Optimisation of transmission systems by use of phase shifting transformers

by use of

by use of

Phase Shifting Transformers

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 13 oktober 2008 om 10:00 uur door

Jody VERBOOMEN

Burgerlijk Werktuigkundig-Elektrotechnisch ingenieur geboren te Leuven, Belgi¨e.

Prof.ir. W.L. Kling

Samenstelling promotiecommissie: Rector Magnificus, voorzitter

Prof.ir. W.L. Kling, Technische Universiteit Delft, promotor Prof.ir. L. van der Sluis, Technische Universiteit Delft

Prof.dr.ir. J.H. Blom, Technische Universiteit Eindhoven

Prof.ir. M. Antal, Technische Universiteit Eindhoven (emeritus) Prof.dr.ir. R. Belmans, Katholieke Universiteit Leuven, Belgi¨e

Prof.dr. G. Andersson, Swiss Federal Institute of Technology, Z¨urich, Zwitserland Dr.ir. P.H. Schavemaker, TenneT Transmission System Operator BV

Dit onderzoek werd uitgevoerd in het kader van het onderzoeksprogramma “In-novatiegerichte Onderzoeksprogramma’s - Elektromagnetische Vermogenstechniek” (IOP-EMVT), dat financieel wordt ondersteund door SenterNovem, een agentschap van het Nederlandse Ministerie van Economische Zaken.

Printed by: W¨ohrmann Print Service, Zutphen, the Netherlands

ISBN 978-90-8570-306-8

Copyright c _{2008 by J. Verboomen}

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilised in any form or by any means, electronic or mechanical, in-cluding photocopying, recording or by any information storage and retrieval system, without permission from the publisher or author.

Optimisation of Transmission Systems by use of Phase Shifting Transformers

In the recent past, power systems have undergone a major transformation as a number of technical and organisational developments have taken place, leading to a change in the way electrical grids are operated and used. An issue that becomes increasingly important is power flow control in meshed systems, which is reflected in the fact that throughout Europe phase shifting transformers (PSTs), which are examples of power flow controllers, are being installed at an increasing number of locations. The specific case of the Dutch and Belgian grid is very interesting in this sense, as several devices are installed in a relatively small area. The operation of a single PST has already an impact on the power flows in the entire network, but if several of these devices are installed, their joint impact is significant.

The combination of remote power generation and international trade gives rise to the problem of transit flows and loopflows, causing congestion problems, especially when the flows are unforeseen, for example due to the uncertain nature of some generation sources. Power flow control in general, and the PST in particular offer a possibility to deal with these congestions and to enhance grid security. The power flow through an area can be controlled and limited if needed. As areas are generally large, several power flow control devices might be needed for adequate control. The coordination between these devices is crucial for obtaining the desired effect, or at least to avoid a decrease in grid security.

As an indicator of how well PST settings are coordinated, the Total Transfer Ca-pacity (TTC) can be adopted. This value indicates the maximum transport that can take place between two neighbouring areas under safe conditions. Every combination of PST settings results in a different TTC. An exhaustive enumeration of all possi-ble combinations is highly impractical as a means for finding the optimum, as every additional PST adds an extra dimension to the search space containing the optimum TTC. Instead, a limited amount of combinations can be randomly selected, and by means of the resulting TTC histogram, an estimation can be made of the best- and worst-case options. This technique is called Monte Carlo Simulation (MCS). The per-formed MCS study indicates that poor coordination can result in internal loopflows,

and can establish an even loading of the interconnectors at the borders if that is the aim.

MCS is not really an optimisation algorithm, but merely gives an indication of the probability of the system states to occur. In order to gain more information on the optimal region of the search space in a reasonable computation time, Multistage Monte Carlo Simulation (MMCS) is adopted. This method consists of performing several subsequent MCS runs in order to zoom into the optimal region. A major drawback is that the required running time is of the order of a few hours, which can be unacceptable for some applications. Furthermore, it is very hard to define the objective function of the optimisation problem as a function of the PST settings, partly due to the required network contingency analysis.

For these kinds of problems, metaheuristic optimisation algorithms, which are optimisation methods based on evaluations of the objective function through simu-lations, offer a solution. They consist of a heuristic local search algorithm guided by a top-level strategy, in order to avoid convergence to local optima. In the thesis, it is shown that not all metaheuristic methods work equally well; it is demonstrated that Particle Swarm Optimisation (PSO) is suited for the TTC optimisation, pro-vided that the algorithm parameters are chosen in a good way. A parameter study is performed to indicate the best values for the inertia and the swarm size, and the resulting tuned algorithm shows a fast convergence.

Next to the optimisation approach with the TTC as the only goal function, for instance also the system losses can be included in a multiobjective optimisation. The result is a Pareto front, offering the possibility to make a trade-off between both goal functions. The simulation results show that the losses increase steeply if the absolute best TTC value is to be obtained. If this is not acceptable, a lower TTC value can be targeted for. Metaheuristic optimisation offers a fast and accurate means to obtain the maximum TTC or the optimal value for any other goal function and it is therefore very useful for the coordination of several PST devices. However, the black-box approach relies on simulations and it does not give any analytical information on the impact and operation of a PST in a meshed grid whatsoever.

By adopting assumptions, like the approximations used in DC load flow calcula-tions, a more profound insight can be obtained in this matter, as the power flow in a transmission line can then be written as a linear function of the PST settings. The coefficients that are found in this linear relation, are referred to as phase shifter dis-tribution factors (PSDFs) and depend only on the system topology. Since the TTC can be calculated as a sum of line flows, it is possible to write it in an analytical form, as a function of the PST settings. It is shown that this expression is piecewise linear, and can therefore be optimised by Linear Programming techniques.

The methodologies for optimal coordination of phase shifters, developed in this thesis, can be applied on a variety of problems. As a first example, it is shown that

as they allow a redirection of the power flows, possibly preventing the dispatch of more expensive peak units in case of network congestions. As a second example, the Linear Least Squares (LLS) method is applied to calculate the optimal PST settings in order to evenly distribute the flows on a set of interconnectors at the border of two systems. The flows on these interconnectors are expressed as a set of linear equations as a function of the PST settings. As a third example, it is shown that in a stochastic context, an optimisation algorithm can be designed to minimise the congestion risk for the whole system taking into account the PST settings. Finally, both PSO and the analytical DC calculation are applied to study different options for the reinforcement of the Dutch-German border.

This thesis shows that a whole range of optimisation strategies can be applied in order to coordinate several PST devices. The result is an improved situation for the grid, be it in terms of transport capacity, system security, or any other criterion. For future research, it would be interesting to focus more on the economic aspect of the optimisation, such as the integration of PSTs in flow based market coupling. Furthermore, a challenge for further research is to find a way to obtain a global optimisation for multiple areas or to make local optimisations within different areas co-exist. Finally, a profound study of the relation between PSTs and system stability still has to be performed.

Jody Verboomen

Optimalisatie van Transportnetten met behulp van

Dwarsregeltransforma-toren

De laatste jaren hebben elektriciteitsvoorzieningsystemen een belangrijke veran-dering ondergaan onder invloed van technische en organisatorische ontwikkelingen, die ertoe hebben geleid dat elektriciteitsnetten op een andere manier worden bedreven en gebruikt. Een aspect dat meer en meer aan belang wint is vermogenssturing in ver-maasde netten, wat tot uitdrukking komt in het feit dat dwarsregeltransformatoren of phase shifting transformers (PSTs), die een voorbeeld zijn van vermogensregelaars, op een toenemend aantal locaties in Europa worden geplaatst. Het specifieke voorbeeld van het Nederlandse en Belgische net is zeer interessant wat dit betreft, aangezien hier meerdere van deze regelaars zijn ge¨ınstalleerd in een relatief klein gebied. Het gebruik van ´e´en enkele PST heeft al invloed op de vermogenstromen in het hele netwerk, en als meerdere van deze toestellen geplaatst worden, dan is hun gecombineerde impact aanzienlijk.

