Optimal Power Flow for an HVDC Feeder Solution for AC Railwayskth.diva-portal.org/smash/get/diva2:565755/FULLTEXT01.pdf · Optimal Power Flow for an HVDC Feeder Solution for AC Railways

Optimal Power Flow for an HVDC FeederSolution for AC Railways

Applied on a Low Frequency AC Railway Power System

JOHN LAURY

Masters’ Degree ProjectStockholm, Sweden 2012

TRITA: XR-EE-E2C 2012:012

Optimal Power Flow for an HVDC FeederSolution for AC Railways

Applied on a Low Frequency AC Railway Power System

JOHN LAURY

Master’s Thesis at Electrical Machines and Power ElectronicsSupervisor: Lars Abrahamsson

Examiner: Stefan Östlund

TRITA: XR-EE-E2C 2012:012

AbstractWith today’s increasing railway traffic, the demandfor electrical power has increased. However, severalrailway systems are weak and are not being controlledoptimally. Thus, transmission losses are high and thevoltage can be significantly lower than the nominallevel.

One proposal, instead of using an extra HVACpower supply system, is to implement a HVDC sup-ply system. A HVDC supply line would be installedin parallel to the current railway catenary system andpower can be exchanged between the HVDC grid andthe catenary through converters.

This thesis investigates different properties andbehaviours of a proposed HVDC feeder solution. AnAC/DC unified Optimal Power Flow (OPF) modelis developed and presented. Decision variables areutilized to obtain proper control of the converters.

The used power flow equations and converter lossfunction, which are non linear, and the use of bi-nary variables for the unit commitment leads to anoptimization problem, that requires Mixed IntegerNon-Linear Programing (MINLP) for solving. Theoptimization problem is formulated in the softwareGAMS, and is solved by BONMIN. In each case in-vestigated, the objective is to minimize the total ac-tive power losses.

The results of the investigated cases presented inthis thesis, show that the proposed OPF-controlledHVDC solution reduces the losses and provides bettervoltage profile at the catenary, compared with today’ssupply systems.

Referat

Med dagens ökande järnvägstrafik har efterfragan paelkraft ökat. Dock är flera järnvägssystem svaga ochkontrolleras inte optimalt. Saledes, är överföringsför-luster höga och spänningen kan vara betydligt lägreän den nominella nivan.

Ett förslag, istället för att använda en extra HVACsystem, är att installera ett HVDC försörjningssy-stem. En HVDC ledning skulle installeras parallelltmed den nuvarande kontaktlednings system och ef-fekt kan utväxlas mellan HVDC nätet och kontakt-ledningen via omriktare.

Detta examensarbete undersöker olika egenska-per och beteenden hos ett föreslaget HVDC försörj-ningssystem. En AC/DC enad Optimal Power Flow(OPF) modell är framtagen och presenteras. Beslutvariabler används för att kontrollera omriktarna.

De använda kraftflödes ekvationer och omriktar-nas förlustfunktion, som inte är linjära, och använd-ningen av binära variabler för att kontrollera omrik-tarenheternas leder till ett optimeringsproblem somkräver Mixed Integer Non-Linear Programmering (MIN-LP) för att lösas. Optimeringsproblemet formulerasi programvaran GAMS, och löses med BONMIN. Ivarje undersökt fall är malet att minimera de totalaaktiva förlusterna.

Resultaten av de undersökta fallen som presente-ras i denna avhandling visar att den föreslagna OPF-kontrollerade HVDC-lösning minskar förlusterna ochger bättre spänningsprofil pa kontaktledningen, jäm-fört med dagens försörjningssystem.

Acknowledgements

First of all I would like to express my gratitude to Professor Stefan Östlundfor allowing me to expand my knowledge in this field.Thanks, Lars Abrahamsson for your support, guidance, patience and under-standing during this time.I also want to thanks my master thesis colleagues for their aid with LaTeXand the fun moments we spent in the master thesis room.I wish to thank my friend Patrik Janus, for his support, aid and the interestingdiscussions we had during our time inside and outside school.I also would like to thank my friend Reinhard Kaisinger for his help and sup-port with the English language, during my work with this thesis.Finally, I want to thanks my family for their endless love and support.

John Laury, September 2012

Contents

1 Introduction 11.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 52.1 Rail Power Supply Systems . . . . . . . . . . . . . . . . . . . 5

2.1.1 Rotary Converters . . . . . . . . . . . . . . . . . . . . 62.1.2 Static Converters . . . . . . . . . . . . . . . . . . . . . 6

2.2 Catenary systems . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 HVDC Transmission . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 General Introduction . . . . . . . . . . . . . . . . . . . 92.3.2 General Advantages and Disadvantages . . . . . . . . . 92.3.3 HVDC converter for the RPSS . . . . . . . . . . . . . . 10

2.4 Optimization theory . . . . . . . . . . . . . . . . . . . . . . . 112.4.1 Types of Optimization Programming . . . . . . . . . . 11

3 Models 153.1 Power Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 AC Power Flows . . . . . . . . . . . . . . . . . . . . . 153.1.2 DC Power Flows . . . . . . . . . . . . . . . . . . . . . 17

3.2 Rotary Converter . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 HVDC Converter . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 The Network model . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 Line Model . . . . . . . . . . . . . . . . . . . . . . . . 223.4.2 Unified AC/DC Load Flow . . . . . . . . . . . . . . . . 22

3.5 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5.1 Power Lines . . . . . . . . . . . . . . . . . . . . . . . 243.5.2 Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5.3 Public Grid to DC Grid Converter . . . . . . . . . . . 24

4 The Optimization Problem 254.1 Minimizing Losses . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . 264.1.2 The Objective Function . . . . . . . . . . . . . . . . . 27

5 Software and Implementation 295.1 Platform for Solution . . . . . . . . . . . . . . . . . . . . . . . 29

5.1.1 GAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.1.2 BONMIN . . . . . . . . . . . . . . . . . . . . . . . . . 295.1.3 Optimization with Rotary Converters . . . . . . . . . . 30

5.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Studied Cases 316.1 Investigated Cases . . . . . . . . . . . . . . . . . . . . . . . . 31

6.1.1 Train Traffic . . . . . . . . . . . . . . . . . . . . . . . . 326.1.2 Investigation of the Cases . . . . . . . . . . . . . . . . 35

7 Results of Simulations 397.1 Dense Traffic Case . . . . . . . . . . . . . . . . . . . . . . . . 39

7.1.1 AT Catenary . . . . . . . . . . . . . . . . . . . . . . . 397.1.2 BT Catenary . . . . . . . . . . . . . . . . . . . . . . . 44

7.2 Light Traffic Case . . . . . . . . . . . . . . . . . . . . . . . . . 507.2.1 AT Catenary . . . . . . . . . . . . . . . . . . . . . . . 507.2.2 BT Catenary . . . . . . . . . . . . . . . . . . . . . . . 56

8 Analysis and Discussion 638.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8.1.1 Different PF at Train Locomotives: Dense Traffic Case 638.1.2 Different PF at Train Locomotives: Light Traffic Case . 648.1.3 OPF sensitivity to Converter Losses: Dense Traffic Case 648.1.4 OPF sensitivity to Converter Losses: Light Traffic Case 658.1.5 Smaller Type of Converters: Dense Traffic Case . . . . 668.1.6 Smaller Type of Converters: Light Traffic Case . . . . 668.1.7 Converter Loss Function . . . . . . . . . . . . . . . . . 67

8.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698.2.1 Impact of Catenary System . . . . . . . . . . . . . . . 698.2.2 Centralized Solution . . . . . . . . . . . . . . . . . . . 708.2.3 Improved Voltage Levels . . . . . . . . . . . . . . . . . 71

9 Conclusions and Future Work 739.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

9.2.1 Moving Trains . . . . . . . . . . . . . . . . . . . . . . . 74

9.2.2 Economical Aspects . . . . . . . . . . . . . . . . . . . . 74

Bibliography 75

A Numerical data 79

B Catenary Transformers 81B.1 Booster Transformer . . . . . . . . . . . . . . . . . . . . . . . 81B.2 Auto Transformer . . . . . . . . . . . . . . . . . . . . . . . . . 82

Chapter 1

Introduction

The history of Swedish railway started around 1856 when the first railwayswere built. They were operated by steam locomotives for approximately 55years. Around 1910, the first electrified railway was introduced. The lineswere electrified with single phase alternating current at a voltage of 15 kVand frequency of 15 Hz1. The power was directly generated from hydro powerplants with a frequency of 15 Hz. Later on, when the railway system sig-nificantly expanded, it was decided to not use direct generation from hydropower plants. Instead, the public grid with rotary converters were used, thusconverting 50 Hz to 16.7 Hz, which became the standard and still is used [12].

Today, the railway is supplied with power from the Swedish public gridvia either rotary converters or static converters, placed at certain distancesfrom each other. The distance between converter stations usually lies between40 to 200 km. Most catenary systems in service have comparatively highimpedances, and the transmission losses are relatively high (between 30-40%).

With increasing railway traffic and expansion of the railway system, morepower is required. Thus, there is an identified need for a technical solution tofulfil the increasing power demand and to reduce the losses.

With the price of semiconductors decreasing, controllable power convertersare available at lower cost. Power converters can be used to convert three phaseAC to single phase AC and vice versa. They can also be used to convert AC toDC or DC to AC. The converters provide control of both active and reactivepower. Therefore, such power converters are suitable for the Rail Power SupplySystem (RPSS). This is further discussed in [6].

The concept of RPSS using High Voltage Direct Current (HVDC) is pre-sented in [4]. The used converter technology is based on medium frequencytransformer, in order to reduce the size of the converter [24].

1 Porjus and Malmbanan

1

CHAPTER 1. INTRODUCTION

This thesis follows the ideas of [4], were the HVDC feeder solution was in-vestigated for an RPSS, where the Overhead Contact Line (OCL) is equippedwith Booster Transformers (BT). However, this thesis compares transmissionlosses and voltage levels for:

• OCL with BT and Auto Transformers (AT) systems.

• Trains operating at power factor 0.8, 0.9 and 1.

• Varied converter size and spatial distribution.

