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Optimisation of Transmission Systems
by use of
Phase Shifting Transformers
Optimisation of Transmission Systems
by use of
Phase Shifting Transformers
PROEFSCHRIFT
ter verkrijging van de graad van doctoraan 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 uurdoor
Jody VERBOOMEN
Burgerlijk Werktuigkundig-Elektrotechnisch ingenieur
geboren te Leuven, Belgie.
Dit proefschrift is goedgekeurd door de promotor:
Prof.ir. W.L. Kling
Samenstelling promotiecommissie:
Rector Magnificus, voorzitterProf.ir. W.L. Kling, Technische Universiteit Delft, promotorProf.ir. L. van der Sluis, Technische Universiteit DelftProf.dr.ir. J.H. Blom, Technische Universiteit EindhovenProf.ir. M. Antal, Technische Universiteit Eindhoven (emeritus)Prof.dr.ir. R. Belmans, Katholieke Universiteit Leuven, BelgieProf.dr. G. Andersson, Swiss Federal Institute of Technology, Zurich, ZwitserlandDr.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 agentschapvan het Nederlandse Ministerie van Economische Zaken.
Printed by: Wohrmann 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 maybe 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.
to my late grandfather Louis Vandermaesen, RIP
Summary
Optimisation of Transmission Systems by use of Phase Shifting Transformers
In the recent past, power systems have undergone a major transformation as anumber of technical and organisational developments have taken place, leading toa change in the way electrical grids are operated and used. An issue that becomesincreasingly important is power flow control in meshed systems, which is reflectedin the fact that throughout Europe phase shifting transformers (PSTs), which areexamples of power flow controllers, are being installed at an increasing number oflocations. The specific case of the Dutch and Belgian grid is very interesting in thissense, as several devices are installed in a relatively small area. The operation of asingle PST has already an impact on the power flows in the entire network, but ifseveral of these devices are installed, their joint impact is significant.
The combination of remote power generation and international trade gives rise tothe problem of transit flows and loopflows, causing congestion problems, especiallywhen the flows are unforeseen, for example due to the uncertain nature of somegeneration sources. Power flow control in general, and the PST in particular offer apossibility to deal with these congestions and to enhance grid security. The powerflow through an area can be controlled and limited if needed. As areas are generallylarge, several power flow control devices might be needed for adequate control. Thecoordination between these devices is crucial for obtaining the desired effect, or atleast 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 cantake place between two neighbouring areas under safe conditions. Every combinationof 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 everyadditional PST adds an extra dimension to the search space containing the optimumTTC. Instead, a limited amount of combinations can be randomly selected, and bymeans of the resulting TTC histogram, an estimation can be made of the best- andworst-case options. This technique is called Monte Carlo Simulation (MCS). The per-formed MCS study indicates that poor coordination can result in internal loopflows,
causing severe overloading. Alternatively, well-coordinated PSTs avoid such loopflowsand can establish an even loading of the interconnectors at the borders if that is theaim.
MCS is not really an optimisation algorithm, but merely gives an indication ofthe probability of the system states to occur. In order to gain more information onthe optimal region of the search space in a reasonable computation time, MultistageMonte Carlo Simulation (MMCS) is adopted. This method consists of performingseveral subsequent MCS runs in order to zoom into the optimal region. A majordrawback is that the required running time is of the order of a few hours, whichcan be unacceptable for some applications. Furthermore, it is very hard to definethe 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 areoptimisation methods based on evaluations of the objective function through simu-lations, offer a solution. They consist of a heuristic local search algorithm guided bya top-level strategy, in order to avoid convergence to local optima. In the thesis, itis shown that not all metaheuristic methods work equally well; it is demonstratedthat Particle Swarm Optimisation (PSO) is suited for the TTC optimisation, pro-vided that the algorithm parameters are chosen in a good way. A parameter studyis performed to indicate the best values for the inertia and the swarm size, and theresulting tuned algorithm shows a fast convergence.
Next to the optimisation approach with the TTC as the only goal function, forinstance also the system losses can be included in a multiobjective optimisation. Theresult is a Pareto front, offering the possibility to make a trade-off between both goalfunctions. The simulation results show that the losses increase steeply if the absolutebest TTC value is to be obtained. If this is not acceptable, a lower TTC valuecan be targeted for. Metaheuristic optimisation offers a fast and accurate means toobtain the maximum TTC or the optimal value for any other goal function and it istherefore 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 onthe 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 atransmission line can then be written as a linear function of the PST settings. Thecoefficients 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 TTCcan 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 thisthesis, can be applied on a variety of problems. As a first example, it is shown that
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a linearised optimal power flow used for unit commitment can be easily extended toinclude PSTs. Optimal use of the PSTs can result in a lower total operational cost,as they allow a redirection of the power flows, possibly preventing the dispatch ofmore expensive peak units in case of network congestions. As a second example, theLinear Least Squares (LLS) method is applied to calculate the optimal PST settingsin order to evenly distribute the flows on a set of interconnectors at the border of twosystems. The flows on these interconnectors are expressed as a set of linear equationsas a function of the PST settings. As a third example, it is shown that in a stochasticcontext, an optimisation algorithm can be designed to minimise the congestion riskfor the whole system taking into account the PST settings. Finally, both PSO and theanalytical DC calculation are applied to study different options for the reinforcementof the Dutch-German border.
This thesis shows that a whole range of optimisation strategies can be appliedin order to coordinate several PST devices. The result is an improved situation forthe 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 aspectof 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 globaloptimisation for multiple areas or to make local optimisations within different areasco-exist. Finally, a profound study of the relation between PSTs and system stabilitystill has to be performed.
Jody Verboomen
iii
Samenvatting in het Nederlands
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, dieertoe hebben geleid dat elektriciteitsnetten op een andere manier worden bedreven engebruikt. Een aspect dat meer en meer aan belang wint is vermogenssturing in ver-maasde netten, wat tot uitdrukking komt in het feit dat dwarsregeltransformatoren ofphase shifting transformers (PSTs), die een voorbeeld zijn van vermogensregelaars, opeen toenemend aantal locaties in Europa worden geplaatst. Het specifieke voorbeeldvan het Nederlandse en Belgische net is zeer interessant wat dit betreft, aangezien hiermeerdere van deze regelaars zijn geınstalleerd in een relatief klein gebied. Het gebruikvan een enkele PST heeft al invloed op de vermogenstromen in het hele netwerk, enals meerdere van deze toestellen geplaatst worden, dan is hun gecombineerde impactaanzienlijk.
De combinatie van afgelegen productielocaties en internationale handel leidt totzogenaamde transit flows en loop flows; hierdoor kan congestie ontstaan, vooral alsdeze flows niet voorzien zijn vanwege bijvoorbeeld het onzekere karakter van bepaaldeopwekkingsbronnen. Vermogenssturing in zijn algemeenheid, en de PST in het bij-zonder maken het mogelijk om iets aan deze congestie te doen en om de netveiligheidte verhogen. De vermogenstroom door een gebied kan geregeld worden, en indiennodig beperkt. Aangezien de gebieden meestal uitgebreid zijn, kan het noodzakelijkzijn om meerdere toestellen te gebruiken om tot een adequate regeling te komen.Coordinatie tussen deze regelaars is cruciaal om het beoogde effect te verkrijgen, oftenminste om een vermindering van netveiligheid te voorkomen.
De Total Transfer Capacity (TTC) kan gebruikt worden om aan te geven hoe goedde instellingen van de PSTs op elkaar zijn afgesteld. Deze indicator geeft aan hoeveelvermogen 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 omde optimale waarde te vinden, aangezien elke PST een extra dimensie toevoegt aan
de zoekruimte waarin de optimale TTC zich bevindt. In de plaats daarvan, kanwillekeurig een beperkt aantal combinaties geselecteerd worden en aan de hand vanhet 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 coordinatiekan leiden tot interne loopflows, waardoor ernstige overbelastingen kunnen ontstaan.Goed gecoordineerde PSTs daarentegen, vermijden zulke loopflows en kunnen, alsdat het doel is, een evenwichtige belasting van de interconnectoren op de grenzenbewerkstelligen.
MCS is niet echt een optimalisatiealgoritme, maar geeft enkel een indicatie vande waarschijnlijkheid dat bepaalde systeemtoestanden voorkomen. Om binnen eenredelijke rekentijd meer informatie te verkrijgen over het optimale gebied van dezoekruimte, 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 omde doelfunctie voor het optimalisatieprobleem als functie van de PST instellingen tedefinieren, mede door de vereiste storingsanalyse.
Voor dit soort problemen vormen metaheuristische optimalisatiealgoritmes, diegebaseerd zijn op evaluaties van de doelfunctie door middel van simulaties, een oplos-sing. Ze bestaan uit een lokaal heuristisch zoekalgoritme dat geleid wordt door eenstrategie op hoger niveau, zodat convergentie naar lokale optima wordt vermeden. Inhet proefschrift wordt getoond dat niet alle metaheuristieken even goed werken; erwordt aangetoond dat Particle Swarm Optimisation (PSO) geschikt is om de TTCte optimaliseren, op voorwaarde dat de parameters van het algoritme op een goedemanier worden gekozen. Een parameterstudie is uitgevoerd om de beste waardes voorde inertie en de zwermgrootte te vinden, en het resulterende algoritme convergeertop 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 metmeerdere doelfuncties. Het resultaat is een Pareto front, waarbij het mogelijk wordteen afweging te maken tussen beide doelen. De simulatieresultaten laten zien datde verliezen zeer sterk toenemen als de allerbeste TTC waarde wordt nagestreefd.Indien dit niet acceptabel is, kan een lagere TTC beoogd worden. Metaheuristischeoptimalisatiealgoritmes maken het mogelijk om op een snelle en accurate manier demaximale waarde te vinden voor de TTC, of de optimale waarde voor welke doelfunc-tie dan ook, en zijn daarom zeer bruikbaar voor de coordinatie van meerdere PSTs.Echter, de black-box aanpak steunt op simulaties, en geeft geen enkele analytischeinformatie over de impact en de bedrijfsvoering van een PST in een vermaasd net.