De combinatie van afgelegen productielocaties en internationale handel leidt tot zogenaamde transit flows en loop flows; hierdoor kan congestie ontstaan, vooral als deze flows niet voorzien zijn vanwege bijvoorbeeld het onzekere karakter van bepaalde opwekkingsbronnen. Vermogenssturing in zijn algemeenheid, en de PST in het bij-zonder maken het mogelijk om iets aan deze congestie te doen en om de netveiligheid te verhogen. De vermogenstroom door een gebied kan geregeld worden, en indien nodig beperkt. Aangezien de gebieden meestal uitgebreid zijn, kan het noodzakelijk zijn om meerdere toestellen te gebruiken om tot een adequate regeling te komen. Co¨ordinatie tussen deze regelaars is cruciaal om het beoogde effect te verkrijgen, of tenminste om een vermindering van netveiligheid te voorkomen.

De Total Transfer Capacity (TTC) kan gebruikt worden om aan te geven hoe goed de instellingen van de PSTs op elkaar zijn afgesteld. Deze indicator geeft aan hoeveel vermogen er tussen twee aangrenzende zones onder veilige condities kan worden uit-gewisseld. Iedere combinatie van PST instellingen resulteert in een andere TTC. Het doorrekenen van alle mogelijke combinaties is een zeer onpraktische manier om de optimale waarde te vinden, aangezien elke PST een extra dimensie toevoegt aan

willekeurig een beperkt aantal combinaties geselecteerd worden en aan de hand van het resulterende histogram van de TTC waardes kan een schatting gemaakt wor-den van de slechtste en beste gevallen. Deze techniek wordt Monte Carlo Simulatie (MCS) genoemd. De uitgevoerde MCS studie laat zien dat een slechte co¨ordinatie kan leiden tot interne loopflows, waardoor ernstige overbelastingen kunnen ontstaan. Goed geco¨ordineerde PSTs daarentegen, vermijden zulke loopflows en kunnen, als dat het doel is, een evenwichtige belasting van de interconnectoren op de grenzen bewerkstelligen.

MCS is niet echt een optimalisatiealgoritme, maar geeft enkel een indicatie van de waarschijnlijkheid dat bepaalde systeemtoestanden voorkomen. Om binnen een redelijke rekentijd meer informatie te verkrijgen over het optimale gebied van de zoekruimte, is een Multistage Monte Carlo Simulatie (MMCS) toegepast. Deze me-thode voert meerdere malen een MCS uit om in te zoomen op het optimale gebied. Een belangrijk nadeel is dat de benodigde rekentijd in de orde van enkele uren ligt, wat voor sommige toepassingen onacceptabel kan zijn. Verder is het erg moeilijk om de doelfunctie voor het optimalisatieprobleem als functie van de PST instellingen te defini¨eren, mede door de vereiste storingsanalyse.

Voor dit soort problemen vormen metaheuristische optimalisatiealgoritmes, die gebaseerd zijn op evaluaties van de doelfunctie door middel van simulaties, een oplos-sing. Ze bestaan uit een lokaal heuristisch zoekalgoritme dat geleid wordt door een strategie op hoger niveau, zodat convergentie naar lokale optima wordt vermeden. In het proefschrift wordt getoond dat niet alle metaheuristieken even goed werken; er wordt aangetoond dat Particle Swarm Optimisation (PSO) geschikt is om de TTC te optimaliseren, op voorwaarde dat de parameters van het algoritme op een goede manier worden gekozen. Een parameterstudie is uitgevoerd om de beste waardes voor de inertie en de zwermgrootte te vinden, en het resulterende algoritme convergeert op een snelle manier.

Naast de optimalisatieaanpak met de TTC als de enige doelfunctie, is het bijvoor-beeld ook mogelijk om de systeemverliezen te betrekken in een optimalisatie met meerdere doelfuncties. Het resultaat is een Pareto front, waarbij het mogelijk wordt een afweging te maken tussen beide doelen. De simulatieresultaten laten zien dat de verliezen zeer sterk toenemen als de allerbeste TTC waarde wordt nagestreefd. Indien dit niet acceptabel is, kan een lagere TTC beoogd worden. Metaheuristische optimalisatiealgoritmes maken het mogelijk om op een snelle en accurate manier de maximale waarde te vinden voor de TTC, of de optimale waarde voor welke doelfunc-tie dan ook, en zijn daarom zeer bruikbaar voor de co¨ordinadoelfunc-tie van meerdere PSTs. Echter, de black-box aanpak steunt op simulaties, en geeft geen enkele analytische informatie over de impact en de bedrijfsvoering van een PST in een vermaasd net.

Door bepaalde aannames te doen, zoals de benaderingen die gebruikt worden in DC load flow berekeningen, kan een dieper inzicht verkregen worden op dit gebied,

gevonden zijn, worden phase shifter distribution factors (PSDFs) genoemd en hangen enkel af van de topologie van het systeem. Aangezien de TTC berekend kan worden als een som van vermogenstromen, is het mogelijk om deze grootheid in een analytische vorm te beschrijven als functie van de PST instellingen. Aangetoond wordt dat deze uitdrukking stuksgewijs lineair is en daardoor met Lineair Programmeren (LP) kan geoptimaliseerd worden.

De methoden voor optimale co¨ordinatie van PSTs die in dit proefschrift zijn ont-wikkeld, kunnen op een breed scala van problemen worden toegepast. Een eerste voorbeeld laat zien dat een gelineariseerde optimal power flow berekening voor de inzet van eenheden op een eenvoudige manier kan worden uitgebreid met PSTs. Op-timaal gebruik van PSTs kan resulteren in lagere totale operationele kosten, omdat ze vermogenstromen kunnen omleiden, waardoor mogelijkerwijs de inzet van dure piekcentrales in geval van congesties in het net wordt vermeden. In een tweede voor-beeld wordt de Linear Least Squares (LLS) methode gebruikt om de optimale PST instellingen te berekenen voor een evenwichtige verdeling van de vermogenstromen over de interconnectoren op de grens tussen twee zones. De vermogenstromen via deze interconnectoren worden uitgedrukt als een stelsel van lineaire vergelijkingen als functie van de PST instellingen. Een derde voorbeeld laat zien dat in een stochastische context, een optimalisatiealgoritme kan worden opgesteld om het congestierisico voor het hele systeem te minimaliseren door middel van PSTs. Tenslotte worden zowel PSO als de analytische DC berekening toegepast om de verschillende opties voor netuitbreiding aan de Nederlands-Duitse grens te bestuderen.

Dit proefschrift laat zien dat een breed spectrum van optimalisatiestrategie¨en kan worden toegepast voor de co¨ordinatie van PSTs. Het resultaat is een verbeterde netsi-tuatie voor wat betreft transportcapaciteit, systeemveiligheid, of welk ander criterium dan ook. Voor toekomstig onderzoek zou het interessant zijn om de aandacht te ves-tigen op het economische aspect van de optimalisatie, zoals de integratie van PSTs in flow based market coupling. Voorts is het een uitdaging voor verder onderzoek, om een manier te vinden om tot een globale optimalisatie voor meerdere zones te komen, of om lokale optimalisaties binnen verschillende zones naast elkaar te laten bestaan. Tenslotte moet nog diepgaande studie worden verricht naar de relatie tussen PSTs en stabiliteit van het systeem.