• OPF sensitivity to converter losses

The OPF HVDC feeder solution is formulated as an optimization problemwhere the objective is to minimize the overall active power losses. However,the solutions in the studied cases are only valid for a specific time instance,since in real-life the consumptions and locations of the trains will vary overtime. Thus, the solutions presented sets a theoretical upper bound on howsmall the losses can be given a specific time instance, if smart control is appliedon the converters.

In the calculations, all equations are expressed in the p.u. system, unlessthe opposite is explicitly stated.

1.1 Outline• Chapter 1 is the Introduction.

• Chapter 2 describes the different existing feeding systems for the railway.A general introduction of HVDC technology is given and a descriptionof the converter for the HVDC feeder solution.

• Chapter 3 presents the mathematical models used to describes the ex-isting feeding systems and the HVDC feeding system.

• Chapter 4 and 5, describes the formulation of the optimization problemand software used.

• Chapter 6 presents the different case studies. The HVDC feeder solutionis, e.g studied with different types of catenary systems and comparedwith todays existing feeder solutions.

• Chapeter 7 present the results of the comparison of the HVDC feedersolution and the existing feeder solutions.

2

1.1. OUTLINE

• Chapter 8 is Analysis and Discussion of the results given in Chapter 7.

• Chapter 9 is Conclusion and Future work.

3

Chapter 2

Background

2.1 Rail Power Supply SystemsIn Sweden there are two types of systems topologies for feeding the catenary,these are

1. Centralized system

2. Decentralized system

The Centralized system has an additional feeder at 132 kV, 1623 Hz, which is

located parallel to the catenary. Transformers rated at 16 MVA and 25 MVAare connected between the OCL and the HVAC feeder according to figure 2.1

Figure 2.1. Centralized feeder system layout.

The decentralized system does not have any extra line [31] and is used in areaswhere it is either not economically motivated or there are societal barriers.Figure 2.2 present the setup of a decentralized system.

5

CHAPTER 2. BACKGROUND

Figure 2.2. Decentralized feeder system layout.

Common for both systems, is that they are fed via rotary converters andstatic converters. These converters are placed individually, or in groups inlarge depots, i.e. converter stations. The number of converters in a converterstation depends on the power demand between converter stations.

2.1.1 Rotary ConvertersRotary converters are equipped with a three phase synchronous motor anda single phase synchronous generator connected via a common shaft. Thefeeding three phase grid will see the converter as a symmetrical three phaseload. Thus, one advantage is that converter can compensate voltage drops inthe three phase grid during operation by producing reactive power [31].

Another major advantage is that the supplying three phase grid and thecatenary are electrically decoupled. However, rotary converters have a consid-erable start-up time and requires synchronization with the supplying publicgrid [31].

2.1.2 Static ConvertersStatic converters are based on power electronics, and the output voltage is con-trolled by appropriate control of the converter. The output voltage does notfollow a purely sinusoidal wave and comes with harmonics. These harmonicsneeds to be filtered [31].

The output voltage can be controlled by delaying the delay angle alphaat the thyristor. Alpha is the angle between the current crossing zero andthe actual firing of the thyristor. By controlling alpha, the mean value of theoutput voltage can be controlled. In particular, this means that active powerflows from the public grid to the railway system can be controlled. In Swedenthe static converters are controlled to mimic rotary converters [3].

6

2.2. CATENARY SYSTEMS

The main advantages of static converters are that they are cheaper, requireless maintenance, have lower installation cost and low short circuit current.Negative aspects include current and voltage harmonics, that have to be fil-tered if the converter rated power is large compared to the short circuit powerof the public grid at the point of connection [31] .

2.2 Catenary systemsIn the railway systems there exists two main types of catenary systems andone possible catenary system:

• Booster transformer (BT)

• Auto transformer (AT)

• A possible combination of AT and BT (AT+BT)

The names are abbreviated from the utilization of transformers at the cate-nary, in order to handle the return currents in the railway system.

BTThe BT-system is the most commonly used catenary system type in Sweden.The utilization of BT was necessary due to the high ground resistivity inSweden, which leads to decreased conductivity via rail and ground [3, 31].

The BTs are normally located 5 km from each other [3]. Between the BTsthere is a ground connection established between the return conductor andthe rail, c.f. Figure 2.3. By using BT on the catenary, the return current fromthe return rail is collected. Thus, the stray currents are reduced [3].

Figure 2.3. A generalized setup of a BT system.

7


A negative aspect according to [31] is that the line impedance of the cate-nary is increased, that will reduce the effective power transfer, since the currentmust flow through every BT. This limitation implies that the distance betweenpower supply stations has to be reduced.

The operation principle of the BT is detailed in Appendix B.1.

AT

The utilization of AT in the Swedish catenary started in 1998. The use ofAT allows a negative conductor instead of having a return conductor. Thus,the transformer is placed between the negative conductor and the overheadcontact line, c.f. Figure 2.4. the midpoint is connected to the rail [31] and thevoltage is doubled [3, 31].

Figure 2.4. A generalized setup of an AT system.

Doubling the voltage means half the current on the catenary if the theactive power is constant. The benefits compared to BT are lower voltage drop,reduced losses and longer possible distances between power supply stations[3, 31].

The current distribution will be almost equal in the AT catenary system.The return current in the negative feeder will be the same as the feedingcurrent in the actual contact line, except at the AT cell where the train islocated. Due to the transformers leakage inductances, stray currents will beflowing along the rail until the next power supply [31].

The operation principle of the AT is detailed in Appendix B.2.

8

2.3. HVDC TRANSMISSION

AT+BTIn [27] a possible combination of AT and BT is presented and investigated.The combination provides the advantages of both catenary systems. InstallingAT in a BT catenary, reduces the line impedance and could increases the max-imum distance between supply stations [27]. Stray current flowing through therail when ATs are used will be collected by the BTs [3].

2.3 HVDC Transmission

2.3.1 General IntroductionWith the introduction of mercury arc converters in 1930, the ground waslaid for the development of HVDC transmission. In 1945, the first HVDCsystem was ready for use in Germany. However, this system never becameoperational due the end of World War II. Instead, the first operational HVDCinstallation was built in Sweden, connecting the island of Gotland with theSwedish mainland in 1954 [16].

Nowadays, HVDC transmission systems take an important role in powersystems, as these systems allow power transfer up to 3000 MW over longdistances. Power system stability, environmental concerns and economicalaspects are the arguments for the use of HVDC [16,30].

2.3.2 General Advantages and DisadvantagesHVDC transmission have several advantages over high voltage AC (HVAC)transmission. One of the most significant advantages is the reduction of trans-mission losses, due to the lack of inductances and capacitances. In HVACtransmission cables, the effects of inductances and capacitances have to becompensated along the line in order to maintain steady voltage.

Other advantages of the HVDC technology are the possibilities for in-terconnection of two asynchronous AC systems, and improvement of powersystem stability. Connecting two asynchronous AC systems with HVDC tech-nology is often beneficial, because the interconnection can provide a backupoption and certain protection against disturbances. If a fault or disturbanceoccurs in one of the system, the interconnection will act as a protection toprevent failure to spread to the other AC system. The interconnection canalso supply damping torque for power oscillations if the HVDC system hasbeen designed accordingly [16,30].

HVDC technology has many advantages, however, there are some disad-vantages that need to be taken into consideration. Converters are required

9


which may be expensive. The ability to handle overcurrents is limited, de-pending on power electronics and the utilized converter topology. Using con-verters, introduces harmonics that need to be filtered, and some harmonicsare difficult to eliminate. These harmonics may affect the AC systems andmay cause additional wear and losses [30].

2.3.3 HVDC converter for the RPSSBy making use of the advantages of HVDC technology, an approach could beto combine of this technology with the RPSS. In [32], a HVDC system for lightrail is presented and in [4] a conversion system based on HVDC technologyfor low frequency AC railways is proposed. It was shown that the systemwith HVDC feeder caused lower losses and better voltage levels compared toconventional power supply system, such as decentralized system.

The key technology of such converter is the use of the Medium Frequency(MF) transformers. In the MF transformer, the excitation frequency is gen-erated by power electronics at significantly higher frequency (in the scale ofkHz). This increased frequency allows the overall size and weight of MF trans-formers to be notably reduced [21].

In order to achieve the high frequency needed, Voltage Source Converters(VSC) are used. These converters consist of power semiconductor switches.Given a DC source, the VSC generates a controllable AC voltage with desiredfrequency.

In order to convert the high frequency required for the MF transformer toa lower desired frequency (16.7 Hz), a cycloconverter is used. A cycloconverterconverts one AC waveform to another AC waveform with lower frequency andvoltage and vice versa. Conversion is done by controlling the triggering angleof the thyristors. Thus, the output voltage will resemble in a sine wave withsuperimposed harmonics. These harmonics have to be filtered properly inorder to extract the fundamental sine wave of desired frequency and voltage.

The setup of VSC, MF transformer and cycloconverters is schematicallyshow in Figure 2.5As shown in Figure 2.5, the cycloconverter is connected to the catenary andthe VSC is connected to the HVDC line. The VSC is controlled so it providesthe desired voltage and frequency for the MF transformer. Thus, the MFtransformer is excited and the energy from the VSC:s is transferred to thecycloconverter. The cycloconverters convert the high frequency to 162

3 Hz.The design and topology of the proposed conversion system are described

more in detailed in [24].

10

2.4. OPTIMIZATION THEORY

Figure 2.5. A generalized setup of a converter station for the HVDCfeeder solution.

2.4 Optimization theoryA major part of this thesis is based on optimization theory for OPF calcula-tions. The principles of optimization theory can be found in [23]. This sectionprovides a brief introduction to optimization theory relevant to this thesis.

2.4.1 Types of Optimization ProgrammingThere are several types of optimization problems. However, in this thesissolving the arising optimization problem requires the use of Mixed IntegerNon-Linear Programming (MINLP), which is a combination of two types ofoptimization programming:

• Non-Linear Programming (NLP), and

• Mixed Integer Liner Programming (MILP)

NLP

NLP is used to solve optimization problems which have non-linear objectivefunction or constrains. In some cases the non-linear functions are non-convex,which makes finding a global optimum a challenge.

