical revenue maximization.

Finally, in the simulations case, additional terminals can be inserted in the MTDC setup and the new

99

systems behavior should be further studied. The GUI interface should also be modified in such a way that

it should allow the simulation of setups with more than 4 terminals.

Regarding the laboratory, future tests can be performed to test the efficiency and sturdiness of the

optimization algorithms. Am mentioned in the previous subchapter, power analyzers can be introduced

in the circuit in order to validate the minimization of the losses.

Moreover, the study cases presented in Chapter 6 can be tested on the real setup. With reference to

the physical setup, a more powerful power source, which can simulate the behavior of a real WPP, should

be used. The RS-232 communication between the computer running the optimization algorithm and

dSpace can be improved by introducing better noise filtering algorithms and by improving the resolution

of the transmitted data.

Furthermore, the introduction of a WPP converter can be an interesting addition to the setup. This

would be extremely useful if a realistic analysis on the converter losses impact over the MTDC system

were to be performed.

Lastly, the system can be extended, and additional WPP and grid converters can be inserted in the

setup, giving the user a better insight on the future supergrid concepts.

100

101

Works Cited

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21st century: building a more sustainable future.

2. Council, Global Wind Energy. GWEC Global Wind Report: Annual Market Update 2010. [Online]

http://www.gwec.net/fileadmin/images/Publications/GWEC_annual_market_update_2010_-

_2nd_edition_April_2011.pdf.

3. Association, European Wind Energy. http://www.wind-energy-the-facts.org/en/part-i-

technology/chapter-3-wind-turbine-technology/evolution-of-commercial-wind-turbine-

technology/growth-of-wind-turbine-size.html. [Online]

4. European Wind Energy Association. Pure power - Wind Energy Targets for 2020 and 2030, Tech. Rep.,

EWEA, 2009. [Online]

http://www.ewea.org/fileadmin/ewea_documents/documents/publications/reports/Pure_Power_Full_R

eport.pdf.

5. Remus Teodorescu, Pedro Rodriguez, Sanjay Chaudhary, Andrzej Adamczyk, Osman S. Senturk.

Aalborg University lecture slides: High Power Converters.

6. Michael Häusler, ABB. Multiterminal HVDC for High Power Transmission in Europe.

7. Transmission and Distribution Networks: AC versus DC. D.M. Larruskain, I. Zamora.

8. HVDC – A key solution in future transmission systems; ABB . Olof Heyman, Lars Weimers, Mie-Lotte

Bohl.

9. ABB. HVDC transmission for lower investment cost. [Online]

http://www.abb.com/industries/db0003db004333/678bb83d3421169dc1257481004a4284.aspx.

10. Craciun, Bogdan. Multilink DC Transmission for Offshore Wind Power Plant Integration.

11. VSC-Based HVDC Power Transmission Systems: An Overview. Florentzou, Nikolas.

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13. Stefan G. Johansson, Gunnar Asplund, Erik Jansson, Roberto Rudervall. Power System Stablilty

Benefits with VSC DC Transmission Systems.

14. Voltage-Source-Converter Topologies for Industrial Medium-Voltage Drives, Industrial Electronics, IEEE

Transactions on, vol. 54, 2007. J. Rodriguez, S. Bernet, Bin Wu, J. Pontt, and S. Kouro.

15. Voltage Source Converter in High Voltage Applications: Multilevel versus Two-Level converters. Yushu

Zhang, G.P. Adam, T.C. Lim, Stephen J. Finney, B.W. Williams.

102

16. MTDC for High Power Transmission in Europe. Hausler, Michael.

17. Multiterminal HVDC for Offshore Windfarms (presentation). Temesgen., Haileselassie.

18. Aggregate Modeling of Wind Farms Containing Full Converter Wind Turbine Generators with PMSM. J.

Conroy, R. Watson.

19. Mohan, Ned. Power Electronics.

20. Develpoment of New VSI Average Model Including Harmonics. S. Ahmed, D. Boroyevich.

21. Filter Optimization for Grid Interactive VSI. Channegowda, Parikshith John, Vinod.

22. Remus Teodorescu, Marco Liserre, Pedro Rodriguez. PERES Course.

23. ABB. HVDC Cable Transmissions.

24. It's Time to Connect with Offshore Wind Suppliment. ABB.

25. Chaudhary, Sanjay K. A layout of Nord E.ON1 HVDC connecting Borkum-II Wind Park with the onshore

grid.