Jody Verboomen

Summary i

Samenvatting in het Nederlands v

Contents ix

1 Introduction 1

1.1 Transmission bottlenecks . . . 2

1.2 Transports due to electricity markets . . . 2

1.2.1 Background . . . 2

1.2.2 Consequences . . . 3

1.2.3 Status in Europe . . . 4

1.3 Variability of intermittent sources . . . 4

1.3.1 Background . . . 4

1.3.2 Consequences . . . 6

1.4 Transit flows and loopflows . . . 6

1.5 Objectives and limitations . . . 7

1.6 Research framework . . . 8

1.7 Outline of the thesis . . . 9

2 Power Flow Control 13 2.1 Power through a transmission line . . . 13

2.2 Classification of active power flow controllers . . . 14

2.2.1 Technology . . . 14

2.2.2 Controlled parameter . . . 15

2.3 Phase shifter technology . . . 16

2.3.1 Direct asymmetrical PSTs . . . 16

2.3.2 Direct symmetrical PSTs . . . 18

2.3.3 Indirect asymmetrical PSTs . . . 19

2.3.4 Indirect symmetrical PSTs . . . 20

2.4.1 Reactance and ideal phase shift . . . 22

2.4.2 Two-port equivalent . . . 23

2.4.3 Non-idealities . . . 24

2.5 Phase shifters in the Netherlands and Belgium . . . 26

2.5.1 The Netherlands . . . 26

2.5.2 Belgium . . . 29

2.6 Static and dynamic operation . . . 30

2.6.1 Load flow analysis . . . 30

2.6.2 Transient stability analysis . . . 31

2.7 Summary . . . 33

3 Search Space Exploration and Path Determination for PST Settings 35 3.1 Transfer capacities . . . 36

3.1.1 Definition . . . 36

3.1.2 Simulation-based calculation . . . 37

3.2 MCS theoretical background . . . 38

3.2.1 Monte Carlo fundamentals . . . 38

3.2.2 Monte Carlo Simulation . . . 39

3.3 Exploration using MCS . . . 40

3.3.1 Search space . . . 40

3.3.2 Simulation setup . . . 41

3.3.3 Relation between TTC and the Meeden PSTs . . . 42

3.3.4 MCS of the TTC with all PSTs . . . 44

3.4 Multistage Monte Carlo Simulation . . . 46

3.5 Sensitivity analysis . . . 48 3.6 Path determination . . . 49 3.6.1 Problem formulation . . . 49 3.6.2 Path strategies . . . 52 3.6.3 Simulations . . . 53 3.7 Summary . . . 54

4 Metaheuristic Optimisation Methods 57 4.1 Introduction to metaheuristics . . . 58

4.1.1 General description . . . 58

4.1.2 Advocates and sceptics . . . 58

4.2 Classification . . . 59

4.2.1 Metaheuristics based on a unique solution . . . 59

4.2.2 Metaheuristics based on a population of solutions . . . 62

4.3 Application of Evolutionary Computation . . . 64

4.3.1 Theoretical background . . . 64

4.3.2 Problem definition . . . 67 x

4.4 Application of Particle Swarm Optimisation . . . 70

4.4.1 Theoretical background . . . 70

4.4.2 Simulations . . . 71

4.5 Transit flow sensitivity . . . 74

4.6 Multiobjective optimisation including losses . . . 76

4.6.1 Definitions . . . 76

4.6.2 Conventional Weighted Aggregation . . . 77

4.6.3 Simulations . . . 78

4.7 Summary . . . 81

5 PSTs in Linearised Equations 83 5.1 Linearised power flow equations . . . 84

5.1.1 DC load flow . . . 84

5.1.2 PSTs in a DC load flow . . . 85

5.1.3 Case study . . . 89

5.2 Analytical expression for TTC . . . 90

5.3 TTC optimisation . . . 91

5.3.1 N secure TTC . . . 91

5.3.2 N − 1 secure TTC . . . 93

5.3.3 Case study . . . 94

5.4 TTC sensitivity . . . 99

5.4.1 Sensitivity of the BCE . . . 99

5.4.2 Sensitivity of the maximum power shift . . . 99

5.4.3 Total sensitivity . . . 100

5.5 Equivalent reactance . . . 101

5.6 Summary . . . 103

6 Application of Optimisation Methods 105 6.1 Generation unit dispatch with PSTs . . . 106

6.1.1 DC OPF without PSTs . . . 106

6.1.2 DC OPF with PSTs . . . 107

6.1.3 Application to the New England system . . . 108

6.2 Border-flow control . . . 111

6.2.1 Linear Least Squares . . . 111

6.2.2 Single border control . . . 113

6.2.3 Combined border control . . . 115

6.2.4 Influence of grid topology changes . . . 116

6.2.5 Modelling of discrete behaviour . . . 118

6.3 Congestion risk minimisation . . . 119

6.3.1 Quantification of uncertainty in power system flows . . . 119 xi

6.3.3 Minimisation of the overall system congestion risk . . . 120

6.3.4 Case study . . . 121

6.4 Reinforcement study on the Dutch-German border . . . 125

6.5 Summary . . . 129

7 Conclusions and Recommendations 131 7.1 Conclusions . . . 131

7.2 Recommendations . . . 132

7.2.1 Economic Aspect . . . 132

7.2.2 Multi- or Interarea Coordination . . . 133

7.2.3 Dynamic Behaviour . . . 133

A Abbreviations, Symbols and Operators 135 A.1 List of abbreviations . . . 135

A.2 List of symbols . . . 136

A.3 List of subscripts . . . 138

A.4 List of superscripts . . . 139

A.5 List of operators . . . 139

B Mathematical Background 141 B.1 Big O notation . . . 141

B.2 Linear Programming . . . 142

C New England Test System Data 145 Bibliography 149 Publications 155 Journal publications . . . 155 Conference publications . . . 155 Other . . . 157 Acknowledgement 159 Biography 161 xii

CHAPTER

1

Introduction

In the recent past, power systems have undergone a major transformation as a num-ber of developments have taken place, leading to a change in the way grids are op-erated. An issue that becomes increasingly important in meshed systems is power flow control, which is reflected in the fact that throughout Europe, phase shifting transformers (PSTs), which are examples of power flow controllers, are installed at numerous locations. There are basically three reasons why these devices are adopted: - The efficiency of grid management can be improved by solving bottlenecks in transmission systems. PSTs are able to shift flows from congested areas to other areas where transmission capacity is still available, offering the possibility to delay grid investments. The devices can either be installed in congested areas to “push” the flow to another area, or in areas with low loading to “pull” the flow.

- A wide range of transmission scenarios induced by market decisions must be handled. The outcome of electrical energy markets can lead to a variety of transport flows, especially when international trade is considered. PSTs offer the possibility to make optimal use of available network capacities and conse-quently to facilitate the market mechanisms as much as possible. Again, this means that investments in the grid can be postponed.

- Large flow variations due to intermittent generation have to be tackled. Re-newable generation is often relying on a stochastic prime mover, such as wind. The penetration of wind energy in the European power system has increased significantly in the last few years, and will continue to do so in the near future. The stochastic nature of the generated power leads to the fact that the flow pattern in the European system can be totally different from one moment to the other.

The combination of intermittent power generation and international trade gives rise to the problem of transit flows and loopflows, causing congestion problems on interconnectors. Power flow control in general, and the PST in particular offer a possibility to deal with these congestions [72]. The power flow through an area can be controlled and limited if needed. As areas are generally large, several power flow control devices might be needed for adequate control. The coordination between these devices is crucial for obtaining the desired effect, or at least to avoid a decrease in grid security.

1.1

Transmission bottlenecks

In a grid, the power flows depend on the location of generating facilities and loads, as well as on the line impedances. For radial lines, the flow distribution is trivial, but the situation is more complex in a meshed grid, as several parallel paths from generation to load may exist. Each line has a maximum capacity determined by a number of constraints, such as thermal loading, stability and object clearance.