Solving a NLP requires the derivatives of the functions, which are used tofind the optimum of the non-linear optimization problem. One of the mostcommon algorithm in NLP for finding the optimum, is the Newton-Raphsonmethod [29].

A non linear optimization problem can be written in its normal form [23]:

11


min f(x) (2.1)subject to: gi(x) ≤ bi, i ∈ Γ (2.2)

hj(x) = bj, j ∈ Ω (2.3)x ∈ Rn

where x is a vector containing the optimization variables, f(x) is the objectivefunction, gi(x) and hj(x) are the constraint functions. The constants bi, ..., bn

are the boundaries of the constrain functions, Γ and Ω are the set of indicesfor inequality and equality constrains, respectively. The vector x∗ holds theoptimal solution, when x causes the smallest deviation and still satisfies allthe constraints of the problem [15,23]. In other words, for any z with gi(z) ≤b1.....gn(z) ≤ bn, then f(z) ≥ f(x∗).

MILPThe MILP is a minimization or maximization of a linear function, where theconstrains or the objective functions are linear. However, the variables inthe optimization problem are restricted to be of integer type at the optimalsolution. Due to the discontinuity, no derivative can be used. Therefore,different approaches have to be utilized to find the optimal solution [22,23].

MINLPCombining non-nonlinearities and integer to one optimization problem, re-quires the use of MINLP. If the non-nonlinearities are non-convex functions,the finding of the solution of the optimization problem can be difficult. It doesnot become obvious when the optimum is found, due to the existence of globalor local optimal solutions. Thus, solving this sort of optimization problemscan be difficult [17]. Global solvers can find a global solution, however thecomputational time can be significant, hence it can be unpractical for cer-tain applications. The general form of a optimization problem that requiresMINLP is:

min f(x) (2.4)subject to: gi(x) ≤ bi, i ∈ Γ (2.5)

hj(x) = bj, j ∈ Ω (2.6)x ∈ Zn, y ∈ Rn

12

2.4. OPTIMIZATION THEORY

where, Γ and Ω are the set of indexes for inequality and equality constrains,respectively, Zn stands for all integer points on Rn. The integer variable hasa useful special case for binary variables, i.e. y ∈ 0, 1.

13

Chapter 3

Models

3.1 Power FlowsTo model the power flows for both the AC and the DC side of the converter,load flow equations are used. Power flow equations are derived from powermismatch and are considered a standard method for analyzing power flows inpower systems. It assumed that the reader is aware of Kirchhoff current andvoltage laws, Ohms law and power flow equations based on those principals.The derivation of the power flow equations can be found in [29].

3.1.1 AC Power FlowsThe AC power flows and AC power mismatch at the buses are expressed byfollowing equations:

PACk= UACk

·NAC∑j=1

UACj[GACjk

cos(θjk) +BACjksin(θjk)] (3.1)

QACk= UACk

·NAC∑j=1

UACj[GACjk

sin(θjk)−BACjkcos(θjk)] (3.2)

PACGk− PACDk

− PACk= 0 (3.3)

QACGk−QACDk

−QACk= 0. (3.4)

Equations (3.1) and (3.2) describe how power flows between each node arecalculated and Equations (3.3), (3.4) refer to the power balance must be ful-filled at each node. The variables in (3.1), (3.2), (3.3) and (3.4) are detailedin Table 3.1

15

CHAPTER 3. MODELS

Table 3.1. Explanations for AC power flow equations.

Denotation DescriptionPACk

the sum of net injected active power in node k, (variable)QACk

the sum of net injected reactive power in node k, (variable)UACj

voltage in node j, (variable)UACk

voltage in node k, (variable)θkj voltage angle difference between node k and j, (variable)

GACkjreal part of admittance matrix, (parameter)

BACkjimaginary part of admittance matrix, (parameter)

PACDkconsumed active power in node k, (variable)

PACGkgenerated active power in node k, (variable)

Ploss−ACjlosses corresponding to AC node j, (variable)

QACGkgenerated reactive power in node k, (variable)

QACDkconsumed reactive power in node k (variable)

NAC number of AC nodesj, k running indeces, j 6=k

In general, the active power losses on power line are according to [28] becalculated as equation (3.5)

Pline,loss = R · I2. (3.5)

where the line losses are dependent of the resistance of the power line andthe line current. However, in this thesis the losses are calculated according toequation [28]

Ploss−ACj=

NAC∑j=1

[U2ACj

GACjk− UACj

UACkGACjk

cos(θACjk)]. (3.6)

16

3.1. POWER FLOWS

3.1.2 DC Power FlowsIn steady state, the flow in the DC grid is dictated by the line resistances andthe difference in voltage magnitude at the ends of respective line. Using thesame load flow equations and power mismatch equations as in Section 3.1, butonly for active power flow and neglecting the angles, i.e. the angles are set tozero, the DC flows and power mismatch can be expressed by

PDCk= UDCk

NDC∑j=1

UDCjGDCj

(3.7)

PDCGk− PDCDk

− PDCk= 0 (3.8)

Ploss−DCj=

NDC∑j=1

[U2DCj

GDCjk− UDCj

UDCkGDCjk

]. (3.9)

The variables in (3.7), (3.8) and (3.9) are given in Table

Table 3.2. Explanations for DC power flow equations.

Denotation DescriptionPDCk

the sum of net injected active power in node k, (variable)UDCj

voltage in node j, (variable)UDCk

voltage in node k, (variable)GDCkj

real part of admittance matrix, (parameter)PDCDk

consumed active power in node k, (variable)PDCGk

generated active power in node k, (variable)Ploss−DCj

losses corresponding to AC node j, (variable)NDC number of AC nodesj, k running indeces, j 6=k

17

CHAPTER 3. MODELS

3.2 Rotary ConverterThe rotary converter used in this thesis is assumed to convert 10 MVA and isreferred as Q48/Q49. The equations originates from [3, 26] and describes therelation between the two systems, i.e. the 162

3 Hz railway grid and the 50 Hzpublic grid. Table 3.3 explains the variables used.

U g = 16.5− QG

#conv · kq

(3.10)

θ0 = θ50 − 13 · arctan X50 · PG

(Um)2 +X50 ·Q50(3.11)

ψ = −13 · arctan

Xmq · PG

#conv

(Um)2 +Xmq · Q50

#conv

− arctanXg

q · PG

#conv

(Ug)2 +Xgq · QG

#conv

(3.12)

θ = θ0 + ψ(PG, QG, U) (3.13)

Table 3.3. Explanation of denotations of rotary converter equations.

Denotation Descriptionθ50 [rad] no-load phase angle, 50 Hz sideXm

q [Ω] quadrature reactance motorXg

q [Ω] quadrature reactance generatorUm [kV] voltage at motor sideU g [kV] voltage at generator side#conv number of convertersψ [rad] phase angle difference between 50 Hz side

and 16.7 Hz side of converterPG [MW] generated active power at generator sideQG [MVAr] generated reactive power at generator sideQ50 [MVAr] absorbed reactive power

The amount of reactive (QG) and active power (PG) injected to the catenaryand the amount of reactive power(Q50) absorbed by the 50 Hz grid is equallydivided by the number of converters, #conv, used in the converter station.The output voltage is controlled such that 16.5 kV are kept at the point ofconnection to the catenary. However, the voltage can drop when the powerdemand is high. This relation is described by Equations (3.10) and (3.12),which also give the phase angle difference between the 50 Hz and 16.7 Hzsides of the converter. Equation (3.11) describes the phase angle of the 50 Hzside of the converter when the converter is utilized, and (3.13) describes thephase angle on the generator side of the the converter [3, 26].

18

3.3. HVDC CONVERTER

3.3 HVDC ConverterFrom a system point of view, the converter is divided into an AC side and aDC side. The DC side of the converter unit is connected to the supplying DCgrid and the AC side is connected to the AC grid, that is the OCL.

The converter unit can operate both in rectifier mode and inverter mode,allowing active power to flow in both directions. Reactive power production isdependent on the AC side voltage of the converter, and is limited by equations(3.18) and (3.19). Equation (3.15) defines the apparent power of the converterunit. If the voltage at the AC side of the converter drops, the converterapparent power output is reduced as described by (3.14). The active powerinput/output of the converter unit is defined by equations (3.16) and (3.17),where the DC voltage is the limiting factor.

The maximum current a converter unit can handle, is calculated with theassumption that the converter unit can provide nominal apparent power at thelowest AC voltage it is designed for [4]. Hence, the nominal apparent powerof the converter unit is set to 1 p.u. and the lowest AC voltage is set to 0.8p.u., and consequently the maximum current is 1.25 p.u.

The converter units electronics and transformer cause losses. Assuminginverter mode operation according to [4], the losses can be modelled as a secondorder polynomial according to equation (3.21), depending of the converter ACcurrent, equation (3.20). The quadratic term and linear terms represents thecurrent dependent losses and the constant term in (3.21) represents the no-loadlosses of the converter [4, 5]. Furthermore, losses caused when the converterunit is operating in rectifier mode are assumed to be 10% less compared toinverter mode operation, and losses caused by harmonics are neglected [4, 5].

The p.u. losses of a converter unit are according to [4, 24] assumed to beindependent of the converter rating.