26. VSC - HVDC for Industrial Power Systems. Du, Cuiquing.

27. Glossary of Terms. Petition no. 39 of 2006. [Online].

28. Remus Teodorescu, Marco Liserre, Pedro Rodríguez. Grid converters for photovoltaic and wind

power systems. ISBN: 9780470057513.

29. R. da Silva, R. Teodorescu, P. Rodriguez. Power delivery in multiterminal VSC-HVDC transmission

system for offshore wind power applications.

30. Multi-terminal DC transmission systems for connecting large offshore wind farms. Lie Xu, Barry W.

Williams, Liangzhong Yao.

31. Grid Monitoring and Advanced Control of Distributed Power Generation. Timbus, A.

32. Control of VSC-based HVDC transmission system for offshore wind power plants. Master Thesis,

Aalborg University, Institute of Energy Technology, Denmark. A. Irina-Stan, D. Ioan-Stroe.

33. DC Grid Management of a Multi-Terminal HVDC Transmision System for Large Offshore Wind Farms.

Lie Xu, Liangzhong Yao, Masoud Bazargan.

34. Chopper Controlled Resistors in VSC-HVDC Transmission for WPP with Full-scale Converters. S. K.

Chaudhary, R. Teodorescu, P. Rodriguez.

35. Pang, Hui. Evaluation of losses in VSC-HVDC transmission system.

36. Daelemans, Gilles. VSC HVDC in meshed networks.

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37. IGBT Power Losses Calculation. Graovac, Dusan.

38. Semiconductor Losses in Voltage Source and Current Source IGBT Converters Based on Analytical

Deviation. M. H. Bierhoff, F. W. Fuchs.

39. IGBT Basics. Part 1. FAIRCHILD Semiconductors.

40. Blaabjerg, Frede. Power Semiconductor Devices Course. s.l. : AAU University.

41. Semiconductor Losses in Voltage Source and Current Source IGBT Converters Based on. Bierhoff, M.H.

42. Capacity for Competition. Investing for an Efficient Nordic Electricity Market . Nordic Competition

Authorities.

43. Electricity Trading in Competitive Power Market: An Overview and Key Issues. Prabodh Bajpai, S. N.

Singh.

44. Kalman, Dan. General Equation of an Ellipse.

104

105

Appendix A – Analytical approach to the converter and line loss

function minimization

Due to the fact that the cost function is too complicated to be computed directly, some simplifications

are made in order to obtain analytical results. The main reason behind the complex nature of the

objective function is represented by the converter stations’ loss functions. A linearization of these

functions will be performed. The linearized functions will be inserted in the simplified objective function

and the objective function will be derived in function of its state variables. In the end, the derivated

expressions will be equaled to 0 and the newly obtained system will be solved.

Firstly, the wind farm converter is analyzed. Because the input power is known, the linearization

approach will be different than in the case of the other two converters. The equations for the converter

losses are shown in equations A-1, A-2. The constraint is represented by equation A-3.

PWFC = a · PDC2 + b · PDC · VDC + c · PDC · IDC + d · PDC + e · IDC A-1

a = 6.2 · 10−6 [

1

W]

b = 2.6 · 10−5 [1

V]

c = 6.4 ∙ 10−4 1

A

d = 2.9 ∙ 10−3 e = 0.037 [V]

A-2

P1 = I1 ∙ V1 A-3

By replacing A-3 in A-1 one obtains:

PWFC (VDC ) = a · PDC2 + b · PDC · VDC + c ·

PDC2

VDC+ d · PDC + e ·

PDC

VDC

A-4

The linearization will take place at the rated voltage: Vrated= 300 kV

PWFC lin= PWFC Vrated +

dP Vrated

dV∙ (V − Vrated )

A-5

PWFC lin= a ∙ P1

2 + b ∙ Vrated ∙ P1 + c ∙P1

2

Vrated+ d ∙ P1 + e ∙

P1

Vrated+ b ∙ P1 − c ∙

P12

Vrated2 − e ∙

P1

Vrated2 ∙ (V

− Vrated )