Door bepaalde aannames te doen, zoals de benaderingen die gebruikt worden inDC load flow berekeningen, kan een dieper inzicht verkregen worden op dit gebied,
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aangezien de vermogenstroom in een transmissielijn dan geschreven kan worden alseen lineaire functie van de PST instellingen. De coefficienten die in deze lineaire relatiegevonden zijn, worden phase shifter distribution factors (PSDFs) genoemd en hangenenkel af van de topologie van het systeem. Aangezien de TTC berekend kan worden alseen som van vermogenstromen, is het mogelijk om deze grootheid in een analytischevorm te beschrijven als functie van de PST instellingen. Aangetoond wordt dat dezeuitdrukking stuksgewijs lineair is en daardoor met Lineair Programmeren (LP) kangeoptimaliseerd worden.
De methoden voor optimale coordinatie van PSTs die in dit proefschrift zijn ont-wikkeld, kunnen op een breed scala van problemen worden toegepast. Een eerstevoorbeeld laat zien dat een gelineariseerde optimal power flow berekening voor deinzet van eenheden op een eenvoudige manier kan worden uitgebreid met PSTs. Op-timaal gebruik van PSTs kan resulteren in lagere totale operationele kosten, omdatze vermogenstromen kunnen omleiden, waardoor mogelijkerwijs de inzet van durepiekcentrales 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 PSTinstellingen te berekenen voor een evenwichtige verdeling van de vermogenstromenover de interconnectoren op de grens tussen twee zones. De vermogenstromen viadeze interconnectoren worden uitgedrukt als een stelsel van lineaire vergelijkingen alsfunctie van de PST instellingen. Een derde voorbeeld laat zien dat in een stochastischecontext, een optimalisatiealgoritme kan worden opgesteld om het congestierisico voorhet hele systeem te minimaliseren door middel van PSTs. Tenslotte worden zowelPSO als de analytische DC berekening toegepast om de verschillende opties voornetuitbreiding aan de Nederlands-Duitse grens te bestuderen.
Dit proefschrift laat zien dat een breed spectrum van optimalisatiestrategieen kanworden toegepast voor de coordinatie van PSTs. Het resultaat is een verbeterde netsi-tuatie voor wat betreft transportcapaciteit, systeemveiligheid, of welk ander criteriumdan 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 inflow based market coupling. Voorts is het een uitdaging voor verder onderzoek, omeen 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 PSTsen stabiliteit van het systeem.
Jody Verboomen
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Contents
Summary i
Samenvatting in het Nederlands v
Contents ix
1 Introduction 11.1 Transmission bottlenecks . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Transports due to electricity markets . . . . . . . . . . . . . . . . . . . 2
1.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Status in Europe . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Variability of intermittent sources . . . . . . . . . . . . . . . . . . . . . 41.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Transit flows and loopflows . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Objectives and limitations . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Research framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Power Flow Control 132.1 Power through a transmission line . . . . . . . . . . . . . . . . . . . . 132.2 Classification of active power flow controllers . . . . . . . . . . . . . . 14
2.2.1 Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.2 Controlled parameter . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Phase shifter technology . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Direct asymmetrical PSTs . . . . . . . . . . . . . . . . . . . . . 162.3.2 Direct symmetrical PSTs . . . . . . . . . . . . . . . . . . . . . 182.3.3 Indirect asymmetrical PSTs . . . . . . . . . . . . . . . . . . . . 192.3.4 Indirect symmetrical PSTs . . . . . . . . . . . . . . . . . . . . 202.3.5 Comparison of the topologies . . . . . . . . . . . . . . . . . . . 20
2.4 Phase shifter modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Reactance and ideal phase shift . . . . . . . . . . . . . . . . . . 222.4.2 Two-port equivalent . . . . . . . . . . . . . . . . . . . . . . . . 232.4.3 Non-idealities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Phase shifters in the Netherlands and Belgium . . . . . . . . . . . . . 262.5.1 The Netherlands . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5.2 Belgium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Static and dynamic operation . . . . . . . . . . . . . . . . . . . . . . . 302.6.1 Load flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . 302.6.2 Transient stability analysis . . . . . . . . . . . . . . . . . . . . 31
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Search Space Exploration and Path Determination for PST Settings 353.1 Transfer capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.2 Simulation-based calculation . . . . . . . . . . . . . . . . . . . 37
3.2 MCS theoretical background . . . . . . . . . . . . . . . . . . . . . . . 383.2.1 Monte Carlo fundamentals . . . . . . . . . . . . . . . . . . . . 383.2.2 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Exploration using MCS . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.1 Search space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.3 Relation between TTC and the Meeden PSTs . . . . . . . . . . 423.3.4 MCS of the TTC with all PSTs . . . . . . . . . . . . . . . . . . 44
3.4 Multistage Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . 463.5 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.6 Path determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . 493.6.2 Path strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.6.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Metaheuristic Optimisation Methods 574.1 Introduction to metaheuristics . . . . . . . . . . . . . . . . . . . . . . 58
4.1.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . 584.1.2 Advocates and sceptics . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.1 Metaheuristics based on a unique solution . . . . . . . . . . . . 594.2.2 Metaheuristics based on a population of solutions . . . . . . . . 62
4.3 Application of Evolutionary Computation . . . . . . . . . . . . . . . . 644.3.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . 644.3.2 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . 67
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4.3.3 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4 Application of Particle Swarm Optimisation . . . . . . . . . . . . . . . 704.4.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . 704.4.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Transit flow sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.6 Multiobjective optimisation including losses . . . . . . . . . . . . . . . 76
4.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.6.2 Conventional Weighted Aggregation . . . . . . . . . . . . . . . 774.6.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5 PSTs in Linearised Equations 835.1 Linearised power flow equations . . . . . . . . . . . . . . . . . . . . . . 84
5.1.1 DC load flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.1.2 PSTs in a DC load flow . . . . . . . . . . . . . . . . . . . . . . 855.1.3 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Analytical expression for TTC . . . . . . . . . . . . . . . . . . . . . . 905.3 TTC optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3.1 N secure TTC . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.3.2 N − 1 secure TTC . . . . . . . . . . . . . . . . . . . . . . . . . 935.3.3 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4 TTC sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.4.1 Sensitivity of the BCE . . . . . . . . . . . . . . . . . . . . . . . 995.4.2 Sensitivity of the maximum power shift . . . . . . . . . . . . . 995.4.3 Total sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.5 Equivalent reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6 Application of Optimisation Methods 1056.1 Generation unit dispatch with PSTs . . . . . . . . . . . . . . . . . . . 106
6.1.1 DC OPF without PSTs . . . . . . . . . . . . . . . . . . . . . . 1066.1.2 DC OPF with PSTs . . . . . . . . . . . . . . . . . . . . . . . . 1076.1.3 Application to the New England system . . . . . . . . . . . . . 108
6.2 Border-flow control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2.1 Linear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . 1116.2.2 Single border control . . . . . . . . . . . . . . . . . . . . . . . . 1136.2.3 Combined border control . . . . . . . . . . . . . . . . . . . . . 1156.2.4 Influence of grid topology changes . . . . . . . . . . . . . . . . 1166.2.5 Modelling of discrete behaviour . . . . . . . . . . . . . . . . . . 118
6.3 Congestion risk minimisation . . . . . . . . . . . . . . . . . . . . . . . 1196.3.1 Quantification of uncertainty in power system flows . . . . . . 119
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6.3.2 Impact of PSTs on system power flows . . . . . . . . . . . . . . 1206.3.3 Minimisation of the overall system congestion risk . . . . . . . 1206.3.4 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Reinforcement study on the Dutch-German border . . . . . . . . . . . 1256.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7 Conclusions and Recommendations 1317.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.2.1 Economic Aspect . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.2.2 Multi- or Interarea Coordination . . . . . . . . . . . . . . . . . 1337.2.3 Dynamic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . 133
A Abbreviations, Symbols and Operators 135A.1 List of abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135A.2 List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136A.3 List of subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138A.4 List of superscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139A.5 List of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B Mathematical Background 141B.1 Big O notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141B.2 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C New England Test System Data 145
Bibliography 149
Publications 155Journal publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Conference publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 powerflow control, which is reflected in the fact that throughout Europe, phase shiftingtransformers (PSTs), which are examples of power flow controllers, are installed atnumerous locations. There are basically three reasons why these devices are adopted:
- The efficiency of grid management can be improved by solving bottlenecks intransmission systems. PSTs are able to shift flows from congested areas toother areas where transmission capacity is still available, offering the possibilityto delay grid investments. The devices can either be installed in congested areasto “push” the flow to another area, or in areas with low loading to “pull” theflow.