19

CHAPTER 3. MODELS

SACjsw≤ UACjsw

· Imax (3.14)S2

ACjsw= P 2

out,ACjsw+Q2

out,ACjsw(3.15)

Pin,DCjsw≥ −Imax · UDCjsw

(3.16)Pin,DCjsw

≤ Imax · UDCjsw(3.17)

Qout,ACjsw≥ −UACjsw

· Imax (3.18)Qout,ACjsw

≤ UACjsw· Imax (3.19)

Iout,ACjsw=SACjsw

UACjsw

(3.20)

Pinvjsw= 0.0135 · I2

out,ACjsw+ 0.0097 · Iout,ACjsw

+ 0.015 (3.21)Prectjsw

= (0.0135 · I2out,ACjsw

+ 0.0097 · Iout,AACjsw+ 0.015) · 0.9 (3.22)

Pout,ACjsw≤ Pmax · αjsw

(3.23)Pout,ACjsw

≥ −Pmax · (1− αjsw) (3.24)Pout,ACjsw

≤ Pmax · γjsw (3.25)Pout,ACjsw

≥ −Pmax · γjsw (3.26)Qout,ACjsw

≤ Qmax · γjsw (3.27)Qout,ACjsw

≥ −Qmax · γjsw (3.28)SACjsw

≤ Smax · γjsw (3.29)Pin,DCjsw

≤ Pmax · γjsw (3.30)Pin,DCjsw

≥ −Pmax · γjsw (3.31)Pswjsw

≤ PL,max · γjsw (3.32)Pswjsw

≥ Pinvjsw− PL,max · (2− αjsw − γjsw) (3.33)

Pswjsw≤ Pinvjsw

+ PL,max · (2− αjsw − γjsw) (3.34)Pswjsw

≤ Prectjsw+ PL,max · (1 + αjsw − γjsw) (3.35)

Pswjsw≥ Prectjsw

− PL,max · (1 + αjsw − γjsw) (3.36)

20

3.3. HVDC CONVERTER

Table 3.4. Descriptions of the denotations.

Denotation DescriptionSACjsw

apparent power at converter AC side, (variable)Pin,DCjsw

active power input at converter DC side, (variable)Qout,ACjsw

reactive power output at converter AC side, (variable)UACjsw

voltage at converter AC side, (variable)Pout,ACjsw

active power output at converter AC side, (variable)Pmax maximum power, (parameter)Imax maximum current, (parameter)PL,max maximum loss of a converter unit (parameter)UDCjsw

voltage at converter DC side, (variable)Pinvjsw

inverter loss function, (variable)Prectjsw

rectifier loss function, (variable)Pswjsw

converter loss function, (variable)γjsw unit commitment variable, (binary variable)αjsw direction of flow variable, (binary variable)jsw the index of the converter

The direction of the active power flow through the converter will impose theoperating state of the converter. The binary variable α is used to model theoperating state of the converter where 1 means inverter operation and 0 meansrectifier operation, c.f. equations (3.23) and (3.24).

The converter no-load losses contributes significantly to the system losses,unless the converter unit is turned off. Thus, the unit commitment is modelledby the binary variable γ, and the converter unit is turned off when γ is valuedzero, c.f. equations (3.25)-(3.32). Assuming maximum current through theconverter unit (Imax) and inverter operation mode, the maximum losses can becalculated by equation (3.21). Furthermore, equations (3.33)-(3.36) ensuresthe right loss function used, when the converter is either rectifying or inverting.

21

CHAPTER 3. MODELS

3.4 The Network model3.4.1 Line ModelWhen modeling power lines at lengths between 100 km and 300 km, the π-model is used. The model consists of an ohmic resistance in series with aninductor. In addition, a shunt capacitance is connected in parallel [29]. Themodel is shown in Figure 3.1, looks like the Greek letter π. Thus, the nameof the model [22,29].

Figure 3.1. The π-model.

The shunt admittance represents the line capacitance to ground, and the seriesinductance represents the reactive losses caused by the leakage current via theinsulator, i.e. the dielectric losses in the insulation material [29].

3.4.2 Unified AC/DC Load FlowThere are two general approaches for the solution of an AC/DC load flow:

• Sequential method

• Unified method

The sequential method solves the AC- and DC power flow equations inde-pendently in each iteration, until convergence is acquired [18]. The methodhas been further developed and adapted for Multi-terminal VSC. In [8], [10]and [9] the solution algorithm is presented and explained. However, accord-ing [18], the sequential method may have problems with convergence in certainsituations.

The unified method combines and solves the AC- and DC power flow equa-tion simultaneously. One of the advantages, according to [18], is that theunified AC/DC power flow convergence is guaranteed. However, for large

22

3.4. THE NETWORK MODEL

systems the computation time can be significant and different solution ap-proaches of the AC power flow are hard to implement, such as decoupled ACloadflow. [18,19].

In this thesis a unified AC/DC power flow approach is chosen to find theoptimal power flows, in order to acquire minimal losses of the HVDC feedersolution

The slack bus of the system is set to the DC side and supplies the powerdemand. Furthermore, in one of the converters the reference angle is set tozero.

The converter has two nodes, an AC node and a DC node,where jconv-AC-node ∈ NAC and jconv-DC-node ∈ NDC. Assuming nolosses, the AC/DC connection of the converters, seen from a system point ofview is described by

Pin,DCjsw− Pout,ACjsw

= 0 (3.37)

For simplicity, when Pout,ACjswis positive, the power flows from the DC side to

the AC side of the system. Thus, according to Equation (3.23) the converter isoperating in inverter mode and the power flows from DC to AC. If Pout,ACjsw

becomes negative the converter is operating in rectifier mode, according to(3.24). Introducing the losses of the converter yields :

Pin,DCjsw− Pout,ACjsw

− Pswjsw= 0 (3.38)

Assuming the converter is operating in inverter mode, the variable Pswjswwill

be equal to Pinvjsw, and in this situation the output power at the AC side of

the converter is positive. If the converter is operating in rectifier mode, thevariable Psw,jsw will be Prectjsw

and the input power at the DC side is negative,thus power flows from AC side to the DC side of the converter.

In order to achieve unified OPF, the power mismatch equations for bothAC side and DC side of the system are used. In the DC side of the system,the consumed active power PDCDk

at the converter DC node is equal to thepower input at the converter DC side. The generated active power PACGk

atthe converter AC node is equal to the power output at the converter AC side.The reactive power output from converter is not directly coupled to activepower, but limited, and is equal to the generated reactive power QACGk

at theAC side node of the converter.

23

CHAPTER 3. MODELS

3.5 SimplificationsThe whole model is a simplification of a real system. The line model andnumber of buses are only approximations. The train models are simplified.These simplifications are explained detailed in the following sections.

3.5.1 Power LinesThere are different types of power lines that can used, and they all come withtheir own characteristics. The DC transmission line is indeed an approxima-tion and, in this thesis, the DC line is only represented by its ohmic resistance.The capacitance in these cables are neglected, as one of the assumptions sincethe system is operating in steady state.

There are two common equivalents of transmission lines for the AC cate-nary system, AT and BT lines, which depends of the transformer used in theOverhead Contact Line (OCL). The BT lines have higher impedance than theAT lines. The AT line used is of type 120 mm2, 2 AT, FÖ and the BT line isof type 109 mm2, 2A, FÖ [20].

It was mentioned that the AC lines are modelled with the π-model. How-ever, the BT lines used in the model do not have such shunt capacitanceand AT shunt capacitance are neglected, due the short distances between theconverter units [20,29].

3.5.2 TrainsThe trains are modelled as static loads, thus the harmonics and other phenom-ena generated by the power electronic based converters on trains are neglected.

3.5.3 Public Grid to DC Grid ConverterThe 3 phase to DC converter, which is the slack bus of the system, is connect-ing the DC grid to the public grid and supplies the active power demand of thesystem. The converter has in reality losses, caused by the power electronicsand other electrical components. However, these losses are disregarded.

An additional assumption is that the active power supply from the con-verter is not limited. Furthermore, the converter is assumed to keep the volt-age at its nominal value. However, in a real system the converter is subjectedto limits, such as active power limitations and reactive power control in the50 Hz public grid.

24

Chapter 4

The Optimization Problem

4.1 Minimizing LossesThere are several ways to reduce losses in the system, such as increasingvoltage and varying the reactive power and reduce the distance between theconverter stations.

However, the optimal system configuration of the HVDC feeder solution isnot considered in this thesis, and, therefore the distance between the convert-ers is kept constant. Instead the operation of the HVDC supply solution isoptimized. However, it can be mentioned that finding an optimal distance be-tween converter stations is a complicated task as it depends on several aspects,such as converter rating, costs etc.

The converters are controlled in such a way that they can be turned onor off in order to minimize the overall losses. This decision is expressed as abinary variable, described in Chapter 3.

The number of installed converters and their ratings will determinate themaximum amount of active power that can be transferred from the HVDCgrid to the catenary.

The known data about the system is the power consumption of the trains,admittance matrix and the converter ratings. It is up to the solver to find thebest converter states, voltage levels, voltage angles and power flows for thesystem, in order to minimize the active power losses.

25

CHAPTER 4. THE OPTIMIZATION PROBLEM

4.1.1 BoundariesIntroducing bounds for the problem is of great importance, as they representthe physical limits of the system and helps the solver to find correct andrealistic solution. As an example, the voltage angles are limited not to spreadmore than 180 degrees. This is due to the properties of the sine and cosinefunction and if the voltage angle is not constrained, it would become moredifficult to find correct solutions or even find a solution at all.

The voltages limit at the catenary are set to an interval around the nominalvalue. Voltage- and power ratings of the converter are set to their physicallimits. The boundaries of the optimizations problem are:

Pmax = 1 p.u. (4.1)Qmax = 1 p.u. (4.2)Smax = 1 p.u. (4.3)PL,max = 0.0482 p.u. (4.4)

0.8 p.u. ≤ UACjsw≤ 1.1 p.u (4.5)

0.4 p.u. ≤ UAC,catenary ≤ 1.3 p.u. (4.6)0.95 p.u. ≤ UDCj

≤ 1.05 p.u. (4.7)0 p.u. ≤ IACjsw

≤ 1.25 p.u. (4.8)−π ≤ θkj ≤ π (4.9)

−Pmax ≤ PACGjsw≤ Pmax (4.10)

−Qmax ≤ QACGjsw≤ Qmax (4.11)

−Smax ≤ SACjsw≤ Smax (4.12)

−Pmax ≤ PDCDjsw≤ Pmax (4.13)

0 ≤ Pinvjsw≤ PL,max (4.14)

0 ≤ Prectjsw≤ 0.9 · PL,max (4.15)

0 ≤ Pswjsw≤ PL,max (4.16)

0 ≤ Plosses ≤ 5 · Smax (4.17)

26

4.1. MINIMIZING LOSSES

4.1.2 The Objective FunctionThe objective function is to minimize the total losses of the HVDC supplysolution. However, the objective function is a simplification of the real system.In a real system, the control of the converters has to be taken into account, andthat measured data is obviously not given in real time. Therefore, it is assumedwith the objective function that the converters are optimally controlled andthat data measured is in real time data. Thus, the objective function of theHVDC feeder solution is expressed by

min Plosses =NAC∑j=1

Ploss−ACj+

NDC∑j=1

Ploss−DCj+

Nsw∑jsw=1

Pswjsw(4.18)

SummaryThe optimization problem can in short be described by the following bullets:

• Minimize Equation (4.18)

• Under the constrain of equations (3.1),(3.2),(3.3),(3.4),(3.7),(3.8),(3.14),(3.15),(3.16),(3.17),(3.18),(3.19),(3.23),(3.24),(3.20),(3.21),(3.22),(3.38),(3.25),(3.26),(3.27),(3.28),(3.30)(3.31),(3.32)(3.33),(3.34),(3.35).