A-6

In the end:

106

PWFC lin= a ∙ P1

2 + 2 · c ∙P1

2

Vrated+ d ∙ P1 + 2 · e ∙

P1

Vrated+ b ∙ P1 − c ∙

P12

Vrated2 − e ∙

P1

Vrated2 ∙ V

A-7

m = a ∙ P12 + 2 · c ∙

P12

Vrated+ d ∙ P1 + 2 · e ∙

P1

Vrated

A-8

n = b ∙ P1 − c ∙P1

2

Vrated2 − e ∙

P1

Vrated2

A-9

Coefficients m and n were introduced in order to simplify the formulas

PWFC V1 = m + n · V1 A-10

For P1=400 MW: m=3.1 and n=0.0085

Figure A.1 presents the comparison between the nonlinear and linear converter loss functions.

Figure A.1. Nonlinear and Linear Converter Loss Function Models at Rated Power

It is to be noticed that the assumed function approximation is only precise for voltages between 250

and 400 kV. This is more than enough for the current purpose (since the voltage can fluctuate between

50 100 150 200 250 300 350 4003

3.5

4

4.5

5

5.5

6

6.5

7

Voltage [kV]

Convert

er

Losses [

MW

]

107

291 and 309 kV). Moreover, if the rated input power changes sensibly form the rated value, the m and n

values have to be recalculated using formulas A-7 and A-8.

The grid side converters slightly differ when linearizing the losses. First of all, the power going into the

converter is unknown, therefore another variable needs to be introduced in the linearization process.

Thus, Plosses will be represented by the current and the voltage on the DC side of the grid converter (V2/3,

I2/3). V2/3 can be further expressed in terms of V1. The point in which the linearization will be performed is

M = (Vrated, Irated/2). M was chosen keeping in mind that the voltages at which the grid side converters

operate are close to the rated voltage. Moreover, the currents are near the Irated/2 value, with small

fluctuations above this value if the line resistance is smaller than the other line’s resistance and vice

versa. The geometrical interpretation of the linearization is a plane that is tangent to the grid side

converter’s Plosses(V,I) surface in M.

PGSC 2lin I2, V1 = a ∙ I2

2 ∙ V12 − 2 ∙ a ∙ I2

3 ∙ V1 ∙ R2 + a ∙ I24 ∙ R2

2 + b ∙ I2 ∙ V12 − 2 ∙ b ∙ V1 ∙ I2

2 ∙ R2 + b ∙ I23 ∙ R2

+ c ∙ I22 ∙ V1 − c ∙ I2

3 ∙ R2 + d ∙ I2 ∙ V1 − d ∙ I22 ∙ R2 + e ∙ I2

A-11

dPGSC 2lin

dI2= 2 ∙ a ∙ I2 ∙ V1

2 − 6 ∙ a ∙ I22 ∙ V1 ∙ R2 + 4 ∙ a ∙ I2

3 ∙ R22 + b ∙ V1

2 − 4 ∙ b ∙ V1 ∙ I2 ∙ R2 + 3 ∙ b ∙ I22 ∙ R2

2

+ 2 ∙ c ∙ I2 ∙ V1 − 3 ∙ c ∙ I22 ∙ R2 + d ∙ V1 − 2 ∙ d ∙ I2 ∙ R2 + e

A-12

dPGSC 2lin

dV1= 2 ∙ a ∙ I2

2 ∙ V1 − 2 ∙ a ∙ I23 ∙ R2 + 2 · b ∙ I2 · V1 − 2 ∙ b ∙ I2

2 ∙ R2 + c ∙ I22 + d ∙ I2

A-13

The equation of the tangent plane is displayed in A-13.

Z = F x0 , y0 +dF

dx x0 , y0 ∙ x − x0 +

dF

dy x0 , y0 ∙ (y − y0)

A-14

Introducing A-10, A-11 and A-12 into A-13, replacing x0 and y0 with Irated/2 and Vrated and by eliminating

the small order terms (coinciding with the terms containing the resistance values), the losses are

represented by the following formula:

PGSC lin 2= p + q ∙ I2 + r ∙ V2 A-15

r = 0.014 kA ; q = 4.24 kV ; p = −4.6 [MW] A-16

At the tangent point, the function has a value of 2.82 MW, more than reasonable, for the

simplifications made. Similarly, the other grid side converter will have the same linearized formula. It is

worth mentioning that the approximations are valid only for small variations around the linearization

point. If the input power for the wind farm decreases (therefore, the current on the circuit will decrease

also), the linearization has to be made around a new (x0, y0) pair.