When power is exchanged between two areas, the flow may spread over different parallel paths, mostly in a way that each path has a different loading. As a con-sequence, the transport capacity between two areas is not equal to the sum of the maximum capacities of the interconnectors between both areas, but it is limited due to the fact that one interconnector reaches its maximum earlier than the others.

If congestion arises, the conventional approach is to perform generation redispatch. This can be done within the area or cross-border in both areas, depending on the situation. This solution is not economically favourable, as more expensive units are dispatched. Instead, power flow control can be used as an alternative.

1.2

Transports due to electricity markets

1.2.1

Background

In the past, the electricity industry was dominated by large utilities that were active in the field of generation, transmission and distribution. These companies are referred to as vertically-integrated utilities, as they are active in every level of the industry [47; 54]. Customers were obliged to buy their electricity from a local utility, and the price was

not the result of a market mechanism, but it was an average of the aggregated cost incurred in generation, transmission and distribution of that utility. Unit commitment and economic dispatch of generation took place on a regional or national level and the capacity of the transmission grid was an integral part of the optimisation process. Since market mechanisms have been introduced in this structure, radical changes have taken place. Instead of one large utility, several specialised companies come into play [12; 33]:

- Many generating companies appear, each owning one or multiple power plants. They can offer power to the market, but they are not involved in the transmis-sion and distribution of electrical energy.

- Transmission companies own a part of the transmission grid. Their task is to facilitate the market by allowing large power transports between producers and distribution grids or large customers (e.g. a steel factory).

- Distribution companies own local distribution networks, linking the transmis-sion network to the customers.

The customers are free to choose their electricity supplier, enabling competition be-tween the different supplying companies.

1.2.2

Consequences

Introducing competition in the electricity sector brings along all kinds of conse-quences:

- Customers are free to choose the supplier they want, even if this means that they buy electric energy from a supplier in another country. They can compare different options and make a choice based upon the most interesting offer. - Customers do not only choose based upon the price, but also based upon the

service they get. Consequently, companies are forced to supply a decent service in order to keep their customers satisfied.

- Power suppliers are free to invest in new power plants, whether the extra power generation is needed by local loads or not [24]. Obviously, the investment cli-mate for power suppliers is related to the need for power in the system. - Transmission companies have to facilitate the market, meaning that they have

to transport the electrical energy agreed upon by the market and they are obliged to solve network constraints [12; 92]. They are also responsible for the security of supply.

1.2.3

Status in Europe

In Europe, the liberalisation process started with the European Directive 96/92/EC, offering guidelines to the EU memberstates [32]. The main principles are:

- Everyone is free to produce electricity and every generator needs to be assured that he will have access to the grid.

- Customers are eligible, meaning that they are free to choose their electricity supplier. This requirement was implemented gradually in time. However, Di-rective 2003/54/EC accelerated the process, and as from the 1st of July 2007, all consumers are eligible [59].

- The vertically-integrated companies have to be split up into several independent entities [33].

1.3

Variability of intermittent sources

1.3.1

Background

-150 -10 -5 0 5 10 15 500 1000 1500 2000 2500 3000

wind speed variation [m/s]

fre qu en cy [-] 15 minutes 1 hour 1 day

Figure 1.1: Variation of wind speed for different time intervals.

The growing concern for the environment has led to a shift from classical, fossil fuel-fired power plants to renewable energy systems [5]. Different technologies are

available, but at the time of writing, wind turbines are the main representatives in the field of renewables [27; 34].

As wind is a stochastic prime mover, the power output of a wind turbine fluctu-ates and can only be predicted with limited accuracy [89]. The degree of variability depends on the time interval that is considered, as shown in Fig. 1.1. Obviously, the variability is much bigger on a day-to-day basis, compared to the limited fluctuations within a 15-minute interval [35].

The wind variability results in significant electrical power output fluctuations be-cause of the power curve of the wind turbine, which amplifies the effect [87]. A typical characteristic is shown in Fig. 1.2 [115]. The influence of wind speed variability on the power output depends on the operating point on the characteristic. In the example curve, the output power in the wind speed range from 4 to 15 m/s increases steeply with increasing wind speed. Also the so-called cut-out wind speed (26 m/s in the figure) can be a problem. At very high wind speeds, a small fluctuation can lead to a shutdown procedure, resulting in an output power change from nominal power to zero in a very limited time span1_.

0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 wind speed [m/s] P /P n o m [ p u ]

Figure 1.2: Power curve of a typical wind turbine.

1_{This problem is tackled in some modern wind turbine designs, where the power output above} the cut-out wind speed decreases gradually. Also, some wind parks are designed in such a way that each wind turbine has a slightly different cut-out speed.

1.3.2

Consequences

Fluctuations within the minute are often caused by turbulence. This phenomenon is very local, and its effect is partly smoothened within a single wind farm due to aggregation. Hence, the effect on the power system is very small and is of no further interest.

Fluctuations within a quarter of an hour do not influence the scheduling of the production park, but they are to be compensated by reserve power [98]. The provision of reserve power is mostly organised by transmission companies within defined zones of the power system. Hence, the fluctuations are compensated locally and this does not influence the power flows in the grid to a great extent.

Hourly variations do have an impact on scheduling, and the forecasting of the wind becomes important. A lot of research is being performed on different forecast methods, but the accuracy is still limited, especially when the wind speed is near the cut-out speed, or between the cut-in and nominal power wind speeds [73]. Com-pensation is mostly performed by the program responsible parties themselves or by using reserve power. The impact on the grid might be bigger as the variations are also larger, but the effects are still local.

Variations on the longer term can be due to change of weather or other climatic effects. These fluctuations are particularly important for planning and for trading, and might be different from region to region. The power flows in the system can be very much influenced, as the injection pattern may change in a very dynamic way. A line flow can be from A to B on one day, and the opposite the other day. This way of thinking is very different from the situation where only central generation is applied.

1.4

Transit flows and loopflows

In the framework of the liberalised electricity markets, consumers or traders can buy their electrical energy by signing a contract with the supplier of their choice, even across the borders. The contractual path is the direct electrical path between producer and consumer. However, in a strongly meshed grid such as the European system, the power flow spreads out over several parallel paths, giving rise to so-called transit flows and loopflows [39]. The distribution of the power exchange over the parallel paths depends on their relative impedance.

Fig. 1.3 shows possible scenarios that can occur. A third party (transit) loopflow occurs when a fraction  of a power exchange P between two areas A and B flows through a third area C. A typical example is the power exchange between Germany and France, of which typically 25 to 30 percent flows through the Netherlands and Belgium. A second party loopflow happens when an internal power transfer partly flows through a second area. This problem can occur when generation and load centres are in geographically distant locations, which can be the case with large wind parks and large conventional power generation at remote sites. An example is the

P

A

B

C

(1 − )P

P

(a) 3rd party loopflow

(1 − )P

B

A

P

P

(b) 2nd party loopflow Figure 1.3: Possible loopflow scenarios.

situation in Germany, where a high concentration of wind parks is installed in the north and a high concentration of load is located in the south [27]. The resulting north-south flows induce a loopflow in the Netherlands. Moreover, this loopflow is not constant, because of the variability of the wind.

A loopflow is a phenomenon that is difficult to handle, as it often comes rather unexpected for the second or third party, and even if it can be predicted, it may still require special actions to prevent congestions [19; 21]. A first step towards better prediction in the European system is the day-ahead congestion forecast (DACF), where possible congestions are predicted for the next day, taking into account the situation in the neighbouring areas [97]. However, the DACF only gives an indication instead of solving the problem itself [101]. The problem can be relieved by using PSTs.

