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
System: Closed Loop r to y
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
Peak amplitude: 1.24
Overshoot (%): 24.4
At time (seconds): 0.0104
System: Closed Loop r to y
I/O: r to y
Settling Time (seconds): 0.0301
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
System: Closed Loop r to y
I/O: r to y
Peak amplitude: 1.09
Overshoot (%): 8.85
At time (seconds): 0.0178 System: Closed Loop r to y
I/O: r to y
Settling Time (seconds): 0.0269
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
System: Closed Loop r to y
I/O: r to y
Peak amplitude: 1.09
Overshoot (%): 8.8
At time (seconds): 0.0177 System: Closed Loop r to y
I/O: r to y
Settling Time (seconds): 0.0268
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
Line Losses, R=2 ohm
Line Losses, R=4 ohm
Line Losses, R=6 ohm
Line Losses, R=8 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
1sec 2sec 3sec 4sec 5sec 6sec 7sec 8sec 9sec 10sec
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
Sharing Factor (I2/I3)
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
Sharing Factor (I2/I3)
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
1sec 2sec 3sec 4sec 5sec 6sec 7sec 8sec 9sec 10sec
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
Figure 7.4 – Current sharing for k=2
Figure 7.5 – DC voltages for k=2
k = 2 Measured Values Simulation Values
Currents I1 = 5 A; I2 = 3.25 A; I3 = 1.75 A; I1 = 5 A; I2 = 3.33 A; I3 = 1.667 A;
Voltages V2 = 613 V; V3 = 615.5 V; V2 = 613 V; V3 = 615.8 V;
Table 7.2 - Laboratory and simulation results for k=2
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
Figure 7.6 – Current sharing for k=3
Figure 7.7 – DC voltages 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
k = 3 Measured Values Simulation Values
Currents I1 = 5 A; I2 = 3.7 A; I3 = 1.3 A; I1 = 5 A; I2 = 3.75 A; I3 = 1.25 A;
Voltages V2 = 612.5 V; V3 = 617 V; V2 = 612.1 V; V3 = 616.9 V;
Table 7.3 - Laboratory and simulation results for k=3
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
Currents I1 = 5 A; I2 = 3.5 A; I3 = 1.5 A; I1 = 5 A; I2 = 1.667 A; I3 = 3.33 A;
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
Currents I1 = 5 A; I2 = 1.15 A; I3 = 3.85 A; I1 = 5 A; I2 = 1.24 A; I3 = 3.76 A;
Voltages V2 = 618 V; V3 = 611 V; V2 = 617.4 V; V3 = 610.6 V;
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
Currents I1 = 5 A; I2 = 3.54 A; I3 = 1.45 A; I1 = 5 A; I2 = 3.54 A; I3 = 1.46 A;
Voltages V2 = 623 V; V3 = 623 V; V2 = 622.6 V; V3 = 622.6 V;
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
Currents I1 = 5 A; I2 = 3.43 A; I3 = 1.49 A; I1 = 5 A; I2 = 3.53 A; I3 = 1.47 A;
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
1. Communities, Commission of the European. Renewable Energy Road Map Renewable energies in the
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.
12. The ABC’s of HVDC Transmission Technology. Michael P. Bahrman, Brian K. Johnson.
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.
103
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
esis
tan
ce
s &
Sim
Tim
e
Mo
dific
atio
ns
Sh
arin
g F
acto
r fo
r M
an
ua
l
Op
tim
iza
tio
n
Po
we
r V
alu
es
Mo
dific
atio
ns
Re
ad
Po
we
r V
alu
es
Fro
m F
ile
Tu
rn O
ptim
iza
tio
n
ON
/OF
F
Nu
mb
er
of T
erm
ina
ls
Se
lectio
n
Typ
e o
f O
ptim
iza
tio
n to
be
Pe
rfo
rme
d
Eq
uiv
ale
nt M
TD
C S
yste
m
Sch
em
e
Ou
tpu
t V
alu
es o
f th
e
Syste
m
Sta
rt S
imu
latio
nZ
oo
m O
ptio
ns
Price
s F
ile P
ath
Se
lectio
n
112