- A wide range of transmission scenarios induced by market decisions must behandled. The outcome of electrical energy markets can lead to a variety oftransport flows, especially when international trade is considered. PSTs offerthe possibility to make optimal use of available network capacities and conse-quently to facilitate the market mechanisms as much as possible. Again, thismeans that investments in the grid can be postponed.
2 Chapter 1
- 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 increasedsignificantly 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 flowpattern in the European system can be totally different from one moment tothe other.
The combination of intermittent power generation and international trade givesrise to the problem of transit flows and loopflows, causing congestion problems oninterconnectors. Power flow control in general, and the PST in particular offer apossibility to deal with these congestions [72]. The power flow through an area canbe controlled and limited if needed. As areas are generally large, several power flowcontrol devices might be needed for adequate control. The coordination between thesedevices is crucial for obtaining the desired effect, or at least to avoid a decrease ingrid 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 fromgeneration to load may exist. Each line has a maximum capacity determined by anumber of constraints, such as thermal loading, stability and object clearance.
When power is exchanged between two areas, the flow may spread over differentparallel 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 themaximum capacities of the interconnectors between both areas, but it is limited dueto 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 thesituation. This solution is not economically favourable, as more expensive units aredispatched. 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 inthe field of generation, transmission and distribution. These companies are referred toas 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
Introduction 3
not the result of a market mechanism, but it was an average of the aggregated costincurred in generation, transmission and distribution of that utility. Unit commitmentand economic dispatch of generation took place on a regional or national level andthe capacity of the transmission grid was an integral part of the optimisation process.
Since market mechanisms have been introduced in this structure, radical changeshave taken place. Instead of one large utility, several specialised companies come intoplay [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 tofacilitate the market by allowing large power transports between producers anddistribution 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 thatthey buy electric energy from a supplier in another country. They can comparedifferent options and make a choice based upon the most interesting offer.
- Customers do not only choose based upon the price, but also based upon theservice they get. Consequently, companies are forced to supply a decent servicein order to keep their customers satisfied.
- Power suppliers are free to invest in new power plants, whether the extra powergeneration 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 haveto transport the electrical energy agreed upon by the market and they areobliged to solve network constraints [12; 92]. They are also responsible for thesecurity of supply.
4 Chapter 1
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 assuredthat he will have access to the grid.
- Customers are eligible, meaning that they are free to choose their electricitysupplier. 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 independententities [33].
1.3 Variability of intermittent sources
1.3.1 Background
-15 -10 -5 0 5 10 150
500
1000
1500
2000
2500
3000
wind speed variation [m/s]
frequ
ency
[-]
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, fossilfuel-fired power plants to renewable energy systems [5]. Different technologies are
Introduction 5
available, but at the time of writing, wind turbines are the main representatives inthe 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 variabilitydepends on the time interval that is considered, as shown in Fig. 1.1. Obviously, thevariability is much bigger on a day-to-day basis, compared to the limited fluctuationswithin 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 typicalcharacteristic is shown in Fig. 1.2 [115]. The influence of wind speed variability on thepower output depends on the operating point on the characteristic. In the examplecurve, the output power in the wind speed range from 4 to 15 m/s increases steeplywith increasing wind speed. Also the so-called cut-out wind speed (26 m/s in thefigure) can be a problem. At very high wind speeds, a small fluctuation can lead toa shutdown procedure, resulting in an output power change from nominal power tozero in a very limited time span1.
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
wind speed [m/s]
P/P
no
m [
pu
]
Figure 1.2: Power curve of a typical wind turbine.
1This problem is tackled in some modern wind turbine designs, where the power output abovethe cut-out wind speed decreases gradually. Also, some wind parks are designed in such a way thateach wind turbine has a slightly different cut-out speed.
6 Chapter 1
1.3.2 Consequences
Fluctuations within the minute are often caused by turbulence. This phenomenonis very local, and its effect is partly smoothened within a single wind farm due toaggregation. Hence, the effect on the power system is very small and is of no furtherinterest.
Fluctuations within a quarter of an hour do not influence the scheduling of theproduction park, but they are to be compensated by reserve power [98]. The provisionof reserve power is mostly organised by transmission companies within defined zonesof the power system. Hence, the fluctuations are compensated locally and this doesnot influence the power flows in the grid to a great extent.
Hourly variations do have an impact on scheduling, and the forecasting of thewind becomes important. A lot of research is being performed on different forecastmethods, but the accuracy is still limited, especially when the wind speed is nearthe 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 byusing reserve power. The impact on the grid might be bigger as the variations arealso larger, but the effects are still local.
Variations on the longer term can be due to change of weather or other climaticeffects. 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 bevery much influenced, as the injection pattern may change in a very dynamic way. Aline flow can be from A to B on one day, and the opposite the other day. This way ofthinking 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 buytheir electrical energy by signing a contract with the supplier of their choice, evenacross the borders. The contractual path is the direct electrical path between producerand consumer. However, in a strongly meshed grid such as the European system, thepower flow spreads out over several parallel paths, giving rise to so-called transit flowsand loopflows [39]. The distribution of the power exchange over the parallel pathsdepends on their relative impedance.
Fig. 1.3 shows possible scenarios that can occur. A third party (transit) loopflowoccurs when a fraction ε of a power exchange P between two areas A and B flowsthrough a third area C. A typical example is the power exchange between Germanyand France, of which typically 25 to 30 percent flows through the Netherlands andBelgium. A second party loopflow happens when an internal power transfer partlyflows through a second area. This problem can occur when generation and loadcentres are in geographically distant locations, which can be the case with large windparks and large conventional power generation at remote sites. An example is the
Introduction 7
εP
A B
C
(1 − ε)P
εP
(a) 3rd party loopflow
(1 − ε)P
B
A
εPεP
(b) 2nd party loopflow
Figure 1.3: Possible loopflow scenarios.
situation in Germany, where a high concentration of wind parks is installed in thenorth and a high concentration of load is located in the south [27]. The resultingnorth-south flows induce a loopflow in the Netherlands. Moreover, this loopflow isnot constant, because of the variability of the wind.
A loopflow is a phenomenon that is difficult to handle, as it often comes ratherunexpected for the second or third party, and even if it can be predicted, it may stillrequire special actions to prevent congestions [19; 21]. A first step towards betterprediction in the European system is the day-ahead congestion forecast (DACF),where possible congestions are predicted for the next day, taking into account thesituation in the neighbouring areas [97]. However, the DACF only gives an indicationinstead of solving the problem itself [101]. The problem can be relieved by usingPSTs.
1.5 Objectives and limitations
The operation of PSTs in a meshed grid has an impact on the power flow in thenetwork. 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 thisthesis, transmission grids with PSTs are investigated. In particular, the followinggoals are put forward:
- The analysis and quantification of the impact of a PST on a meshed grid. Thisincludes the development of models for the device.
- The development of methods to obtain optimal coordination of several PSTs ina meshed grid. An objective function should be formulated, and an optimisationmethod must be adopted to solve the problem.
- The investigation of different strategies to use a PST.
8 Chapter 1
The research that is presented in this thesis focuses on the steady state of thesystem. Although a brief introduction to PST operation under transient conditionsis given in chapter 2, this issue is not further elaborated. Furthermore, the study islimited to PSTs only; (other) FACTS devices are not considered. Finally, the researchfocuses on large transmission grids, and although distribution networks are taken intoaccount 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 EconomicalAffairs. The “Intelligent Power Systems” project is initiated by the Electrical PowerSystems and Electrical Power Electronics Groups of the Delft University of Technol-ogy and the Electrical Power Systems and Control Systems Groups of the EindhovenUniversity of Technology. In total 10 PhD students are involved and work closelytogether. The research focuses on the effects of the structural changes in generationand 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).
Inherentlystable
transmissionsystem
Manageabledistribution
networks
Optimalpowerquality
Self-controlling
autonomousnetworks
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 dynamicbehaviour 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 moregeneration takes place in the distribution networks, whereas the remainder is possiblygenerated further away in neighbouring systems. Solutions investigated include the
Introduction 9
control of centralised and decentralised power, the application of power electronicinterfaces and monitoring of the system stability.
The second part (research part 2), manageable distribution networks, focuses onthe distribution network, which becomes “active”. Technologies and strategies haveto be developed that can operate the distribution network in different modes andsupport the operation and robustness of the network. The project investigates howthe power electronic interfaces of decentralised generators or between network partscan be used to support the grid. Also the stability of the distribution network andthe effect of the stochastic behaviour of decentralised generators on the voltage levelare 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 partof the distribution network is sufficient to supply a local demand, the network canbe operated autonomously but as a matter of fact remains connected to the rest ofthe grid for security reasons. The project investigates the control functions neededto operate the autonomous networks in an optimal and secure way. The interactionbetween the grid and the connected appliances has a large influence on the powerquality.