27

Chapter 5

Software and Implementation

5.1 Platform for SolutionOptimization problems that require MINLP are difficult to solve, and efficientsolvers and algorithms are being developed. BONMIN (Basic Open-sourceNonlinear Mixed INterger) [1], distributed by COIN-OR (Computational In-frastructure for Operations Research) has been selected as the solver for thisthesis, as this solver is free and open-source, and, has reached academic ac-ceptance [22].

5.1.1 GAMSGAMS (General Algebraic Modeling System) is a high-level modeling systemfor mathematical programming and optimization [2].

GAMS has a compiler and several integrated solvers for different optimiza-tion problems. It is made for large, complex modeling applications. Furtherinformation can be found in [2].

5.1.2 BONMINThe solver BONMIN is programmed in C++, and have several algorithms inorder to solve optimization problems that require MINLP.

The solver program BONMIN uses IPOPT1 and CBC2 to solve the NLPand MILP that are needed to solve this thesis optimization problem. IPOPTis an interior point algorithm for solving the NLP and CBC is Branch andBound algorithm to solve the MILP. The interior point algorithm and Branch

1 Interior Point OPTimizer, URL: https://projects.coin-or.org/Ipopt/2 Coin-or branch and cut, URL: https://projects.coin-or.org/Cbc/

29

CHAPTER 5. SOFTWARE AND IMPLEMENTATION

and Bound algorithm are described in detail in [7], [14], [17] and [25]. Furtherdetails about BONMIN can be found in [33].

5.1.3 Optimization with Rotary ConvertersFor solving the optimization problem with rotary converters, a light versionof a program that is presented in [3] and applied in [13] is utilized. Theequations of the rotary converter and power flows are described in Chapter 3.In order to solve the problem, which is a system of equations and in GAMS iscalled Constrained Nonlinear System (CNS), the solver CONOPT is used [2].Further information can be found in [3] and [13].

5.2 ImplementationThe parameters of the HVDC feeder solution are modeled in MATLAB. Theseparameters are the power demand of the trains, the values of the admittancematrices, voltage level of the slack bus and reference angle. The data is trans-ferred to GAMS, were BONMIN solves the optimization problem. The datawith the solution variables of the optimization problem is then transferredback to MATLAB for analysis.

30

Chapter 6

Studied Cases

6.1 Investigated CasesThere are two main cases that are investigated:

1. Dense traffic

2. Light traffic

Figure 6.1. Flow chart of the studied cases.

31

CHAPTER 6. STUDIED CASES

Furthermore, these cases are investigated for a) both AT and BT catenarysystems, b) different power factors at train locomotive, and c) smaller anddensely installed converters of the rating of 2 MVA. From these case studies,knowledge should be gathered on how the different RPSS behaves and thedifferent cases are compared to each other. See Figure 6.1 for the flow chartof the studied cases.

The reference case of the HVDC feeder solution is:

• Converters have a rating of 5 MVA.

• The distance between the converters is 33.3 km.

• The power factor of the train locomotives is 1.

6.1.1 Train TrafficThe locomotives are assumed to consume/regenerate 4 MW in all cases. Allsystems investigated have a total installed apparent power of 20 MVA. How-ever, the decentralized system with BT catanery system has a total installedpower of 40 MVA due to high transmission losses, and Train III consumptionin this case is reduced by 4% when the power factor is 0.8 in order to obtainfeasible solution for the system. The length of the systems are 100 km andthe train traffic setup is:

• Dense traffic case: Four trains in total, three consuming and onebraking, see Figure 6.2 and Table 6.1 for train actions.

• Light traffic case: Two trains in total, one consuming and one braking,see Figure 6.3 and Table 6.2 for train actions.

32

6.1. INVESTIGATED CASES

Figure 6.2. Dense traffic case.

Figure 6.3. Light traffic case.

33


Train Number of locomotives ActionI 1 consuming (inductive load)II 1 regenerating (capacitive load)III 2 consuming (inductive load)IV 1 consuming (inductive load)

Table 6.1. Dense traffic case.

Train Number of locomotives ActionI 1 consuming (inductive load)II 1 regenerating (capacitive load)

Table 6.2. Light traffic case.

34


6.1.2 Investigation of the CasesThe node numbers of the systems are described in Tables 6.3 and 6.4. Theconverters are divided in a DC- and an AC side, with different node numbers,as seen in Table 6.5. Comparison is done with today’s Swedish RPSS and thelosses of the rotary converters are neglected. In the centralized solutions forthe HVDC reference case and Smaller Converters, the converters have beenreplaced with High Voltage Transformers (henceforth denoted HVT). The firstconverter at 0 km and last converter at 100 km are replaced with 25 MVAtransformers (ZT 225) and the rest of the converters are replaced with 16 MVAtransformers (ZT 216). Furthermore, the rotary converters are placed at 0 kmand 100 km.

Table 6.3. Heavy traffic case.

Nodes Descriptionjac ∈ 1...8, 17...20 AC nodes index

jac,train nodes ∈ 17, 18, 19, 20 Train nodes indexjdc ∈ 9...16 DC nodes index

jdc,3AC-DC ∈ 13 DC slack node index

Table 6.4. Light traffic case.

Nodes Descriptionjac ∈ 1...8, 17...20 AC nodes indexjac,train nodes ∈ 17, 18 Train nodes index

jdc ∈ 9...16 DC nodes indexjdc,3AC-DC ∈ 13 DC slack node index

Table 6.5. AC and DC converter nodes.

Converter number jsw AC node index DC node index1 5 92 6 103 7 114 8 12

35


Different power factors at trains

Today’s locomotives can operate at unity power factor. However, there aremany older locomotives still in use that do not operate at unity power factor.Thus, investigating different power factors of is relevant for the study of theHVDC solution. Increased reactive power demand will increase the currenton the AC side of the systems, thus increasing losses in the systems.The basepower of the systems is set to 5 MVA.

OPF sensitivity to converter losses

It is of interest to investigate how increased losses in the converters affect thesystem. The converter loss function is therefore multiplied with different fac-tors, in order to investigate how sensitive the OPF solution is if the converterlosses changes. The base power of the systems is set to 5 MVA.

Smaller converters

Smaller converters, in the order of 2 MVA, are placed at a distance of ap-proximately 11 km from each other. It is only investigated for locomotivesoperating at power factor of 1. This is to investigate how the system behaveswith smaller and more densely installed converters. It can lead to that lossesare further reduced, and indicate a direction of which system configurationcould be developed. The base power of the system is set to 2 MVA in thiscase. For the used system layout, see Figures 6.4 and 6.5. For the nodenumbering of the systems c.f. Tables 6.6, 6.7 and 6.8.

Figure 6.4. Dense traffic case.

36


Figure 6.5. Light traffic case.

Table 6.6. Heavy traffic case with 2 MVA converters.

Nodes Descriptionjac ∈ 1...20, 41...44 AC nodes indexjac,train nodes ∈ 41...44 Train nodes index

jdc ∈ 21...40 Total DC nodesjdc,3AC-DC = 31 DC slack node index

Table 6.7. Light traffic case with 2 MVA converters.

Nodes Descriptionjac ∈ 1...20, 41...42 AC nodes indexjac,train nodes ∈ 41, 42 Train nodes index

jdc ∈ 21...40 Total DC nodesjdc,3AC-DC = 31 DC slack node index

37


Table 6.8. AC and DC converter nodes for 2 MVA converters

Converter number jsw AC node index DC node index1 11 212 12 223 13 234 14 245 15 256 16 267 17 278 18 289 19 2910 20 30

38

Chapter 7

Results of Simulations

7.1 Dense Traffic Case

7.1.1 AT CatenaryDifferent power factors

Comparing the transmission losses on the AC side for the different systemsshown in Table 7.1, there is a comparably small difference on the AC systemlosses between HVDC- and a centralized solution. For both in the HVDC- andthe centralized solution, the current flows via the high voltage transformersand the converters, thus, minimizing the current in the catenary.

Table 7.1. AC transmission losses for the three supply systems, [MW].

PF at Train AT, HVDC AT, Centralized AT, Decentralizedcosφ = 1 0.1524 0.1760 0.3205

cosφ = 0.9 0.1973 0.2245 0.4095cosφ = 0.8 0.2228 0.2739 0.5290

Table 7.2 illustrates the detailed losses of the HVDC feeder solution with 5MVA converters, with AT catenary system.

Table 7.2. Detalied losses of the HVDC fedeer solution with AT, [MW].

PF at Train Converter losses AC losses DC losses Total lossescosφ = 1 0.4452 0.1524 0.0273 0.6249

cosφ = 0.9 0.4856 0.1973 0.0273 0.7102cosφ = 0.8 0.5753 0.2228 0.0282 0.8263

39

CHAPTER 7. RESULTS OF SIMULATIONS

When the train locomotives are operating at power factor of 1, the voltagelevels at the train locomotives with AT catenary are at acceptable levels forthe three different feeding systems. However, if the train locomotives startconsuming reactive power, the voltages at the train locomotives and catenarystart to decrease. The HVDC solution and a centralized solution, has theability to maintain almost constant voltage levels at the train locomotivesand catenary, compared to the decentralized system. See Figures 7.1, 7.2 and7.3.

Figure 7.1. Voltage levels at trains, with three different supply solutions.PF at trains are 1.