1.5

Objectives and limitations

The operation of PSTs in a meshed grid has an impact on the power flow in the network. If several of these devices are installed, their joint impact is significant. As was discussed earlier, they might help to overcome problems in the grid. In this thesis, transmission grids with PSTs are investigated. In particular, the following goals are put forward:

- The analysis and quantification of the impact of a PST on a meshed grid. This includes the development of models for the device.

- The development of methods to obtain optimal coordination of several PSTs in a meshed grid. An objective function should be formulated, and an optimisation method must be adopted to solve the problem.

The research that is presented in this thesis focuses on the steady state of the system. Although a brief introduction to PST operation under transient conditions is given in chapter 2, this issue is not further elaborated. Furthermore, the study is limited to PSTs only; (other) FACTS devices are not considered. Finally, the research focuses on large transmission grids, and although distribution networks are taken into account in some of the simulations, they are not the topic of this thesis.

1.6

Research framework

The research presented in this work has been performed within the framework of the “Intelligent Power Systems” project. The project is part of the IOP-EMVT program (Innovation Oriented Research Program - Electro-Magnetic Power Technology), fi-nancially supported by SenterNovem, an agency of the Dutch Ministry of Economical Affairs. The “Intelligent Power Systems” project is initiated by the Electrical Power Systems and Electrical Power Electronics Groups of the Delft University of Technol-ogy and the Electrical Power Systems and Control Systems Groups of the Eindhoven University of Technology. In total 10 PhD students are involved and work closely together. The research focuses on the effects of the structural changes in generation and demand taking place, like for instance the large-scale introduction of distributed (renewable) generators [78]. The project consists of four parts (illustrated in Fig. 1.4).

Inherently stable transmission system Manageable distribution networks Optimal power quality Self-controlling autonomous networks

Figure 1.4: The four parts of the “Intelligent Power Systems” research project. The first part (research part 1), inherently stable transmission system, investi-gates the influence of uncontrolled decentralised generation on stability and dynamic behaviour of the transmission network. As a consequence of the transition in the gen-eration, less centralised plants will be connected to the transmission network as more generation takes place in the distribution networks, whereas the remainder is possibly generated further away in neighbouring systems. Solutions investigated include the

control of centralised and decentralised power, the application of power electronic interfaces and monitoring of the system stability.

The second part (research part 2), manageable distribution networks, focuses on the distribution network, which becomes “active”. Technologies and strategies have to be developed that can operate the distribution network in different modes and support the operation and robustness of the network. The project investigates how the power electronic interfaces of decentralised generators or between network parts can be used to support the grid. Also the stability of the distribution network and the effect of the stochastic behaviour of decentralised generators on the voltage level are investigated.

In the third part (research part 3), self-controlling autonomous networks, au-tonomous networks are considered. When the amount of power generated in a part of the distribution network is sufficient to supply a local demand, the network can be operated autonomously but as a matter of fact remains connected to the rest of the grid for security reasons. The project investigates the control functions needed to operate the autonomous networks in an optimal and secure way. The interaction between the grid and the connected appliances has a large influence on the power quality.

The fourth part (research part 4), optimal power quality, of the project analyses all aspects of power quality. The goal is to provide elements for the discussion between polluter and grid operator who has to take measures to comply with the standards and grid codes. Setting up a power quality test lab is an integral part of the project. The research described in this thesis is within research part 1: inherently stable transmission systems.

1.7

Outline of the thesis

This thesis is structured in the following way:

- Chapter 2 gives a short overview of active power flow controlling devices. They can be classified in different ways depending on the characteristics, such as switching technology and the controlled parameter. As the focus of this thesis is on PSTs, a more profound review of PST technology is presented, discussing different possible topologies. Furthermore, the modelling of PSTs is discussed, including the non-idealities that are neglected in many cases. Next, an overview of the current and future PST devices in the Netherlands and Belgium is given. Finally, the static and dynamic operation of a PST is demonstrated with a test case. It is shown that the impact of such a device on the transient stability is very hard to quantify, but the effect on the steady-state flows is clear.

- In chapter 3, a first step towards optimal PST coordination is taken. Differ-ent transfer issues between areas are discussed and the Total Transfer Capacity

(TTC) is chosen as the optimality indicator. The search space can be described by a multidimensional function that connects the TTC to the different PST settings. The space is very large due to the high number of possible sets of PST settings, making an exhaustive search highly impractical. A Monte Carlo Simulation can serve as a tool to explore the search space, although it is not an optimisation method. The application of Monte Carlo Simulation to the Dutch-Belgian grid gives, amongst others, insight in a worst-case and a best-case scenario regarding the TTC. In order to gain more information on the op-timal region in the search space, Multistage Monte Carlo Simulation is adopted and applied to the Dutch-Belgian grid. It is also shown that the sensitivity around the optimal point is very different for each PST. Finally, the problem of avoiding unfavourable intermediate states when switching between two sets of PST settings is addressed. This problem can be redefined as a shortest path problem.

- In chapter 4, metaheuristic optimisation methods are discussed. Metaheuristics are algorithms that only rely on evaluations of the objective function, which makes them very suitable for simulation-based optimisation. A classification is given based on the number of intermediate solutions that are used. A few of these methods are applied to the PST coordination problem in the Netherlands and Belgium, namely Meta-Evolutionary Programming, Evolution Strategies and Particle Swarm Optimisation. Without PSTs, transit flows have a large impact on the Dutch and Belgian grid. The optimal coordination of PSTs results in a much lower sensitivity to transits within the control capabilities of the devices. Besides the Total Transfer Capacity, system losses can be incorporated in the objective function as well, resulting in a multiobjective optimisation formulation. The concept of the Pareto front offers a solution for this problem, allowing for a trade-off between transfer capacity and losses.

- Chapter 5 introduces DC load flow approximations, leading to analytically-closed equations that describe the relation between PST settings and active power flows. It is shown that this relation is linear, and that it is characterised by sensitivity factors called phase shifter distribution factors. These factors depend on the network topology only and not on the power injection patterns. Based on these linear equations, an analytical expression for the Total Transfer Capacity can be derived. It is shown that this is a piecewise linear function of the different PST settings, allowing a linear optimisation approach. The method offers a very fast optimisation, at the cost of inaccuracy caused by the approximations. The derived expressions lead to a possibility to model a PST as an equivalent reactance that is a function of the PST setting and depending also on the network topology.

chapters are presented. The first application is the unit commitment problem, where the goal is to distribute the total required power over the production facilities in the most cost-effective way. The PSTs can be integrated in an op-timal power flow formulation, so that they are opop-timally controlled in order to relieve congestions. A second application is to make the relative line loadings of the interconnectors on a certain border equal. Depending on the number of PSTs in relation to the number of interconnectors, the set of equations can be either solved exactly, or by using a Linear Least Squares approach. As a third application, a stochastic approach is adopted to quantify the risk of con-gestion in the network. It is known that with a PST, this risk can be altered in a straightforward manner for a single line. However, in a meshed system, the minimisation of the global congestion risk requires a global optimisation algorithm for the calculation of the optimal settings. In the final application, the reinforcement of the interconnectors on the Dutch-German border is con-sidered. Several possibilities are taken into account and for each of them the maximum attainable Total Transfer Capacity (with optimal PST coordination) is calculated.

- Chapter 7 contains the conclusions of this thesis, as well as recommendations for future work.

CHAPTER

2

Power Flow Control

As discussed in chapter 1, electrical energy transports have increased over the last years in the European grid due to the liberalisation of the electricity markets and the increased penetration of wind energy. Congestion in the transmission grid is a phenomenon that is encountered more often than before, and it is in this framework that power flow control becomes key. This chapter is devoted to power flow control in general and phase shifting transformers in particular. Different aspects of phase shifters are discussed in detail, such as possible configurations, modelling issues for the integration in load flow studies, and their impact on transient system stability.