The fourth part (research part 4), optimal power quality, of the project analyses allaspects of power quality. The goal is to provide elements for the discussion betweenpolluter and grid operator who has to take measures to comply with the standardsand 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 stabletransmission 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. Theycan be classified in different ways depending on the characteristics, such asswitching technology and the controlled parameter. As the focus of this thesisis on PSTs, a more profound review of PST technology is presented, discussingdifferent possible topologies. Furthermore, the modelling of PSTs is discussed,including the non-idealities that are neglected in many cases. Next, an overviewof 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 testcase. It is shown that the impact of such a device on the transient stability isvery 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
10 Chapter 1
(TTC) is chosen as the optimality indicator. The search space can be describedby a multidimensional function that connects the TTC to the different PSTsettings. The space is very large due to the high number of possible sets ofPST settings, making an exhaustive search highly impractical. A Monte CarloSimulation can serve as a tool to explore the search space, although it is notan optimisation method. The application of Monte Carlo Simulation to theDutch-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 adoptedand applied to the Dutch-Belgian grid. It is also shown that the sensitivityaround the optimal point is very different for each PST. Finally, the problemof avoiding unfavourable intermediate states when switching between two setsof PST settings is addressed. This problem can be redefined as a shortest pathproblem.
- In chapter 4, metaheuristic optimisation methods are discussed. Metaheuristicsare algorithms that only rely on evaluations of the objective function, whichmakes them very suitable for simulation-based optimisation. A classification isgiven based on the number of intermediate solutions that are used. A few ofthese methods are applied to the PST coordination problem in the Netherlandsand Belgium, namely Meta-Evolutionary Programming, Evolution Strategiesand Particle Swarm Optimisation. Without PSTs, transit flows have a largeimpact on the Dutch and Belgian grid. The optimal coordination of PSTs resultsin a much lower sensitivity to transits within the control capabilities of thedevices. Besides the Total Transfer Capacity, system losses can be incorporatedin the objective function as well, resulting in a multiobjective optimisationformulation. 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 activepower flows. It is shown that this relation is linear, and that it is characterisedby sensitivity factors called phase shifter distribution factors. These factorsdepend on the network topology only and not on the power injection patterns.Based on these linear equations, an analytical expression for the Total TransferCapacity can be derived. It is shown that this is a piecewise linear functionof the different PST settings, allowing a linear optimisation approach. Themethod offers a very fast optimisation, at the cost of inaccuracy caused by theapproximations. The derived expressions lead to a possibility to model a PSTas an equivalent reactance that is a function of the PST setting and dependingalso on the network topology.
- In chapter 6, some applications of the methods that are developed in earlier
Introduction 11
chapters are presented. The first application is the unit commitment problem,where the goal is to distribute the total required power over the productionfacilities in the most cost-effective way. The PSTs can be integrated in an op-timal power flow formulation, so that they are optimally controlled in order torelieve congestions. A second application is to make the relative line loadingsof the interconnectors on a certain border equal. Depending on the numberof PSTs in relation to the number of interconnectors, the set of equations canbe either solved exactly, or by using a Linear Least Squares approach. As athird 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 alteredin a straightforward manner for a single line. However, in a meshed system,the minimisation of the global congestion risk requires a global optimisationalgorithm 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 themaximum attainable Total Transfer Capacity (with optimal PST coordination)is calculated.
- Chapter 7 contains the conclusions of this thesis, as well as recommendationsfor future work.
CHAPTER 2
Power Flow Control
As discussed in chapter 1, electrical energy transports have increased over the lastyears in the European grid due to the liberalisation of the electricity markets andthe increased penetration of wind energy. Congestion in the transmission grid is aphenomenon that is encountered more often than before, and it is in this frameworkthat power flow control becomes key. This chapter is devoted to power flow controlin general and phase shifting transformers in particular. Different aspects of phaseshifters are discussed in detail, such as possible configurations, modelling issues forthe 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, omittingshunt elements1, as shown in Fig. 2.1. The complex current through this line isdepending on the voltage levels and angles on each side of the line and the impedancebetween both nodes:
I =V1 − V2
ZL=V1∠δ1 − V2∠δ2RL + jXL
(2.1)
1This approximation is valid for short overhead lines. For longer lines and cables, the shuntelements must be included.
14 Chapter 2
I
V2 6 δ2
RL jXL
V1 6 δ1
(a) Circuit diagram
jXLIV1 6 δ1
V2 6 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
R2L+X2
L
XL
sin δ +V 2
1 − V1V2 cos δR2L+X2
L
RL
+ j
V 21 − V1V2 cos δ
R2L+X2
L
XL
− V1V2
R2L+X2
L
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
XLsin δ Q1 =
V 21
XL− V1V2
XLcos 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 indirectlychange the line power, provided a meshed grid is considered. Conventional meansto achieve this include the addition of series impedances (inductances or capacitors)and the use of phase shifting transformers (PSTs). Modern developments in powerelectronics add new functionalities to these devices and even have led to a wholefamily 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 theother. Based on different technologies used for switching, a classification can be made[99].
Power Flow Control 15
- Mechanical switching is used for conventional devices where speed is not anissue. It is clear that control on dynamics is not possible. However, this tech-nology shows clear advantages, such as simplicity, relatively low cost and highreliability.
- Thyristor based devices are able to switch at a faster rate, but they can notperform 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 classicalhigh 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 forpulse 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 possibilitiesare discussed hereafter.
Line reactance
The active line power is inversely proportional to the line reactance. It is impossibleto 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 moreflexible control, a thyristor controlled reactor (TCR) in parallel with a fixed capacitorcan 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 seriescompensator 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 usedfor power flow control. Of course, voltage control is a very important topic, but it isnot the issue of this research.
16 Chapter 2
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 quadraturevoltage. This is established in a PST by injecting a part of the voltage between twophases in the third phase by using a tap variable transformer [52]. This kind of powerflow 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 unifiedpower flow controller (UPFC), which have capabilities that reach much further thanpower 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 byconnecting the windings in an appropriate manner to each other.
- Indirect PSTs are based on a construction using two separate transformers: onevariable tap exciter to regulate the amplitude of the quadrature voltage and oneseries transformer to inject the quadrature voltage in the line.
- Asymmetrical PSTs create an output voltage with an altered phase angle andamplitude 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. Eachcategory 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 terminalis magnetically coupled with the winding between the other two terminals. By doing
Power Flow Control 17
S2
L1
L2
L3
S1
S3
(a) Circuit diagram
∆V3
VL3
VS3
VS2
∆V2
VL2
VS1
∆V1
α
VL1
(b) Phasor diagram
Figure 2.3: Direct asymmetrical PST.
so, a quadrature voltage that can be regulated by means of the variable tap is addedto the input voltage in order to obtain a phase shift α. The direction of the phaseshift can be changed by using switches. In this way, the power flow in the line canbe increased or decreased. The relation between the tap position and the angle α isnonlinear 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 voltageis given by:
VS1 =∆V1
sinα(2.7)
Using (2.6), equation (2.7) becomes:
VS1 =∆V1
sin(
arctan ∆V1VL1
) (2.8)
The secondary voltage VS1 is always larger than the input voltage VL1 (for anonzero phase shift). The fact that voltage levels are changed, also influences thetransmitted power over the line:
P =VS1V2
Xl +Xpstsin(δ + α) (2.9)
where Xl is the line reactance and Xpst is the series reactance of the PST.
18 Chapter 2
Using (2.6) and (2.8), equation (2.9) can be rewritten as:
P =V2
Xl +Xpst· ∆V1
sin(
arctan ∆V1VL1
) sin(δ + arctan
∆V1
VL1
)(2.10)
=V2
Xl +Xpst·
∆V1
(sin δ cos
(arctan ∆V1
VL1
)+ cos δ sin
(arctan ∆V1
VL1
))sin(
arctan ∆V1VL1
) (2.11)
=V2
Xl +Xpst· (VL1 sin δ + ∆V1 cos δ) (2.12)
For constant δ, the relation between the quadrature voltage and the active linepower is linear. Equations (2.6) and (2.12) are plotted in Fig. 2.4 for δ = π
6 = 30,with VL1 and V2
Xl+Xpst= 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 2.50
1
2
3
∆ V1 [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 thedevice. The advantages are that the voltage amplitudes remain unchanged and thatthe maximum attainable angles are larger.
The relation between the quadrature voltage and the angle α is again nonlinearand can be derived from the phasor diagram (Fig. 2.5b):
α = 2 arcsin∆V1
2VL1(2.13)
Power Flow Control 19
M1
S1
L1
S3
L3
S2
L2
M3
M2
(a) Circuit diagram
∆V2
VS3
VS1
α
∆V1
VL1
VL3
VL2
VM1
VM3∆V3
VM2
VS2
(b) Phasor diagram
Figure 2.5: Direct symmetrical PST.
Using (2.13), the transferred active power becomes:
P =VL1V2
Xl +Xpstsin(δ + 2 arcsin
∆V1
2VL1
)(2.14)
Equations (2.13) and (2.14) are plotted in Fig. 2.6 with VL1 = 1 pu and V2Xl+Xpst
=1 pu. The active line power now reaches a maximum and decreases to zero. The reasonfor this behaviour is that, in contrast to the asymmetrical configuration, the angleα 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 inFig. 2.7a. In every phase, a controllable voltage is injected proportional to the voltagebetween the primary and secondary terminal of the two other phases. The resultingphasor 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 separatetanks 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 thesame as the one depicted in Fig. 2.3b.
20 Chapter 2
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
1.2
1.4
∆ V1 [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 symmetricalPST with δ = π
6 = 30.
L2L1 L3S1 S2 S3
(a) Circuit diagram
∆V3
α
∆V1
∆V2
VS3
VS2
VL2
VS1VL1
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 serieswinding in two and tapping the voltage for the exciter from the middle. Fig. 2.9shows 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. Thisfact is also reflected in terms of cost. However, the fact that the voltage amplitude isaltered is a major issue, making symmetrical topologies more popular. Furthermore,
Power Flow Control 21
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 asymmetriccounterparts.