40

7.1. DENSE TRAFFIC CASE

Figure 7.2. Voltage levels at trains, with three different supply solutions.PF at trains are 0.9.


41



In the interval B in Figure 7.4 the system is not sensitive to increased losseson the converter and the total AC losses constant. The state of the systemchanges only when the converter losses are multiplied with the factors 0.4 or1.6, thus either the AC losses increases or decreases. The converter states aredescribed in Table 7.3. Looking more in detail at the interval C, converter 2is turned on to supply the power demand as the converter consumes a highamount of active power. The AC transmission losses decreases due to lessdistance for the active power to flow.

Figure 7.4. Converter losses, AC transmission and DC transmission lossesdepending of different factors that the converter loss function is multipliedwith.

Table 7.3. Converter states for Figure 7.4.

Description Converter 1 Converter 2 Converter 3 Converter 4A ON ON ON ONB ON OFF ON ONC ON ON ON ON

In Figure 7.5, the total losses of the HVDC system increases more or lesslinearly with increasing factor, which indicates that global optimums havebeen found for every factor.

42


Figure 7.5. OPF solutions for different factors.

Smaller Converters

The transmission losses on the AC side of the system are approximately 50%less with smaller converters compared with the reference HVDC feeder solu-tion. The AC transmission losses have been reduced by 78% of the trans-mission losses of the decentralized system, and approximately 40% of thecentralized system with 10 HVT, c.f. 7.4

Considering the total losses, the both HVDC feeder solutions have compar-atively high converter losses. However, a great improvement has been madeconsidering the AC transmission losses, shown in Table 7.5

Table 7.4. AC transmission losses with 2 MVA converters, and centralizedsystem with 10 HVT

PF at Trains AT, OPF HVDC AT, Centralized AT, Decentralizedcosφ = 1 0.0700 0.1200 0.3205

Table 7.5. Detalied losses of the two types of HVDC feeder systems, powerfactor at trains are equal to 1.

Type of converters Converter losses AC losses DC losses Total2 MVA 0.4340 0.0700 0.032 0.5365 MVA 0.4452 0.1524 0.0273 0.6249

43


The voltage levels are all above the nominal value of 15 kV, c.f. Figure 7.6,and the trains tractive force is unaffected.

Figure 7.6. Voltages at train with different RPSS.

7.1.2 BT CatenaryDifferent Power Factors

With a BT catenary system, the impedance is much higher compared to anAT catenary system. When the locomotives are operating at a power factor ofcos(φ) of 0.8, the AC transmission losses of the decentralized system are high(around 30%). Using the HVDC feeder solution the AC transmission lossesare reduced by approximately 80% compared to the decentralized system.

Table 7.6. AC transmission losses for the three supply systems, [MW].

PF at Train BT, HVDC BT, Centralized BT, Decentralizedcosφ = 1 0.6813 1.1615 2.4300

cosφ = 0.9 0.9327 1.6330 4.8400cosφ = 0.8 1.3275 2.2620 7.3000

The total losses for the HVDC feeder solution, are very low or low compared tothe centralized and decentralized system transmission, where in fact the lossesof rotary converters are neglected, c.f. Tables 7.6 and 7.7. As described inChapter 6, the consumption of Train III in the decentralized system is reducedby 4% when the power factor is 0.8 at the train, and the in this case the totalinstalled power is doubled.

44


Table 7.7. Detalied losses of the HVDC solution, [MW].

PF at Train Converter losses AC losses DC losses Total lossescosφ = 1 0.5163 0.6813 0.0321 1.2298

cosφ = 0.9 0.5648 0.9327 0.0333 1.5308cosφ = 0.8 0.6312 1.3275 0.0345 1.9932

Voltage levels at the trains with the OPF HVDC feeder solution are at ac-ceptable level compared to the decentralized solution, where the voltage levelsare significantly low. The OPF HVDC solution can maintain better voltagelevels than centralized and decentralized systems, since the HVDC system canproduce/absorb reactive power at every 33 km, which increases the voltage.See Figures 7.7, 7.8 and 7.9.


45




46



With a BT system, all converters are ON, since the higher impedance of asuch system. The DC side losses are constant, even with the increase of lossesat the converters. It is clear from Figure 7.10 that the line losses starts toincrease quadratically as the converters losses are high and more currents isneeded, thus the increase of AC transmission losses.

Figure 7.10. Converter losses, AC transmission and DC transmissionlosses depending of different factors that the converter loss function is multi-plied with.

47


Optimal solutions have been found for all factor as shown in Figure 7.11,


Smaller converters

The AC transmission losses have been reduced approximately by 50% com-pared to the HVDC solution with 5 MVA converters, as shown in Table 7.9.Total losses have also decreased due to the converter rating, meaning less cur-rent through each individual converter unit, compared to 5 MVA converters.

Table 7.8. Transmission losses with 2 MVA converters, and centralizedsystem with 10 HVT

PF at locomotive BT, HVDC BT, Centralized BT, Decentralizedcosφ = 1 0.3440 0.5250 2.4300

Compared to the decentralized system, the OPF controlled HVDC systemwith 2 MVA converters, provide a reduction of the AC transmission losses by65% and a 34% compared to centralized with 10 HVT, c.f. Table 7.8.

48


Table 7.9. Detalied losses of the two types of HVDC feeder systems, powerfactor at trains are equal to 1.


Figure 7.12, shows the voltage levels for all five methods investigated. BothOPF HVDC systems can maintain voltage levels. However, the centralizedsys-tem with 10 HVT can maintain similar voltage level as the HVDC solutions.


49


7.2 Light Traffic Case7.2.1 AT CatenaryDifferent power factors

Losses on the catenary are shown in Table 7.10. With decreasing power factor,the HVDC solution provide 44% less AC losses, compared to the decentralizedsystem. However, when replacing the converters with HVT, the AC lossesdecrease with 77% compared to the decentralized system, due to more powerwill flow in the HVAC line.

Table 7.10. AC losses for the three supply systems, [MW].

PF at Train AT, HVDC AT, Centralized AT, Decentralizedcosφ = 1 0.0985 0.0397 0.1760

cosφ = 0.9 0.1216 0.0488 0.2245cosφ = 0.8 0.1540 0.0616 0.2739

Table 7.11 illustrates the contribution of the the different parts of the systemto the total losses of the HVDC solution for an AT catenary, with 5 MVAconverters.

Table 7.11. Detalied losses of the HVDC feeder solution with BT, [MW].

PF at Train Converter losses AC losses DC losses Total lossescosφ = 1 0.0763 0.0985 1 · 10−8 0.1748

cosφ = 0.9 0.0766 0.1216 1 · 10−8 0.1982cosφ = 0.8 0.0770 0.1540 1 · 10−8 0.2310

The voltages at the trains are at acceptable levels, for all three system. SeeFigures 7.13, 7.14 and 7.15.

50

7.2. LIGHT TRAFFIC CASE



51



52



At A, when the converter losses are low, all converter are turned on. At B,converters 2 and 3 are turned off and the power will flow via the catenary andHVDC grid. However, at point C only converter 1 is ON in order to supplythe system losses. The power will flow via the catenary, as activating moreconverters will increase the overall losses. As consequence the AC losses areconstant, see Figure 7.16 and Table 7.12.


Table 7.12. Converter states for Figures 7.16

.

Description Converter 1 Converter 2 Converter 3 Converter 4A ON ON ON ONB ON OFF OFF ONC ON OFF OFF OFF

53


For the light traffic case, OPF solutions have been found for every factor, c.f.Figure 7.17.


54


Smaller converters

The losses in the catenary are the same, even with a change of converter type,c.f. Table 7.13. However, the conversion losses are smaller due to the use ofsmaller converters, c.f. Table 7.14.


PF at Trains AT, OPF HVDC AT, Centralized AT, Decentralizedcosφ = 1 0.098 0.030 0.1760

Table 7.14. Detalied losses of the two types of HVDC feeder systems,power factor at trains are equal to 1.

Type of converters Converter losses AC losses DC losses Total2 MVA 0.032 0.098 1 · 10−7 0.1305 MVA 0.076 0.098 1 · 10−8 0.174

The voltage at the trains for all RPSS is presented in Figure 7.18. All voltagesabove the nominal value of 15 kV.


55


7.2.2 BT CatenaryDifferent power factors

With BT catenary, the losses at the AC side are approximately 51% less thanthe decentralized system, and losses are reduced approximately 64% with theuse of a centralized system. Thus, the centralized solution grants the lowestAC losses of the three systems. However, it should be noted that the losses ofthe rotary converters are neglected. See Table 7.15.

Table 7.15. AC losses for the three supply systems, [MW].

PF at Train BT, HVDC BT, Centralized BT, Decentralizedcosφ = 1 0.2273 0.1660 0.4733

cosφ = 0.9 0.2856 0.2055 0.5466cosφ = 0.8 0.2967 0.2606 0.6547

The larger share of the power will flow on the AC side of the HVDC system,due to more power flows through the converters and will lead to increasedoverall losses. This can be seen in Table 7.16, where the DC losses are lowif compared to the other previous cases investigated, e.g DC side losses whenstudying the Dense traffic case.

A decreasing power factor at the train locomotives implies higher converterlosses because more current is needed to supply the increased reactive powerdemand. Thus, the DC side losses and AC side losses increase, see Table 7.16.

Table 7.16. Detailed losses of the HVDC solution, [MW].

PF at locomotive Converter losses AC losses DC losses Total lossescosφ = 1 0.2474 0.2273 0.0015 0.4762

cosφ = 0.9 0.2763 0.2856 0.0015 0.5635cosφ = 0.8 0.3552 0.2967 0.0018 0.6536

When Train II is operating at power factor 0.9 and 0.8, the decentralizedsolution and centralized solution have voltages that are quite high and cancause serious damage to the equipment at the train. The HVDC solution hasalso quite high voltage levels when Train II is operating at power factor 0.8.The converters closest to Train II should absorb the reactive power to lowerthe voltage. However, this leads to increased losses on the converters, thusincreasing the overall losses.