2.1

Power through a transmission line

A transmission line can be modelled as a reactance in series with a resistance, omitting shunt elements1_{, as shown in Fig. 2.1. The complex current through this line is}

depending on the voltage levels and angles on each side of the line and the impedance between both nodes:

I = V1− V2 ZL

=V1∠δ1− V2∠δ2

RL+ jXL

(2.1) 1_{This approximation is valid for short overhead lines. For longer lines and cables, the shunt} elements must be included.

I

V

26

δ

2

R

L

jX

L

V

16

δ

1

(a) Circuit diagram

jXLI V16 δ1 V26 δ2 I RLI ∆V (b) Phasor diagram Figure 2.1: Series equivalent of a transmission line.

The single-phase complex line power at node 1 can be obtained by:

S1= V1I∗ (2.2)

where V1 is the phase voltage.

If (2.1) is substituted in (2.2), the complex power becomes: S1= V1V2 R2 L+XL2 XL sin δ +V12− V1V2cos δ R2 L+XL2 RL + j   V2 1 − V1V2cos δ R2 L+XL2 XL − RV21V2 L+XL2 RL sin δ   (2.3)

where δ = δ1− δ2, the phase angle difference between both nodes.

For a lossless transmission line (RL = 0), the active and reactive line powers

become: P1= V1V2 XL sin δ Q1= V2 1 XL − V1V2 XL cos δ (2.4)

Both the active and reactive line power are a function of four variables: line reactance, phase angle and bus voltages. Any of these parameters can be altered to indirectly change the line power, provided a meshed grid is considered. Conventional means to achieve this include the addition of series impedances (inductances or capacitors) and the use of phase shifting transformers (PSTs). Modern developments in power electronics add new functionalities to these devices and even have led to a whole family of controllers, all categorised under the term flexible AC transmission systems (FACTS) [45]. Their control capabilities go further than power flow control alone, and are therefore not relevant as such in this research [43].

2.2

Classification of active power flow controllers

2.2.1

Technology

All power flow control devices are based on a switching technology in one way or the other. Based on different technologies used for switching, a classification can be made [99].

- Mechanical switching is used for conventional devices where speed is not an issue. It is clear that control on dynamics is not possible. However, this tech-nology shows clear advantages, such as simplicity, relatively low cost and high reliability.

- Thyristor based devices are able to switch at a faster rate, but they can not perform multiple switch operations within one half period of the line voltage, as they are line commutated [58]. An example of this can be found in classical high voltage direct current (HVDC) schemes.

- Fast switching components are for instance used in so-called voltage-source con-verters (VSCs). Components such as insulated gate bipolar transistors (IGBTs) are able to be switched on and off independent of the line voltage, allowing for pulse width modulation (PWM) control schemes [60]. This technology is rela-tively recent in transmission systems and consequently experience is limited.

2.2.2

Controlled parameter

As shown in equation (2.4), the active power through a transmission line can be con-trolled by line reactance, phase angle and bus voltages [70]. These control possibilities are discussed hereafter.

Line reactance

The active line power is inversely proportional to the line reactance. It is impossible to directly control the reactance, but it can be compensated by a series capacitor. The total line reactance then becomes:

XL= j  ωL₋ 1 ωC  (2.5) Fixed series capacitors are sometimes used for permanent compensation. For a more flexible control, a thyristor controlled reactor (TCR) in parallel with a fixed capacitor can be used, combined in a thyristor controlled series capacitor (TCSC) (Fig 2.2a) [46]. This device behaves like a controllable capacitor, enabling adequate line com-pensation and can also be used for mitigation of oscillations. An example of a series compensator based on VSC technology is the solid state series compensator (SSSC) (Fig 2.2b) [99].

Voltages

The bus voltages can not deviate much from 1 pu. Therefore this quantity is not used for power flow control. Of course, voltage control is a very important topic, but it is not the issue of this research.

C L (a) TCSC VSC C (b) SSSC

Figure 2.2: Two examples of series compensation.

Phase angle

Phase angle control can be attained in a relatively easy way by injecting a quadrature voltage. This is established in a PST by injecting a part of the voltage between two phases in the third phase by using a tap variable transformer [52]. This kind of power flow control gives rise to nonlinearities, because of the sine function in (2.4).

There are also devices that are based on power electronics, such as the unified power flow controller (UPFC), which have capabilities that reach much further than power flow control alone.

2.3

Phase shifter technology

PSTs exist in many different forms. They can be classified by the following charac-teristics [106].

- Direct PSTs are based on one three-phase core. The phase shift is obtained by connecting the windings in an appropriate manner to each other.

- Indirect PSTs are based on a construction using two separate transformers: one variable tap exciter to regulate the amplitude of the quadrature voltage and one series transformer to inject the quadrature voltage in the line.

- Asymmetrical PSTs create an output voltage with an altered phase angle and amplitude compared to the input voltage.

- Symmetrical PSTs create an output voltage with an altered phase angle com-pared to the input voltage, but with the same amplitude.

The combination of these characteristics results in 4 categories of PSTs. Each category is discussed in the following paragraphs.

2.3.1

Direct asymmetrical PSTs

Fig. 2.3a shows the configuration of a direct and asymmetrical PST. The input termi-nals are L1 to L3. The winding with a variable tap connected to the input terminal

S2 L1 L2 L3 S1 S3

(a) Circuit diagram

∆V

3

V

_L3

V

_S3

V

_S2

∆V

2

V

_L2

V

_S1

∆V

1

α

V

_L1 (b) Phasor diagram Figure 2.3: Direct asymmetrical PST.

so, a quadrature voltage that can be regulated by means of the variable tap is added to the input voltage in order to obtain a phase shift α. The direction of the phase shift can be changed by using switches. In this way, the power flow in the line can be increased or decreased. The relation between the tap position and the angle α is nonlinear and can be derived from the phasor diagram (Fig. 2.3b):

α = arctan∆V1 VL1

(2.6) The relationship between the secondary voltage and the injected quadrature voltage is given by:

VS1=

∆V1

sin α (2.7)

Using (2.6), equation (2.7) becomes: VS1=

∆V1

sinarctan∆V1

VL1

 (2.8)

The secondary voltage VS1 is always larger than the input voltage VL1 (for a

nonzero phase shift). The fact that voltage levels are changed, also influences the transmitted power over the line:

P = VS1V2 Xl+ Xpst

sin(δ + α) (2.9)

Using (2.6) and (2.8), equation (2.9) can be rewritten as: P = V2 Xl+ Xpst · ∆V1 sinarctan∆V1 VL1  sin  δ + arctan∆V1 VL1  (2.10) = V2 Xl+ Xpst ·

∆V1sin δ cos arctan∆V_VL11 + cos δ sin arctan∆V_VL11

 sinarctan∆V1 VL1  (2.11) = V2 Xl+ Xpst · (V L1sin δ + ∆V1cos δ) (2.12)

For constant δ, the relation between the quadrature voltage and the active line power is linear. Equations (2.6) and (2.12) are plotted in Fig. 2.4 for δ = π

6 = 30◦,

with VL1 and _Xl+XV2_pst = 1 pu. The curve of α is relatively linear up to a value of

about α = 0, 6 rad ≈ 34◦_. 0 0.5 1 1.5 2 2.5 0 1 2 3 ∆ _V 1 [pu] P [ p u ] 0 0.5 1 1.5 2 2.50 0.5 1 1.5 α [ ra d ] P α

Figure 2.4: Relation between P, α and the quadrature voltage for a direct asymmet-rical PST with δ =π

6 = 30◦.

2.3.2

Direct symmetrical PSTs

With some modifications, the direct asymmetrical PST can be made symmetrical (Fig. 2.5a). An additional tap changer is needed, which increases the total cost of the device. The advantages are that the voltage amplitudes remain unchanged and that the maximum attainable angles are larger.