A direct configuration is easier to construct and hence cheaper compared to theindirect implementation, as no exciter is needed. A major drawback, however, isthe fact that the tap changer and the regulating winding are directly exposed tosystem disturbances, making them very vulnerable. Also, the indirect topology allows
22 Chapter 2
more flexibility in the design phase, as the regulator circuit can be dimensionedindependently, 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 +Xpstsin(δ + α) (2.15)
On first sight, the relation between the active power that is transported over theline and the phase shift angle seems straightforward. However, if a change in α isapplied, this has an influence on δ, which contributes to the change in P . Hence, if acontrol action is performed, the active power does not follow a constant δ curve. Anextreme case is when a PST would be installed in a line connecting a single generatorto the rest of the grid. In this case, the change in α is fully compensated by thechange in δ, as the line power must stay constant and equal to the output power ofthe generator.
In order to illustrate the operating principles of a PST in a meshed grid, twoparallel 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 alarger 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 linesis equal, line 2 will reach its rated power much faster, leading to an inefficient use ofline 1. This can be solved by installing a PST. In this example, the device is placedin line 1, but it could be installed in either of the lines. It can be easily verified that(2.16) then becomes:
Power Flow Control 23
0 π δ
P
P1
P2
δ
(a) Without PST
0 π δ
P
−α
P1
P2
δ′
(b) With PST
Figure 2.11: P − δ graphs for two parallel lines.
P1
PG=
1
1 +(X1 +Xpst
X2
)(sin δ′
sin(δ′ + α)
) P2
PG=
1
1 +(
X2
X1 +Xpst
)(sin(δ′ + α)
sin δ′
)(2.17)
where δ′ is the phase angle difference between the bus of the generator and the busof 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 powerthrough line 1. Since the total transported power PG must be constant, the powertransported over line 2 decreases, resulting in a smaller value for δ. This is a veryimportant 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 linewithout 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 PSTintroduces a quadrature shift in voltage instead of a longitudinal one. As the phaseof the voltage is altered, and possibly the magnitude, the winding ratio t must beregarded as a complex number [22].
If Ohm’s law is applied to YSC , the following equation follows:
Ik = YSC(Vk − tVl) = YSCVk − tVlYSC (2.18)
24 Chapter 2
Il
t : 1YSC
VlVk
Ik
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 tapchanger: [
IkIl
]= YSC
[1 −t−t∗ |t|2
]·[VkVl
](2.20)
If t is written in its polar form, this equation becomes:[IkIl
]= YSC
[1 −(|t|∠α)
−(|t|∠− α) |t2|
]·[VkVl
](2.21)
t = 1 for symmetrical PSTs. It can be seen that the admittance matrix becomesasymmetrical for a complex winding ratio. This is a major drawback of this modellingtechnique 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 causesa 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 thecorresponding single-phase diagram. The sending end voltage Vs is shifted ideallyby 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 adeviation ∆α from no-load. The angle deviation is calculated by taking the tangent:
tan(∆α) = tan((∆α+ ϕ)− ϕ) =tan(∆α+ ϕ)− tanϕ
1 + tan(∆α+ ϕ) tanϕ(2.22)
Furthermore, by applying basic geometry:
tan(∆α+ ϕ) =XpstI + V ′s,l sinϕ
V ′s,l cosϕ(2.23)
Power Flow Control 25
Xl
Vr 6 0Vs 6 δ
V ′s,0
6 (δ + α) V ′s,l
6 (δ + α−∆α)
α
IXpst
(a) Single-line diagram
α
I
VrVs
V ′s,0
V ′s,l
Vinj,0jXpstI
∆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
V ′s,lcosϕ
1 +XpstI
V ′s,lsinϕ
(2.24)
If the reactance of the PST is written in pu, the following holds:
XpstI
V ′s,l= Xpst,puXbase
I
V ′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 cosϕ
1 +I
InXpst,pu sinϕ
(2.26)
This result is valid for an inductive load as seen from the PST terminals (I lagsV ′s,l). For a capacitive load, the angle ϕ can be regarded as negative. If this conventionis 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 isused to reduce the power, the voltage change phasor does not change direction, sincethis is only determined by the power flow direction. In this power-limiting mode, thevoltage change actually contributes to the phase shifting angle. This is an importantdesign issue, since the increased voltage across the PST compared to the no-load casecan cause saturation in certain parts of the device.
26 Chapter 2
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.
-25 -20 -15 -10 -5 0 5 10 15 20 251
1.05
1.1
1.15
1.2
1.25
1.3
1.35
no-load angle [degrees]
one-
phas
e re
acta
nce
[pu]
Measured dataQuadratic fit
Figure 2.14: Relation between the single-phase reactance and the phase shift anglefor 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 Belgiumand Germany (Fig. 2.15). The southern part of the country is closer to the centre ofthe meshed continental European grid (UCTE zone) than the northern part. As aconsequence, import of power causes a heavy loading of the southern interconnectors,especially on the line Maasbracht-Rommerskirchen/Siersdorf, compared the loadingof 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 systemoperator (TSO) TenneT, studied various solutions to solve this problem. Additionaltransmission lines did not offer instant relief, as such projects would take far too longbecause 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
Power Flow Control 27
Figure 2.15: Interconnectors of the Netherlands with its neighbouring countries.
Three-phase through rating 1000 MVAApplicable standards IECType of cooling ONANType of regulation SymmetricalNumber of steps +/- 16 stepsNo-load phase angle 37.2
Load phase angle at 1000 MVA 30
Short circuit impedance <12% at 1000 MVARated 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 beplaced in the Maasbracht interconnector with Germany, as this would only shift theimport to the Belgian border, not affecting the Meeden interconnector. Locating thePST in Meeden offered the possibility to increase the import in the northern part ofthe 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 thatwere installed are given in Table 2.1 [90]. The series transformer and the exciter could
28 Chapter 2
have been constructed as two separate three-phase transformers. This is the two-tankdesign. The problem with this design is the fact that the connections between bothtanks 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 exciterper phase, resulting in a three-tank design. More interconnections are needed, butthey 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 thetwo tap-changers.
The scheme of a unit is drawn in Fig. 2.16. The series transformer consists of twosmaller transformers of 133 kV primary and 70 kV secondary each. The exciter isconstructed out of two parallel transformers with variable tap. The primary windingis rated at 208 kV and the secondary winding at 38.5 kV. The secondary windingsare connected in series, resulting in a rated regulating voltage of 77 kV.
The Meeden transformers, that came into operation in 2002 and 2003, are theonly PSTs installed in the Dutch transmission system. However, at the German sideof the Hengelo-Gronau interconnector, a PST is installed in Gronau with a range ofabout -12 to +12, and a rating of 1200 MVA.
Power Flow Control 29
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 increasedpower flows between countries. Transit flows induced by increased trade betweenGermany 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 ofFrance) cause additional loading of the Belgian and Dutch grid, leading to criticaloperational situations. It is in this framework that the Belgian TSO Elia has madethe 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 otherto the Gramme substation. Two PSTs are needed to gain total control over thisinterconnector. 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
30 Chapter 2
of -25 to +25. They are of the indirect symmetrical type. In this research, thesetwo PSTs are referred to as Van Eyck 1 and 2. The former is installed in the lineMaasbracht-Meerhout and the latter in the line Maasbracht-Gramme. An identicalPST is installed in the Zandvliet substation.
Furthermore, a PST is placed near the Belgian-French border, in the Monceausubstation, in order to alleviate local problems [79]. The 220 kV line coming fromChooz (France) is extended to Monceau [30], and in this substation a transformer isinstalled, coupling the 220 kV line with the 150 kV grid and at the same time actingas 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 ofphase shifters in a grid. The study system is depicted in Fig. 2.18. For this study, aPST is installed between busses 6 and 9.
230 kV
1
2 3
4
5 6
7
8
9
2
1
3
Load C
Load BLoad 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 activepower flows are given. It can be seen that the line powers change linearly with the
Power Flow Control 31
phase shift angle. This behaviour is investigated later on in this research. The powerbalance is not changed by the phase shifter operation (except from the fact that thelosses can slightly differ), but the power flows are redistributed.
-20 -15 -10 -5 0 5 10 15 20-100
-50
0
50
100
150
PST settings [degrees]
Line
Pow
er [M
W]
Line 7-8 Line 8-9 Line 7-5 Line 5-4 Line 4-6 Line 6-9
Figure 2.19: Line flows in the 9-bus system for different PST settings.
2.6.2 Transient stability analysis
The relation between PST operation and stability in general and transient stabilityin particular is very difficult to comprehend. For a single machine connected to aninfinite bus, some relations can be derived [105]. However, there are no publicationsthat really describe the relation in analytical terms for meshed systems.
When it comes to the analysis of transient stability, three classes of methods canbe distinguished [71]:
- Time-domain simulations
- Direct methods, based on the Lyapunov criterion for stability
- Automatic learning
In the field of direct methods, some work has been done to develop an energyfunction accounting for PSTs [38]. In the area of time-domain simulations, the criticalclearing time is an indicator often used. In other work, the maximum rotor speeddeviation (MRSD) is used as an indicator [77; 88]. If a fault occurs somewhere in thesystem, the rotor speed of every synchronous generator shows an oscillation in time.