Going back to the objective function and introducing equations A-15 and A-9, equation A-16 results:

108

Plosses = R2 ∙ I22 + R3 ∙ I3

2 + m + n · V1 − 2 ∙ p + q ∙ I1 + I2 + r ∙ (V1 − I2 ∙ R2 + V1 − I3 ∙ R3) A-17

I1 =P1

V1; I1 = I2 + I3

A-18

By replacing A-17 in A-16:

Plosses = R2 ∙ I22 + 2 ∙ R2 ∙ I1 ∙ I3 + I3

2 ∙ R2 + R3 + m + n ·P1

I1− 2 ∙ p + q ∙ I1 +

2 ∙ P1 ∙ r

I1+ r ∙ I1 ∙ R2 − r

∙ I3 ∙ (R2 − R3)

A-19

dPlosses

dI1= 0

dPlosses

dI3= 0

A-20

By solving system A-19 the following formulas are obtained:

I3 =2 ∙ R2 ∙ I1 + r ∙ (R2 − R3)

2 ∙ (R2 + R3)

A-21

I2 =2 ∙ R3 ∙ I3 + r ∙ (R3 − R2)

2 ∙ (R2 + R3)

A-22

While I1 respects the following equation:

I13 ∙ 2 ∙ R2 2 ∙ R3 − r ∙ R2 − R3 ∙ R2 − I2

2 ∙ r ∙ 4 ∙ R22 − r2 ∙ R2 − R3

2 + 2 ∙ q ∙ R2 + R3 ∙ I1 − 2 ∙ P1

∙ R2 + R3 ∙ n + 2 ∙ r = 0

A-23

The last equation can either be solved with Lagrange’s Method or with the General Formula of the

Roots. Since the methods are complex and solving them is not the topic of discussion. The variables were

replaced with their values and the function was introduced in a cubic function root calculator.

For the system: R2=1.8 ohm, R3=3.6 ohm, P1=400 MW the following results are obtained:

I1=1.35 kA, V1=296.29 kV, I2=0.91 kA, I3=0.45 kA.

The results obtained while running the simulation are the following:

V1 V2 V3 I1 I2 I3

291 kV 289.5 kV 289.4 kV 1.36 kA 0.86 kA 0.50 kA

Table A.1 – Simulation Results Converter+Line Loss Minimization

109

Appendix B – Laboratory Setup Parameters

DC SOURCE

Rated Power: 6 kW Maximum DC Voltage: 600 V Maximum DC Current: 10 A

DC CABLE RESISTANCES

Terminal 1 Resistance: 2.1 Ω Terminal 2 Resistance: 2.5 Ω

VSC PARAMETERS

Rated Power: 15 [kW] Supply Voltage: 380 – 500 [V] Power Factor: >0.98

FILTER PARAMETERS

Rated Power: 15 [kW] Rated Voltage: 500 [V] Rated Current: 38 [A] Filter Inductance: 1.6 [mH] Filter Capacitance: 10 [μF]

TRANSFORMER PARAMETERS

Rated Power: 10 [kVA] Rated Voltage: 400 [V] Rated Current: 38 [A] Short Circuit Impedance: 3% Connection: DYn11

CURRENT TRANSDUCER’S PARAMETERS

Primary Nominal RMS Current: 50 [A] Primary Current Measuring Range: 0 – 70 [A] Conversion Ratio: 1:1000 Supply Voltage: ±12…15 *V+

VOLTAGE TRANSDUCER’S PARAMETERS

Primary Nominal RMS Current: 10 [mA] Primary Current Measuring Range: 0…±14 [mA] Conversion Ratio: 2500:1000 Supply Voltage: ±12…15 *V+

110

111

Appendix C – The MTDC Optimization GUI R

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Documents

OPTIMIZATION OF MULTILINK DC TRANSMISSION FOR … · maximization) in a MTDC (Multi Terminal HVDC) transmission system and consists of eight chapters. The first chapter presents a