The relation between the quadrature voltage and the angle α is again nonlinear and can be derived from the phasor diagram (Fig. 2.5b):

α = 2 arcsin ∆V1 2VL1

M1 S1 L1 S3 L3 S2 L2 M3 M2

(a) Circuit diagram

∆V2 V_S3 V_S1 α ∆V1 V_L1 V_L3 V_L2 V_M1 V_M3 ∆V3 V_M2 V_S2 (b) Phasor diagram Figure 2.5: Direct symmetrical PST.

Using (2.13), the transferred active power becomes: P = VL1V2 Xl+ Xpst sinδ + 2 arcsin ∆V1 2VL1  (2.14) Equations (2.13) and (2.14) are plotted in Fig. 2.6 with VL1= 1 pu and _Xl+XV2_pst =

1 pu. The active line power now reaches a maximum and decreases to zero. The reason for this behaviour is that, in contrast to the asymmetrical configuration, the angle α can become larger than 90 degrees, and can theoretically even reach 180 degrees. The quasi-linear range of the α-curve has thus doubled in comparison with Fig. 2.4. An alternative implementation of a direct and symmetrical PST is depicted in Fig. 2.7a. In every phase, a controllable voltage is injected proportional to the voltage between the primary and secondary terminal of the two other phases. The resulting phasor diagram (Fig. 2.7b) has a hexagonal shape.

2.3.3

Indirect asymmetrical PSTs

The indirect asymmetrical PST consists of an exciter and a series transformer. De-pending on the rating of the system, these two transformers are housed in separate tanks or in a single tank. The two-tank system has the advantage of an easier trans-port.

Fig. 2.8 shows the configuration of the system. The phasor diagram is exactly the same as the one depicted in Fig. 2.3b.

0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ∆ V 1 [pu] P [ p u ] 0 0.5 1 1.5 2 2.50 0.5 1 1.5 2 2.5 3 3.5 α [ ra d ] P α

Figure 2.6: Relation between P, α and the quadrature voltage for a direct symmetrical PST with δ = π

6 = 30◦.

L₂

L1 S1 S2 L3 S3

(a) Circuit diagram

∆V3 α ∆V1 ∆V2 VS3 VS2 VL2 VS1 VL1 VL3 (b) Phasor diagram Figure 2.7: Direct symmetrical PST in hexagonal configuration.

2.3.4

Indirect symmetrical PSTs

The indirect asymmetrical PST can be made symmetrical by splitting the series winding in two and tapping the voltage for the exciter from the middle. Fig. 2.9 shows this configuration. The phasor diagram is the same as the one in Fig. 2.5b.

2.3.5

Comparison of the topologies

Asymmetric PSTs are obviously less complex when it comes to construction. This fact is also reflected in terms of cost. However, the fact that the voltage amplitude is altered is a major issue, making symmetrical topologies more popular. Furthermore,

Series Transformer Exciter L₁ L₂ L3 S₁ S₂ S3

Figure 2.8: Indirect asymmetrical PST.

Series Transformer Exciter L₁ L2 L₃ S₁ S2 S₃

Figure 2.9: Indirect symmetrical PST.

these symmetrical configurations can attain a larger angle than their asymmetric counterparts.

A direct configuration is easier to construct and hence cheaper compared to the indirect implementation, as no exciter is needed. A major drawback, however, is the fact that the tap changer and the regulating winding are directly exposed to system disturbances, making them very vulnerable. Also, the indirect topology allows

more flexibility in the design phase, as the regulator circuit can be dimensioned independently, which is a major asset when selecting the tap changer.

2.4

Phase shifter modelling

2.4.1

Reactance and ideal phase shift

A PST can be modelled as a reactance Xpst in series with an ideal phase shift α.

Equation (2.4) becomes:

P = V1V2 Xl+ Xpst

sin(δ + α) (2.15)

On first sight, the relation between the active power that is transported over the line and the phase shift angle seems straightforward. However, if a change in α is applied, this has an influence on δ, which contributes to the change in P . Hence, if a control action is performed, the active power does not follow a constant δ curve. An extreme case is when a PST would be installed in a line connecting a single generator to the rest of the grid. In this case, the change in α is fully compensated by the change in δ, as the line power must stay constant and equal to the output power of the generator.

In order to illustrate the operating principles of a PST in a meshed grid, two parallel lines are considered (Fig. 2.10). Line 1 has a larger reactance than line 2

G Load PL= PG X1 X2 P1 P2 PG

Figure 2.10: Scheme of two parallel lines with a PST in one line (X1> X2).

(X1 > X2); resistances are neglected. Without power flow control, line 2 carries a

larger share of the total power PG= P1+ P2:

P1 PG = X2 X1+ X2 P2 PG = X1 X1+ X2 (2.16) This is illustrated in the P − δ plane in Fig. 2.11a. If the rated power of the lines is equal, line 2 will reach its rated power much faster, leading to an inefficient use of line 1. This can be solved by installing a PST. In this example, the device is placed in line 1, but it could be installed in either of the lines. It can be easily verified that (2.16) then becomes:

0 π _δ P P₁ P₂ δ (a) Without PST 0 π _δ P −α P₁ P₂ δ0 (b) With PST Figure 2.11: P − δ graphs for two parallel lines.

P1 PG = 1 1 + X1+ Xpst X2   sin δ₀ sin(δ0_{+ α)}  P2 PG = 1 1 + X2 X1+ Xpst  sin(δ₀+ α) sin δ0  (2.17) where δ0 _{is the phase angle difference between the bus of the generator and the bus}

of the load and α is the phase shift angle of the PST.

The situation with a PST is illustrated in Fig. 2.11b. The PST boosts the power through line 1. Since the total transported power PG must be constant, the power

transported over line 2 decreases, resulting in a smaller value for δ. This is a very important conclusion: the increase in α is partly counter-acted by a decrease in δ, resulting in a smaller power boost on the line with the PST. In case of a single line without parallel paths, the increase in α is fully counteracted by the decrease in δ, resulting in a constant active power flow.

2.4.2

Two-port equivalent

If a grid is considered in pu quantities, a transformer can be modelled by its short-circuit admittance YSC [116]. If the transformer has a tap changer, an ideal

trans-former with winding ratio t : 1 has to be added to the model (Fig. 2.12). A PST introduces a quadrature shift in voltage instead of a longitudinal one. As the phase of the voltage is altered, and possibly the magnitude, the winding ratio t must be regarded as a complex number [22].

If Ohm’s law is applied to YSC, the following equation follows:

I_l t : 1 YSC Vl Vk I_k

Figure 2.12: Equivalent network for a tap-changing transformer.

If the property of conservation of complex power in the ideal transformer is applied, an expression for Il can be found:

Il= −t∗Ik= −t∗YSCVk+ |t|2VlYSC (2.19)

The two equations make up the two-port description of a transformer with a tap changer:  Ik Il  = YSC  1 −t −t∗ _|t|2  ·  Vk Vl  (2.20) If t is written in its polar form, this equation becomes:

 Ik Il  = YSC  1 −(|t|∠α) −(|t|∠ − α) |t2 |  ·  Vk Vl  (2.21) t = 1 for symmetrical PSTs. It can be seen that the admittance matrix becomes asymmetrical for a complex winding ratio. This is a major drawback of this modelling technique and the reason why it is not used in this research.