32 Chapter 2
The maximum of all deviations from the rated speed is a measure for the systemstability. The larger the maximum deviation, the less stable the system is and viceversa.
-20 -15 -10 -5 0 5 10 15 200.022
0.024
0.026
0.028
0.03
0.032
0.034
0.036
PST setting [degrees]
Max
imum
roto
r spe
ed d
evia
tion
[pu]
Fault bus 5Fault bus 6Fault bus 8
Figure 2.20: Maximum rotor speed deviation for different fault locations. The faultis a three-phase bus fault with a duration of 200 ms.
Simulations are performed on the 9-bus system in the Power Systems AnalysisToolbox (PSAT) in Matlab. Three-phase bus faults are applied to the busses 5, 6and 8 for a duration of 200 ms. The MRSD is monitored for different PST settings,and plotted in Fig. 2.20.
If a fault occurs on bus 5, a large positive setting of the PST is unfavourablefor stability as can be seen in the graph. A possible explanation is the fact that atsuch a setting, a large power flow occurs on line 7-5, causing generator 2 to becomeless stable and therefore losing synchronism more easily. On the other hand, whenthe same type of fault occurs at bus 6, a high positive setting causes an improvedstability. This may be explained by the limited amount of power on line 9-6 in thissituation, causing generator 3 to be more stable. The conclusion is that stability willbe affected (in a positive or a negative way), depending on different factors such asfault location and PST setting, but this limited case-study shows that even for such asmall system, it is hard to draw straightforward conclusions on the relation betweenthe PST setting and transient stability. This aspect of PST behaviour is not treatedfurther in this research, but it is certainly an interesting topic for future research.
Power Flow Control 33
2.7 Summary
Active power flow depends on the phase angle difference between the busses at bothends of the line, on the voltage magnitude at these busses and on the reactance ofthe line. Each of these three parameters can be influenced by a power flow controller.Recent innovations in power electronics have opened many doors in this particularfield, but the PST is still a valuable alternative.
PSTs come in various implementations: a distinction can be made between sym-metrical and asymmetrical types and between direct and indirect configurations.Classical modelling approaches include two-port equivalents, resulting in an asym-metrical admittance matrix, and the model of an ideal phase shift with a seriesreactance.
In the Netherlands and Belgium, PSTs are a very important topic. Two devicesare already installed in Meeden, on the Dutch-German border and one in Gronau, onthe German side of the Hengelo-Gronau interconnector. Furthermore, four devicesare planned in the Belgian grid; one near the Belgian-French border, and one in eachinterconnector at the Belgian-Dutch border.
Load flow analyses show that active power flows can be controlled in a quasi-linearway by PSTs. This aspect is studied in more detail further in this research. Transientstability analyses on a small network with a PST indicate that a relation betweentransient stability and the setting of the PST exists, but it is hard to comprehendthis. It might be an interesting topic for future research.
CHAPTER 3
Search Space Exploration and PathDetermination for PST Settings
In the previous chapter, it was stated that in the Netherlands and Belgium severalPSTs are being installed. It is obvious that coordination is needed in order to getthe best performance of these devices. Maximising the transfer capabilities of theinterconnectors between countries without constraining the security of the system isan important goal. In this chapter the influence of the PSTs on the so-called TotalTransfer Capacity (TTC) is investigated. This TTC is defined in section 3.1.
A multidimensional search space (TTC as a function of PST settings) can beexplored by Monte Carlo Simulation (MCS). This technique is able to extract a certainamount of information with a rather limited number of samples. The mathematicalbackground is discussed in section 3.2. The actual simulations and the correspondingresults are shown in section 3.3.
The MCS algorithm is adapted in section 3.4, resulting in the Multistage MonteCarlo Simulation (MMCS) method. The aim is to get a closer look at the region of thesearch space where the TTC is at its maximum. Clearly, this leads to optimisationmethods, discussed in later chapters. The sensitivity of the TTC to changes in PSTsettings is discussed in section 3.5.
Finally, the topic of path determination is discussed in section 3.6. A method is
36 Chapter 3
developed to determine a transition scheme between two sets of PST settings, avoidingunfavourable intermediate states.
3.1 Transfer capacities
3.1.1 Definition
ETSO (the organisation of European Transmission System Operators) provides pro-cedures on how to calculate transfer capacities between countries [31]. The maximumamount of power that can be transferred between countries A and B without violatingany security criterion is called the Total Transfer Capacity (TTC) between A and B.
However, the exact operational conditions can not be predicted on beforehandwith full accuracy. The information that a TSO receives (be it indications frommarket players or measurements from the past) and uses to predict future conditionsis mostly uncertain and hard to guarantee. Hence, a security margin is introduced:the Transmission Reliability Margin (TRM). The TRM is determined by the TSOsat the planning stage. It is mostly a fixed value, but it can be adapted according toseasonal variations or network configuration changes.
The transfer capacity that can be offered to the market is the Net Transfer Capac-ity (NTC). It is the maximum amount of power that can be transferred across a borderwithout violating any security constraints and taking into account the uncertainty inthe planning process:
NTC = TTC − TRM (3.1)
Hence, the NTC should be used as an optimality criterion. However, one of theproblems with the use of the NTC is the fact that its value depends on the TRM. ATSO of one country can decide to make the TRM 300 MW, but another TSO canuse a value of 500 MW. This is the reason why the TTC is more appropriate to maketransfer capacity calculations. Of course, in practice, the NTC should be used foroperation (as it is the capacity that is available for the market), but for optimisation,the TTC is a more objective parameter (as it does not depend on local conventions).
For the calculation of the TTC, a base case load flow is used, including a certainamount of base case exchange (BCE) between both areas considered. In order toincrease the power exchange between the areas, the generation in one country isincreased and decreased by the same amount in the other. The change in generationlevel is called the power shift (PS). The power shift is increased or decreased until asecurity constraint is violated. The maximum increase in generation in a given area isdesignated as PS+
max, the maximum decrease as PS−max. A graphical representationof the transfer capacities is shown in Fig. 3.1.
Search Space Exploration and Path Determination for PST Settings 37
TTC−
0MW
TRM+
NTC+
NTC−
TRM−
PS+max
BCE
PS−
max
TTC+
Figure 3.1: Transfer capacities according to ETSO.
3.1.2 Simulation-based calculation
The TTC is mostly determined through load flow calculations, by using a methodcalled the linear projection technique [85], as will be shown here. Suppose that thecalculations are performed for the import capacity of the Netherlands from Germany.As a first step, a PS is applied, i.e. the total generation level in Germany is increasedby a certain amount (typically 100 MW), and decreased by the same amount in theNetherlands. By doing so, an additional flow is created between the two countries. Forimport capacity studies, PS is negative for the system studied (i.e the Netherlands)and positive for the exporting system (i.e. Germany). For export capacity studies,the signs are reversed.
The PS can be obtained in different ways, depending on how it is distributedamongst the different generators in the area. A common method is to assign a fractionof the PS to every generator in the area, proportional to its rating. In special cases,it is also possible to select a limited number of generators to accomplish the PS.
After the PS has been performed, the active power flow on every interconnectoris calculated and compared to the pre-shift situation. The sensitivity sl of every lineflow (sometimes also referred to as the power transfer distribution factor or PTDF)is taken as this difference divided by the PS:
sl =∆PlPS
(3.2)
This sl is assumed to be constant for every value of the PS (assuming the sameapproach for obtaining the PS is maintained). In this way, the power on every linecan be expressed as a function of the power shift:
Pl = Pl,0 + sl · PS (3.3)
The line power should not exceed the rated value:
∀l : |Pl| ≤ Pl,r (3.4)
38 Chapter 3
line2
P
PS
−Prate
Prate
PS−
max PS+
max
line1 lin
e3
IMPORT EXPORT
Figure 3.2: Graphical representation of the linear projection technique.
From both equations, the maximum allowable PS (PS+max or PS−max, depending
on whether it is an import or export calculation) can be determined. This principle isa linear approximation of the problem and is shown in Fig 3.2 for three interconnectorswith the same rated power. For a positive PS, the upper limit of line 2 is the mostrestrictive constraint; in the negative direction, it is the upper limit of line 3.
The N secure TTC is obtained as the sum of the BCE and the maximum PS. Ifa contingency analysis is performed, the most limiting capacity is the N − 1 secureTTC.
3.2 MCS theoretical background
3.2.1 Monte Carlo fundamentals
Originally, Monte Carlo techniques were developed for the numerical calculation ofintegrals. The principle is easy to understand by considering the Hit-or-Miss MonteCarlo method [80].
Suppose an integral I has to be calculated:
I =∫ b
a
g(x)dx (3.5)
where the function g(x) and the variable x are bounded:
0 ≤ g(x) ≤ c a ≤ x ≤ b (3.6)
The rectangle Ω, as indicated in Fig. 3.3, is described by:
Ω = (x, y)|a ≤ x ≤ b, 0 ≤ y ≤ c (3.7)
Let (X,Y ) be a random vector uniformly distributed over Ω. The probability density
Search Space Exploration and Path Determination for PST Settings 39
MISS
HIT
x
Ω
S
a b
g(x)c
Figure 3.3: Graphical representation of the Hit-or-Miss Monte Carlo method.
function (PDF) is given by:
fXY (x, y) =
1
c(b−a) , if (x, y) ∈ Ω0, otherwise.