The voltage level for Train I are at acceptable levels for the three RPSSinvestigated, see Figures 7.19, 7.20 and 7.21.

56




57



58



At A, all converters are turned on and losses on the AC side are low. At B,converter number one is turned off, due increased losses on the converters.

Converters 2 and 3 are turned on at C. These converters are close to thetrain, and certain power will flow via the converter and rest via the catenary.Further increasing converter losses, at D, converters 2 and 3 are turned offand converter 1 and 4 are turned on. Train II is closer to converter number3, however if it is turned on, more power will flow via this converter, thusincreasing the overall losses.

At E, the AC losses are highest and only converter 1 is turned on. Turningon more converters, will increase the overall losses. Thus, more current isflowing via the catenary, the losses increase quadratically and the influence ofthe quadratic term of equation (3.5) is clear now, see Figure 7.22 and Table7.17


Furthermore, at F, converter 3 is turned on, the AC transmission lossesare reduced at the cost of increased converter losses. At G converter 1 and 2are turned on and the rest of the converters are off. Due to the high converter

59


losses, it is more beneficial to transfer the power via the catenary and onlyactive converter 1 and 2 to compensate for the transmission losses.

Table 7.17. Converter states for Figure 7.22

Description Converter 1 Converter 2 Converter 3 Converter 4A ON ON ON ONB OFF ON ON ONC OFF ON ON OFFD ON OFF OFF ONE ON OFF OFF OFFF ON OFF ON OFFG ON ON OFF OFF

OPF solutions have been found for all factors, as shown in Figure 7.23


60


Smaller Converters

The AC losses are reduced by 78 %, with the use of 2 MVA converters com-pared to the decentralized system, c.f. Table 7.18. However, accounting forall losses of the 2 MVA system, the losses are reduced by 33%. The converterlosses increase due to the use of more converters. This means, more powerflows via the DC grid, thus reducing losses on the AC side, c.f. 7.19.


PF at Trains BT, OPF HVDC BT, Centralized BT, Decentralizedcosφ = 1 0.100 0.110 0.4733

Table 7.19. Detalied losses of the two types of HVDC feeder systems,power factor at trains are equal to 1.


The resulting voltages in this case are presented in Figure 7.24, along withthe other voltages of the other systems.


61

Chapter 8

Analysis and Discussion

8.1 Analysis

8.1.1 Different PF at Train Locomotives: Dense TrafficCase

AT

By utilizing an AT catenary system, the AC transmission losses are reducedby approximately 51% with HVDC solution compared to the decentralizedsystem. The converter losses account for a large part of the total losses of theHVDC solution and the advantages are not obvious. However, it should benoted that the converter losses of the decentralized system are neglected.

The impact of the small ohmic resistance of the AT catenary lines and thedoubling of the voltage explains why the voltage level are at adequate levels forthe decentralized system, even with several trains operating. However, for allthree RPSS, the voltages are at acceptable levels, and there are small differencein voltage levels between the trains. Thus, the voltage at the catenary is aboveits nominal values.

BT

With the HVDC solution installed in a BT catenary system, the losses decreaseby approximately up 81% compared to the decentralized system, where thetotal installed power is doubled and Train III consumption is reduced by 4%int the case when the power factor is 0.8. It outperforms the centralizedsolution, even with converter losses included.

With decreasing power factors, the voltages at train locomotives are con-siderably low with decentralized solution and the tractive force of the locomo-

63

CHAPTER 8. ANALYSIS AND DISCUSSION

tives is significantly affected, implying that the train have to drive at slowerspeed or even have to stand still [11]. However, the HVDC can maintain avoltage level just around the nominal value due to the ability to produce reac-tive power at several locations in the system. Thus, the tractive force of thelocomotives is not affected. However, for Train III the tractive force is slightlyreduced according to [11], due to the Rc4 locomotive needs at least 14.5 kVfor full tractive force.

8.1.2 Different PF at Train Locomotives: Light TrafficCase

AT

The HVDC solution, with AT catenary, gives a significant reduction of losses.However, the centralized solution performs better, as power will flow via the132 kV line. In the HVDC solution, most power flows via the catenary, asturning on converters would increase the overall losses. Thus, the centralizedsolution performs better than the HVDC solution in this cases, as the trans-former losses are low, compared to the converter losses. For all power factorsinvestigated, the voltage levels are at acceptable levels.

BT

With a BT catenary system, the AC transmission losses with the HVDC solu-tion are reduced by approximately 52% compared to the decentralized system.The use of HVT reduces the AC losses more than the HVDC solution, sincethe HVT in this case causes less losses, compared to the HVDC converters.

The voltages are actually high for Train II with both centralized and de-centralized systems, when the power factor decreases at the trains, as to highvoltage may damage the equipment on the train. These systems have limitedability to compensate for the increased reactive power at the catenary. In op-posite to these systems, the HVDC can compensate for the increased reactivepower generation due the ability of the converters to absorb reactive powerand to keep the voltage around the nominal value.

8.1.3 OPF sensitivity to Converter Losses: Dense TrafficCase

AT

With the AT catenary system, the HVDC solutions are sensitive to differentfactors of converter losses. However, this sensitivity depends on how high the

64

8.1. ANALYSIS

converter losses are and if the converter losses increase or decrease. It wasshown in Figure 7.4 that even with increased losses in the converters the ACtransmission losses were almost constant. This is due to the ability of theconverter to absorb reactive power and there is no point to turn-on or turn-offa converter in the interval B. If a converter is turned off at the factor of 0.4,the current on the other converters would increase. Thus, the converter losseswould increase and the AC transmission losses would increase also, leading tohigher overall losses.

This is further discussed in section 8.1.7, where the loss function is anal-ysed.

BT

With the BT catenary, the AC transmission losses of the OPF solution arenot sensitive to increased converter losses, due to the fact that BT having highimpedance and the AC transmission losses are then high. Thus, all convert-ers are needed, in order to supply the power demand. When the converterlosses are high, the influence of the quadratic term of Equation (3.5) becomesdominating due to the increased current in the AC system.

8.1.4 OPF sensitivity to Converter Losses: Light TrafficCase

AT

With the AT catenary, the system is not sensitive to increased converter losses,and only one converter is on, since all the power will flow via the AC grid duethe small impedance of the AT catenary.

The AC losses can be reduced by turning on more converters. However,turning on a converter leads to increased overall losses that are higher thanwhen the converter is turned off. Thus, the power have to flow via the catenary.

BT

The BT catenary with light traffic is sensitive to increased converter losses. Ifthe converter losses are 50% higher than normal, the AC transmission lossesincreases due to more current in the system, and at the factor of 2.5 thelosses increases quadratically due to Equation (3.5) and the line losses ofstarts to become dominant. When converter losses starts to become high, theturning off converters is required at the cost of increased AC transmissionlosses. However, the total losses are less compared if converters are turned on.

65


8.1.5 Smaller Type of Converters: Dense Traffic CaseAT

The AC transmission losses with AT catenary are reduced by 78% of the lossesof the decentralized system. However, the converters still stands for a largepart of the losses, as more converters have to be used. The power have moreoptions to flow via the DC grid, as more converters are used. Thus, the ACtransmission are significantly reduced.

With the utilization of 2 MVA converters, the voltage level are in generalno better than for the other supply solutions, as the voltage level for all supplysolutions are above the nominal level.

BT

With BT catenary, the AC transmission losses are reduced by 85 % comparedto the decentralized system, due to the ability to extract power at differenceplaces along the catenary. Thus, reducing the current on the catenary.

Comparing the overall losses of the 5 MVA converters to 2 MVA convertersshows that the losses were further reduced by 32%.

The impact on voltage levels with BT catenary can be observed. Thedecentralized solution cannot keep the voltage at acceptable levels as the cen-tralized and the HVDC systems do. However, the HVDC solution is evenmore superior than the centralized system, due the ability of reactive powercompensation.

Comparing the 2 MVA system with the 5 MVA system, the 2 MVA systemcan better maintain voltage levels at the trains and catenary, as shown by thevoltage difference between the trains. This is due to more places to drawpower from the DC grid and shorter distances between the converters.

8.1.6 Smaller Type of Converters: Light Traffic CaseAT

The AC transmission losses, with AT catenary, are not influenced when al-tering from 5 MVA converters to 2 MVA converters, c.f Table 7.14. However,converter losses are higher with 5 MVA converters due to their size and rating.

The power flows via the catenary, from Train II to Train I, see Figure 6.5.The converter closest to Train I transfers the necessary power from the slackbus to keep power balance.

Replacing the converters with 10 HVT, the AC transmision losses are lowcompared to the 2 MVA HVDC converter solution. This is reasonable as the

66

8.1. ANALYSIS

high voltage transformers cause lower losses compared to the converters.The voltages at the trains are about the same and no large voltage drops

occur. All systems with AT catenary perform rather well. This is due to thesmall load in the system and the impedance of the AT catenary system.

BT

With BT catenary system, the losses are reduced by 78% compared to thedecentralized system. The AC losses are small, however converter losses andDC losses have increased compared to the AT catenary system. This impliesthat a larger portion of the power flow will flow via the DC grid, compared tothe AT catenary system.

With densely and smaller installed converters, the HVDC system has theability to maintain voltage levels at the catenary above the nominal value of15 kV, as the voltage difference between the train are small.

8.1.7 Converter Loss FunctionIn Chapter 3, the converter loss function was presented. The losses weremodelled as a second order polynomial. The quadratic term represents andthe linear term represent the current dependent losses and, the constant termrepresent the no-load losses. The equation is dependent of the AC currentmagnitude at the converter AC side, see Equation (8.1).

Pinvjsw= 0.0135 · I2

out,ACjsw+ 0.0097 · Iout,ACjsw

+ 0.015 (8.1)

67


Figure 8.1. Influence of the different terms of the converter loss function

From Figure 8.1 it becomes clear that the constant term, representing theno-load losses, contributes most to the losses of the converter. The quadraticterm begins to affect the losses at 1.1 p.u current, and the linear term doesnot have significant influence on the converter losses. It can be concluded thatmost of the losses on the converters are no-load losses.

In Figure 8.2, a comparison is done between the use of one converter unitand two converter units, were the base power is set to 5 MVA and base voltageis set to 15 kV.