2.4.3

Non-idealities

Load dependence

A PST adds an extra series reactance to the transmission line. This reactance causes a voltage change, altering the total phase shift angle, depending on the load [83]. Fig. 2.13 shows the phasor diagram for a transmission line with a PST and the corresponding single-phase diagram. The sending end voltage Vs is shifted ideally

by an angle α by injecting a voltage Vinj,0, assuming no-load conditions. However,

under load, a voltage drop jXpstI has to be taken into account. This results in a

deviation ∆α from no-load. The angle deviation is calculated by taking the tangent: tan(∆α) = tan((∆α + ϕ) − ϕ) = _{1 + tan(∆α + ϕ) tan ϕ}tan(∆α + ϕ) − tan ϕ (2.22) Furthermore, by applying basic geometry:

tan(∆α + ϕ) = XpstI + Vs,l0 sin ϕ

Vs,l0 cos ϕ

Xl Vr 6 0 Vs6 δ V0 s,06 (δ + α) Vs,l0 6 (δ + α − ∆α) α I Xpst

(a) Single-line diagram

α I Vr Vs V0 s,0 V_s,l0 Vinj,0 jXpstI ∆α ϕ δ (b) Phasor diagram

Figure 2.13: Single-phase diagram and phasor diagram for a PST under load. If (2.23) is combined with (2.22), an expression for the angle deviation results [84]:

∆α = arctan     XpstI V0 s,l cos ϕ 1 + XpstI V0 s,l sin ϕ     (2.24) If the reactance of the PST is written in pu, the following holds:

XpstI V0 s,l = Xpst,puXbase I V0 s,l = Xpst,pu I In (2.25) where In is the rated PST current (current for Xpst = Xbase)

This equation can be combined with (2.24):

∆α = arctan     I InXpst,pu cos ϕ 1 + I InXpst,pu sin ϕ     (2.26) This result is valid for an inductive load as seen from the PST terminals (I lags Vs,l0 ). For a capacitive load, the angle ϕ can be regarded as negative. If this convention

is taken into account, (2.26) remains valid.

In the case studied here, a PST is used to boost the power through the line. If it is used to reduce the power, the voltage change phasor does not change direction, since this is only determined by the power flow direction. In this power-limiting mode, the voltage change actually contributes to the phase shifting angle. This is an important design issue, since the increased voltage across the PST compared to the no-load case can cause saturation in certain parts of the device.

Dependence of the PST reactance on the phase shift angle

The internal PST reactance varies as a function of the phase shift angle. Measure-ments show that this relation can be approximated by a quadratic curve (Fig. 2.14). In practice, this dependence is often omitted for the sake of simplicity.

-251 -20 -15 -10 -5 0 5 10 15 20 25 1.05 1.1 1.15 1.2 1.25 1.3 1.35

no-load angle [degrees]

on e-ph as e re ac ta nc e [p u] Measured data Quadratic fit

Figure 2.14: Relation between the single-phase reactance and the phase shift angle for an existing PST.

2.5

Phase shifters in the Netherlands and Belgium

2.5.1

The Netherlands

The Netherlands has five interconnections with its neighbouring countries Belgium and Germany (Fig. 2.15). The southern part of the country is closer to the centre of the meshed continental European grid (UCTE zone) than the northern part. As a consequence, import of power causes a heavy loading of the southern interconnectors, especially on the line Maasbracht-Rommerskirchen/Siersdorf, compared the loading of the most northern interconnector Meeden-Diele. In order to maintain N − 1 se-curity, the import capacity has to be limited there. The Dutch transmission system operator (TSO) TenneT, studied various solutions to solve this problem. Additional transmission lines did not offer instant relief, as such projects would take far too long because of negotiations with the German TSOs RWE-Netz and Eon Netz and proce-dures to obtain all necessary permits. A better solution is to install a phase shifter

Figure 2.15: Interconnectors of the Netherlands with its neighbouring countries. Three-phase through rating 1000 MVA

Applicable standards IEC

Type of cooling ONAN

Type of regulation Symmetrical

Number of steps +/- 16 steps

No-load phase angle 37.2◦

Load phase angle at 1000 MVA 30◦

Short circuit impedance <12% at 1000 MVA

Rated voltage 380 kV

Table 2.1: Main design parameters of the Meeden PSTs.

at an appropriate location in the transmission grid [50]. The device should not be placed in the Maasbracht interconnector with Germany, as this would only shift the import to the Belgian border, not affecting the Meeden interconnector. Locating the PST in Meeden offered the possibility to increase the import in the northern part of the country, distributing the loading of the interconnectors more equally. In practice, two PSTs are required as the interconnector consists of a double circuit.

The most important design parameters of the symmetrical indirect PSTs that were installed are given in Table 2.1 [90]. The series transformer and the exciter could

have been constructed as two separate three-phase transformers. This is the two-tank design. The problem with this design is the fact that the connections between both tanks are at the highest voltage level.

Figure 2.16: A single-phase unit of a Meeden PST.

For the Meeden PSTs, it was decided to group the series transformer and exciter per phase, resulting in a three-tank design. More interconnections are needed, but they are on a lower voltage level.

Each single-phase unit consists of:

- A single-phase series transformer, rated at 213 MVA

- A single-phase exciter with two tap-changers, rated at 202 MVA

- Two current control transformers to distribute the current equally between the two tap-changers.

The scheme of a unit is drawn in Fig. 2.16. The series transformer consists of two smaller transformers of 133 kV primary and 70 kV secondary each. The exciter is constructed out of two parallel transformers with variable tap. The primary winding is rated at 208 kV and the secondary winding at 38.5 kV. The secondary windings are connected in series, resulting in a rated regulating voltage of 77 kV.

The Meeden transformers, that came into operation in 2002 and 2003, are the only PSTs installed in the Dutch transmission system. However, at the German side of the Hengelo-Gronau interconnector, a PST is installed in Gronau with a range of about -12 to +12◦_{, and a rating of 1200 MVA.}

2.5.2

Belgium

Figure 2.17: PSTs in the Netherlands and Belgium. Source: UCTE

As discussed before, new developments in the power system lead to increased power flows between countries. Transit flows induced by increased trade between Germany and France and loopflows because of surplus of power in noncentral gener-ating sites (wind energy in the north of Germany or nuclear energy in the north of France) cause additional loading of the Belgian and Dutch grid, leading to critical operational situations. It is in this framework that the Belgian TSO Elia has made the decision to install PSTs in the year 2007 and 2008.

There are two interconnectors between the Maasbracht substation in the Nether-lands and the Belgian grid. One line is connected to the Meerhout and the other to the Gramme substation. Two PSTs are needed to gain total control over this interconnector. Both devices are installed in a new substation in the vicinity of Kin-rooi called Van Eyck. Both PSTs have a rating of 1400 MVA and an angle range

of -25 to +25◦_{. They are of the indirect symmetrical type. In this research, these}

two PSTs are referred to as Van Eyck 1 and 2. The former is installed in the line Maasbracht-Meerhout and the latter in the line Maasbracht-Gramme. An identical PST is installed in the Zandvliet substation.

Furthermore, a PST is placed near the Belgian-French border, in the Monceau substation, in order to alleviate local problems [79]. The 220 kV line coming from Chooz (France) is extended to Monceau [30], and in this substation a transformer is installed, coupling the 220 kV line with the 150 kV grid and at the same time acting as a PST. It has a rating of 400 MVA and a range of -15 to +3◦ _{at full load (-12 to}

+12◦_{at zero-load). An overview of the location of all devices can be seen in Fig. 2.17.}

2.6

Static and dynamic operation

In this section, the 9-bus system described in [7] is used to illustrate the operation of phase shifters in a grid. The study system is depicted in Fig. 2.18. For this study, a PST is installed between busses 6 and 9.

230 kV

1

2

3

4

5

6

7

8

9

2

1

3

Load C

Load B

Load A

230 kV 18 kV 13.8 kV 16.5 kV 230 kV

Figure 2.18: 9-bus system.

2.6.1

Load flow analysis

The setting of the PST is varied from -20 to +20◦_{. In Fig. 2.19, the resulting active}