(3.8)
The probability p that (X,Y ) falls within the region S (area under g(x)) is:
p =area Sarea Ω
=I
c(b− a)(3.9)
The aim is to estimate the integral I, corresponding to the estimation of the areaunder the function g(x). Suppose that N independent points (x, y) are generated. IfNH represents the number of “hits”, designating points falling within the area S, theprobability p can be approximated by p:
p =NHN
(3.10)
From this, an estimator θ for the integral can be derived:
I ≈ θ = c(b− a)NHN
(3.11)
It can be proven that the standard deviation of the estimator is [80]:
σθ = N−12 (I[c(b− a)− I])
12 (3.12)
This means that the estimator converges to the integral I as the number of samplesincreases. This is a very important property of Monte Carlo methods: already acertain amount of information can be extracted with a limited amount of samples.
3.2.2 Monte Carlo Simulation
In a lot of applications, an output parameter of a process has to be studied as afunction of several input parameters. Sometimes, the equations that govern this
40 Chapter 3
process are absolutely unknown, and measurements are needed. It can also occurthat a simulation model is available, but not an analytically-closed equation thatdescribes the relation between output and inputs. In that case, multiple simulationruns can provide information.
In a Monte Carlo Simulation (MCS), the distributions of the input parametersare sampled numerous times. These distributions can have any shape (e.g. Weibulldistributions in case of wind speed [96] or negative exponential ones in case of failuretime of components [13]), and can be independent or correlated to a certain degree[36; 65]. The resulting histogram of the output is an estimator of its PDF, becomingmore accurate as the number of samples increases. This approach is mathematicallyclosely related to Monte Carlo Integration: for an output variable X, the numberof output samples below a certain threshold X ′ is counted (hits) and expressed as arelative number in the interval [0, 1]. If this is done for multiple values of X ′, theirrelative number of hits constitute an estimator for the cumulative density function(CDF) of the output variable X [68]. The histogram obtained converges to the PDF(or CDF, depending on the representation) as the number of samples increases.
3.3 Exploration using MCS
3.3.1 Search space
As a PST changes the flows in the entire system, the TTC has a different value forevery combination of PST settings [113]. If only one device is installed, the TTC canbe calculated for all possible settings and the optimum can be identified easily. Ifa second PST is added, the problem becomes two-dimensional, and the calculationtime increases dramatically if all possibilities are to be checked. In general, a searchvolume SV can be defined, containing all possible values of the input variables:
SV =d∏i=1
(maxi −mini) (3.13)
with d the number of input variables or the dimension of the problem and mini andmaxi the minimum and maximum values for variable i.
Clearly, for problems with high dimensionality, the approach of enumerating allpossible combinations becomes unrealistic. This dimensionality problem, and the factthat the TTC is calculated by simulations and no exact analytical model is available,makes MCS an attractive technique in this framework. The aim is to explore andoptimise the TTC as a function of PST settings. Therefore, the distributions of thedifferent PST settings are taken as uniform between minimum and maximum values.In this way, the search volume is covered in a uniform way, as illustrated for thetwo-dimensional case in Fig. 3.4.
The output of the simulation is a histogram of the TTC, converging towards thePDF as the number of samples increases. From this histogram, an estimate can be
Search Space Exploration and Path Determination for PST Settings 41
Figure 3.4: Search area for two variables with MCS.
network element numberbusses 1272
generators 289loads 980lines 2689
transformers 640
Table 3.1: Network data of the simulation model.
made regarding the best- and worst-case TTC to be expected. Also, the shape of thehistogram can yield valuable information.
3.3.2 Simulation setup
The network model used for TTC calculation consists of a snapshot of the situationof the grid of the Netherlands, Belgium, and the neighbouring countries on the 19th
of January 2000 at 10h30, representing a typical state of the grid. A short summaryof the network data is listed in Table 3.1. The Belgian PSTs of Zandvliet, Van Eyck1 and 2, and Monceau are added to the network model.
For the load flow calculations, PSS/E is used [85]. Because of the high numberof calculations required, some form of automation is needed. This is realised using aPython script, enabling automated execution of PSS/E commands [1]. A major partof the script in Python is the random number generator used to sample the uniforminput distributions of the PST settings (i.e. non-discrete real numbers). In everysample, the random settings are applied to the PSTs and a load flow calculation isperformed. As the subsequent sets of settings are uncorrelated, the resulting situation
42 Chapter 3
of the grid can be very different from the previous one, possibly resulting in a divergingAC load flow. Therefore, a DC load flow is performed first in order to initialise thesystem for the subsequent AC load flow. Next, the TTC calculation by means of thelinear projection technique is performed as described in section 3.1.2. This calculationis implemented as a separate module in PSS/E. Finally, the results are written to atext file, for post-processing in Matlab. The pseudo-code of the program can be foundin Algorithm 3.1.
Algorithm 3.1 Pseudo-code for MCS of TTC as a function of PST settings.Initialise()while sample < max samples dosettings ←GenerateRandomSamples(limits)networkmodel ←ApplySettings(settings)DCLoadFlow(networkmodel)ACLoadFlow(networkmodel)sens ←CalculateSensitivities(networkmodel)ttc ←CalculateTTC(sens,networkmodel)outputfile ←WriteResults(ttc)
end whilePostProcessing(outputfile)
3.3.3 Relation between TTC and the Meeden PSTs
In order to demonstrate the influence of a PST on the TTC, the import TTC ofthe Netherlands from Germany and Belgium is calculated for different settings ofthe Meeden PSTs. The other PSTs in the region are set to 0, while the setting ofthe Meeden PSTs varies from -30 to +30. The results are shown in Fig. 3.5. Byconvention, import capacities are shown as negative values.
Clearly, the TTC is a piecewise linear function. Every linear part is characterisedby a “limiting element - contingency” pair. The limiting element is the line that isoverloaded first, which occurs when the specified outage takes place.
In Fig. 3.5a, it can be seen that a large negative setting draws power to the Meeden-Diele interconnectors, making them the limiting elements. A high positive settingpushes power to the Hengelo-Gronau interconnectors. The optimum, a maximumimport TTC, occurs at a moderate negative setting of around −11.
Fig. 3.5b shows that the import TTC from Belgium is not influenced much by theMeeden PSTs in a wide range of settings. However, an extreme negative setting makesthe Meeden-Diele interconnectors the limiting elements, as is the case for the TTCfrom Germany. An extreme positive setting pushes the power to the Hengelo-Gronauinterconnectors.
Search Space Exploration and Path Determination for PST Settings 43
−30 −20 −10 0 10 20 30−7000
−6500
−6000
−5500
−5000
−4500
−4000
−3500
−3000
−2500
PST setting [degrees]
impo
rt T
TC
[MW
]
lim: HGL − Gro 1con: Mee − Diele 2lim: Mee − Diele 1
con: MBT − Rokilim: MBT − Sier
con: HGL − Gro 2
(a) Netherlands from Germany
−30 −20 −10 0 10 20 30−6000
−5500
−5000
−4500
−4000
−3500
−3000
−2500
−2000
PST setting [degrees]
impo
rt T
TC
[MW
]
lim: MBT − Lixhecon: GT − ZV
con: Mee − Diele 2lim: Mee − Diele 1
con: HGL − Gro 2lim: HGL − Gro 1
(b) Netherlands from Belgium
Figure 3.5: Import TTC of the Netherlands from Germany (a) and Belgium (b). Thelimiting line is designated as “lim” and the outaged line as “con”.
44 Chapter 3
-8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 00
10
20
30
40
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80
import TTC [MW]
fre
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cy [-]
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import TTC [MW]
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-8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 00
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import TTC [MW]
fre
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cy [-]
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-8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 00
500
1000
1500
2000
2500
import TTC [MW]
fre
qu
en
cy [-]
(d) 50000 samples
Figure 3.6: Histograms of import TTC of the Netherlands and Belgium from Germanyand France resulting from an MCS with different amounts of samples.
3.3.4 MCS of the TTC with all PSTs
An MCS is performed on the study system that was discussed in 3.3.2, and differentamounts of samples are used. For each sample, the import TTC of the Netherlandsand Belgium from Germany and France is calculated. All the interconnectors onthe Dutch-German, Dutch-Belgian and Belgian-French borders are monitored andtaken into consideration as contingencies. No other outages are considered for thecontingency analysis for simplicity reasons.
The required calculation time for the complete MCS (50000 samples) is about 8hours and 12 minutes on a 3.2 GHz Intel P4. The resulting histograms are shown inFig. 3.6. As mentioned in the previous section, import TTC values are negative byconvention.
Search Space Exploration and Path Determination for PST Settings 45
Mee Gro Zv VE1 VE2 Monbest -21.81 -10.92 -13.01 8.02 23.98 -4.73
worst 15.56 6.49 24.04 24.43 23.85 -10.40
Table 3.2: PST settings [] for the best and worst-case situations.
By studying these graphs, the basic property of an MCS is evident: as the amountof samples increases, the resulting histogram becomes a more accurate estimate ofthe PDF of the output parameter, i.e. the TTC in this case.
The x-axis is cut off at 0 MW in order to show the best-case region clearly, but thehistograms actually have a long tail into the positive TTC area. This means that forsome combinations of PST settings, no power can be imported in the Netherlands andBelgium due to overloading problems. These problems can only be solved by exportingpower to Germany and/or France, resulting in positive values for the import TTC.