68

8.2. DISCUSSION

Figure 8.2. 1 converter unit vs 2 converter units

Utilizing two converter units, the current is halved. However, the no-loadlosses are high and the benefit of having two converter units to decrease theconverter- and AC transmission losses is not obvious. The converter losseswill affect the total system losses more than the AC losses. Therefore, thesolver will turn off converter units, unless the AC transmission losses becomehigh enough that it will effect the total system losses more than the converterlosses.

8.2 Discussion8.2.1 Impact of Catenary SystemThe impact of the catenary layout is obvious. The AT catenary in a decen-tralized system in dense traffic case and light traffics case is sufficient enoughand voltages were above nominal values.

Opposed to the AT catenary system, the BT catenary system caused largevoltage drops due its high ohmic impedance. In the dense traffic case, thevoltage level at the trains were considerably low, which influences the train

69


operation. In the light traffic case the opposite happens; when the trainlocomotive is braking, the voltage level is high and can cause damage to thetrain locomotive equipment. Thus, the BT catenary system will impose acondition on the power factor of the operating trains. The HVDC feedersolution offers the ability to compensate for the reduced or increased voltagelevels, and at the same time decrease the AC transmission losses of the system.

8.2.2 Centralized SolutionThe comparison between the centralized feeder system layout and the HVDCfeeder solution may be unfair, since in reality the high voltage transformersused have ratings of 16 MVA and 25 MVA, and are normally placed at longerdistances from each other than the 33 km and 11 km used in the cases inves-tigated. In certain cases the total active power losses of the OPF controlledreference HVDC feeder solution were less than the transmission losses of thecentralized and decentralized feeder solutions, where the conversion losses ofthe rotary converter were neglected.

Furthermore, in order to achieve about the same voltage levels at trainlocomotives and reduced losses as the 2 MVA HVDC solution proposed, 10HVTs were needed for the centralized system. This is unrealistic, by remem-bering that HVTs are considerably large and have a weight of 70 tons, and anextra 132 kV power line is needed. In addition, permissions are required fromthe landowners along the railway to install the 132 kV power lines, and quiteoften it is hard to acquire these permissions, especially in densely populatedareas.

Opposed to the centralized solution, the HVDC supply solution have thebenefit of small land usage. Thus, there is less need for permissions fromlandowner along the railway as the cable can be buried along the railway.

70

8.2. DISCUSSION

8.2.3 Improved Voltage LevelsThe objective function of the optimization problem was to reduce the totalactive power losses of the HVDC supply system. The voltages obtained bythe OPF solution were acceptable, as the voltages level were relatively closeto the nominal value. However, the voltages could be improved even more ifthe objective function was set to minimize the voltage difference at the trainlocations, that’s mean that the objective functions is:

min (Utrain − 1)2 (8.2)

This would most definitely lead to improved voltage levels, compared to thevoltages of the original optimization problem. However, the results are higherAC transmission losses, as more reactive power would be required.

71

Chapter 9

Conclusions and Future Work

9.1 Conclusions

An OPF HVDC feeder solution, with converters rated at 5 MVA and installed33 km from each other, has been investigated for two cases of a given timeinstance of train traffic with AT and BT catenary system. The convertersrated at 5 MVA were placed equidistantly, and transmission losses and voltagelevels were investigated for different types of cases and compared with existingfeeder solutions.

An Unified AC/DC load flow approach was chosen and binary variablefor the unit commitment was used. Optimal power flows were achieved forall cases investigated. Thus, setting a theoretical upper bound on how muchlosses can be reduced with a OPF controlled HVDC feeder solution.

The OPF controlled HVDC feeder solution with a BT catenary systemsexcels when trains are operating at a lower power factor than 1 compared tothe existing feeding systems. In one of the cases the transmissions losses were3-4% compared to 11-30% for decentralized system and 7-13% for centralized.Thus, a reduction of transmission losses up to 80% was achieved. With ATcatenary system the transmission losses were reduced up to 50% with HVDCfeeder solution, compared to other feeding systems. Voltage levels were atacceptable levels with the two types of catenary system. However, the HVDCfeeder can maintain better voltage levels at the train locations, compared withother solutions, implying acceptable voltage levels on the overhead contactline.

A HVDC feeder solution with densely and smaller converter of the rating2 MVA installed at 11 km from each other was also investigated, and ACtransmission losses were further reduced by approximately 50% compared tothe HVDC reference case, and the voltage levels were further improved.

73

CHAPTER 9. CONCLUSIONS AND FUTURE WORK

Depending on which catenary system is installed and depending on thetraffic density is high or low, proper control of the converters would lead to asubstantial reduction of the AC transmission losses as it has been presentedthat AC transmission losses in certain cases are sensitive to changes in theconverter loss function.

In all, this thesis presents some advantages of the HVDC feeder solution.It shows that the OPF controlled HVDC feeder solution offers a significantreduction of transmission losses and improved voltage levels on the overheadcontact line compared with existing feeding solutions.

9.2 Future WorkThis thesis is a step in an area of a new method for supplying the railwaysystem. However, several aspects have be to investigated, in order for thissystem to be implemented in near future.

9.2.1 Moving TrainsThe trains were modelled as static loads, thus giving only an instant view ofthe systems properties. Using a dynamic approach, where the train positionschange during time, would give a more realistic view of the system. A programcalled TPPS, presented in [3], could be altered to fit the HVDC solution. Thus,providing a more realistic view of the behaviours of the system.

However, the unit commitment presented in this thesis is modelled witha binary variable. Due to the nature of the load flow equations and theuse of binary variables makes the simulations in some cases time consuming.Improvements could be done in order improve overall speed, especially if thebinary variables could be reduced.

9.2.2 Economical AspectsThis thesis has investigated some technical and mathematical properties, andbehaviours of the HVDC feeder solution. However, the economical aspect is ofcourse important. At the moment, there is no investigation of the costs of thissystem. Comparison should be done with existing feeder systems. However,it is believed that the HVDC feeder solution has a reasonable investment costand has the potential to be competitive.

74

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[3] Lars Abrahamsson. Railway Power Supply Models and Methods for Long-term Investment Analysis, 2008. Royal Institute of Technology, ElectricPower Systems, Trita:2008:036.

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[5] Lars Abrahamsson, Stefan Östlund, and Lennart Söder. OPF Modelsfor Electric Railways Using HVDC. In International Conference on Elec-trical Systems for Aircraft, Railways and Ship Propulsion, october 2012.Accepted for publication.

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[20] Edward Friman. Impedanser för KTL och 132 kV, 30 kV och 15 kV ML.Technical report, Swedish Transport Administration, 2006.

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[24] Tommy Kjellqvist. On Design of a Compact Primary Switched Con-version System for Electric Railway Propulsion. PhD thesis, Royal Insti-tute of Technology, Electrical Engineering, Electrical Machines and PowerElectronics, 2009.

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77

Appendix A

Numerical data

Denotation Value Additional informationUbase,AC 15 kV Base voltage of the AC part of the systemUbase,DC 160 kV Base voltage DC part of the systemUbase,132 132 kV Base voltage of HV line, centralized systemSbase 5 MVA Used for HVDC, 5 MVA converters,

centralized and decentralized simulationsSbase 2 MVA Used for HVDC, 2 MVA convertersZAT 0.0335 + 0.031i Ω/km 120 mm2, 2AT, FÖ

Zinit,F C 0.189 + 0.293i Ω AT initial impedanceZBT 0.2 + 0.2i Ω/km 109 mm2 2A, FÖ

ZDC,Cable 0.1175 Ω/km 240 mm2, bipolarZT 216 0.065 + 0.85i Ω Impedance of 16 MVA transformerZT 225 0.037 + 0.54i Ω Impedance of 25 MVA transformerUg 16.5 kV generator output voltageθ50 0 rad angle of the 50 Hz sideX50 15 % grid reactance, 50 Hz gridQ50 0 p.u absorbed reactive power 50 Hz sideXm

q 1.8176 Ω quadrature reactance, motorXg

q 1.4331 Ω quadrature reactance, generatorTable A.1. Numerical data.

79

Appendix B

Catenary Transformers

B.1 Booster Transformer

Booster transformer, termed BT, have an 1:1 winding. The transformer cur-rent, Ifeeding, generated from feeding station, will set up a flux in the core. Thisflux will oppose the flux set by the winding connected to the return conductor.The winding connected to the return conductor will draw the current from therail, which set up a electromotive force (emf) equal and opposite to the emfof Ifeeding, see Figure (B.1).

The sum of the currents in a BT has to be zero, which is derived from theAmpere turn balance:

N1Ifeeding −N2Ireturn = 0 (B.1)[N1 = N2]⇒ Ifeeding − Ireturn = 0 (B.2)

Figure B.1. Booster Transformer.

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APPENDIX B. CATENARY TRANSFORMERS

B.2 Auto TransformerAuto transformers, termed AT, have two windings connected in series in onesingle leg. If the load is connected to secondary side and applying Ampereturn balance, the equation becomes :

U2

U1= N2

(N1 +N2) = I1

I2(B.3)

Applying Kirchhoff current law, the currents, see Figure (B.2), can bedecided.

i1 = I1 (B.4)i2 = i1 − I2 = I1 − I2 (B.5)

where I1 and I2 are the primary and secondary current in the system, re-spectively. In other words, the negative feeding current and positive feedingcurrent. The currents i1 and i2 are the current through the windings.

Assuming that N1 = N2 and applying equations (B.3) and (B.5), i2 = −I1implies the following:

• That U1 is double the voltage of U2

• The current that goes through the winding will be the same, but oppositevalue.

In this case, U1 is the voltage of the negative feeder of the AT catenary systemand U2 is the voltage at the catenary. The characteristics of this transformermakes good choice for the catenary, as the voltage will be double and thecurrent will be reduced.

Figure B.2. Auto transformer.

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Documents

Optimal Power Flow for an HVDC Feeder Solution for AC Railwayskth.diva-portal.org/smash/get/diva2:565755/FULLTEXT01.pdf · Optimal Power Flow for an HVDC Feeder Solution for AC Railways