HGL
Mee
Mee
Diele
HGL
Gro
MBTBSL GT
Sier
Roki
ZV VE1 VE2
AVM Jam Ach Aub
AVN Chooz Lon Moul
206
205
398
398
1201
1064
194 1857 1273 346
850 15 24 152
NL
B
D
F
Figure 3.7: Border flows [MW] for the worst-case situation (worst import TTC).
Fig. 3.7 shows the flows on the interconnectors in the worst-case situation (basecase with the worst import TTC), corresponding to the PST settings on the sec-ond row of Table 3.2. The interconnector Zandvliet-Geertruidenberg is already over-loaded, even before any contingency occurs. Due to poor coordination of the Van Eyck
46 Chapter 3
1 and Zandvliet PSTs, an internal loopflow is created (Zandvliet-Geertruidenberg-Eindhoven-Maasbracht-Van Eyck 1-Zandvliet), creating severe congestion problems.It can also be seen that the flows on the Dutch-German border are not distributedequally, indicating an inefficient use of the PSTs (more in particular, of the Mee-den PSTs). In fact, a loopflow is induced from north to south, through Germany.Furthermore, an uneven loading occurs at the interconnectors on the Belgian-Frenchborder.
HGL
Mee
Mee
Diele
HGL
Gro
MBTBSL GT
Sier
Roki
ZV VE1 VE2
AVM Jam Ach Aub
AVN Chooz Lon Moul
679
676
477
477
646
441
122 39 12 222
267 22 202 54
NL
B
D
F
Figure 3.8: Border flows [MW] for the best-case situation (best import TTC).
The flows in the best-case situation (base case with the best import TTC) arepresented in Fig. 3.8, corresponding to the PST settings shown on the first row of Ta-ble 3.2. The flows on the Dutch-German and Belgian-French borders are distributedmore evenly, and no loopflow is created at the Dutch-Belgian border. Clearly, this isa much more efficient use of the PSTs.
3.4 Multistage Monte Carlo Simulation
The MCS approach as presented in the previous sections is very useful to obtain anestimate of the boundaries of the TTC. However, as an optimisation tool it is notvery well suited, as the information on the optimal region of the histogram is not
Search Space Exploration and Path Determination for PST Settings 47
sufficiently detailed. In a Multistage Monte Carlo Simulation (MMCS) approach,several subsequent MCS runs are performed in order to zoom in on the optimalregion [18]. In each of these runs, fewer samples are used in comparison to a full-scaleMCS (e.g. 10000 for each run compared to 50000 for a full MCS). In the first stage,an MCS is performed, and the results are sorted from the best to the worst TTCvalues. A minimal volume that contains the first n samples of this sorted list is takenas the search volume (SV) for the next MCS, designated as the first zoomlevel (ZL).The new SV is smaller than the previous one and covers the optimal region. It isdefined by the minimum and maximum values of the variables in the best n samplesof the previous MCS:
SVz =d∏i=1
(maxi,z −mini,z) (3.14)
where z is the zoomlevel.This procedure of zooming in is repeated until the required accuracy is reached.
An obvious disadvantage of the method is the calculation time, that increases linearlywith the number of zoomlevels.
A schematic representation of the method is shown in Fig. 3.9 for the two-dimensional case. Since the number of samples is kept constant in every zoomlevel,the sample density increases as the SV decreases.
Figure 3.9: Search area for two variables with MMCS.
The MMCS procedure is performed on the simulation model described in 3.3.2.For each zoomlevel, a 10000 sample MCS is performed, and every time the best 20samples determine the next SV. The simulation stops after zoomlevel 5, i.e the totalcalculation time is 6 times longer than a 10000 sample MCS or 1.2 times longer thana 50000 sample MCS as performed in section 3.3.4. This means that about 10 hoursare required for the computation.
48 Chapter 3
-8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 00
50
100
150
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250
Import TTC [MW]
fre
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-8000 -7500 -7000 -6500 -6000 -5500 -50000
100
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import TTC [MW]
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ency
[-]
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-8000 -7500 -7000 -65000
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-8000 -7800 -7600 -7400 -7200 -70000
50
100
150
200
250
300
350
400
450
500
import TTC [MW]
frequ
ency
[-]
(d) zoomlevel 5
Figure 3.10: Histograms of import TTC of the Netherlands and Belgium from Ger-many and France resulting from an MMCS at different zoomlevels.
The resulting histogram for the base MCS and zoomlevels 1, 3 and 5 are shownin Fig. 3.101. It is important to note the different x-axis scales. While the baseMCS only indicates a maximum TTC of about 7500 MW, zoomlevel 5 shows that theoptimum is actually around 7700 MW. The exact values of the maximum TTC withthe corresponding PST settings for all the zoomlevels are listed in Table 3.3.
3.5 Sensitivity analysis
If an optimal set of PST settings is found through an arbitrary optimisation algorithm,the question remains how sensitive the TTC is to the setting of every single PST. If,
1Fig. 3.10b should be the same as Fig. 3.6b. However, the frequency axis looks different as adifferent amount of frequency bins is chosen for both plots.
Search Space Exploration and Path Determination for PST Settings 49
TTC Mee Gro Zv VE1 VE2 Monbase -7512 -20.47 -11.83 -22.97 2.04 23.15 -0.66ZL1 -7566 -20.14 -11.64 -23.36 1.52 24.54 -3.80ZL2 -7646 -21.33 -11.41 -20.5 8.52 22.08 10.01ZL3 -7689 -21.68 -11.66 -22.76 3.68 24.64 8.27ZL4 -7698 -21.78 -11.66 -20.15 6.29 23.90 10.23ZL5 -7714 -21.63 -11.62 -22.33 3.13 24.67 10.23
Table 3.3: Import TTC [MW] and PST settings [] for every zoomlevel of the MMCSalgorithm.
for some reason, the exact obtained optimal settings can not be applied in practice(for example due to the limited number of tap changer positions), this informationcan be vital.
As optimal settings, the values found in section 3.4 are used. The setting of onePST is varied by steps of 1, and the resulting TTC is monitored. This is performedfor all six PSTs, resulting in six separate graphs, shown in Fig. 3.11.
What stands out from those graphs is that the sensitivity values are quite differentwhen looking at various devices. For the Meeden and Gronau PSTs, it is very impor-tant to apply the optimal setting as precisely as possible, as the sensitivity is high.The Zandvliet and Van Eyck PSTs show moderate sensitivity values. The sensitivityof the TTC to the setting of the Monceau PST is low.
It can also be seen from the graphs that for some PSTs the optimal TTC is notreached at the settings that were computed as being “optimal”. This means that theMMCS method used to obtain these values did not fully converge to the optimum.Clearly, an optimisation algorithm is needed to accomplish full convergence. Thistopic is discussed in the next chapter.
3.6 Path determination
3.6.1 Problem formulation
After having determined the optimal or near-optimal phase shifter settings, the ques-tion is how to go from the current settings to this point [111]. For technical reasons,this is a stepwise process taking some time due to the limited speed of the mechanicaltap changer of the PSTs. A strategy could be to set one PST to its optimal position,then the second one, and so on. However, it is possible that some of the intermediatestates are (very) unfavourable for the power system, for instance for system security.To illustrate this problem, four different transition sequences are applied to the modeldescribed in section 3.3.2, starting from a random initial configuration. The settingsfound in the MMCS in section 3.4 are used as the target settings (rounded to thenext integer). The transition strategy is to switch one PST one degree at a time to
50 Chapter 3
-25 -24 -23 -22 -21 -20 -19 -18-7800
-7600
-7400
-7200
-7000
-6800
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PST setting [degrees]
impo
rt TT
C [M
W]
360 MW/deg
51 MW/deg
(a) Meeden
-12 -11.5 -11 -10.5 -10 -9.5 -9 -8.5-7800
-7700
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PST setting [degrees]im
port
TTC
[MW
]
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(b) Gronau
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PST setting [degrees]
impo
rt TT
C [M
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31 MW/deg
(c) Zandvliet
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-7720
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rt TT
C [M
W]
37 MW/deg
18 MW/deg
5 MW/deg
-29 MW/deg
(d) Van Eyck 1
21.5 22 22.5 23 23.5 24 24.5 25-7740
-7720
-7700
-7680
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PST setting [degrees]
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rt TT
C [M
W]
-67 MW/deg
20 MW/deg
(e) Van Eyck 2
7 7.5 8 8.5 9 9.5 10 10.5-7730
-7728
-7726
-7724
-7722
-7720
-7718
-7716
-7714
-7712
PST setting [degrees]
impo
rt TT
C [M
W] 5 MW/degree
(f) Monceau
Figure 3.11: Sensitivities of the import TTC to changes in PST settings (vertical lineindicates the base setting).
Search Space Exploration and Path Determination for PST Settings 51
Meeden Gronau Zandvliet Van Eyck 1 Van Eyck 2 Monceauseq. 1 1 2 3 4 5 6seq. 2 6 1 2 3 4 5seq. 3 4 5 6 1 2 3seq. 4 3 4 5 6 1 2
Table 3.4: Order of switching for the four examples.
its target setting, followed by the next PST, and so on. The order in which the PSTsare set to their final setting is shown in Table 3.4. Fig. 3.12 shows the evolution ofthe import TTC for the different transition sequences. Clearly, sequence 2 should beavoided at all times, as the import TTC becomes even positive for some intermediatestates, indicating very unfavourable conditions.
0 20 40 60 80 100 120 140-8000
-6000
-4000
-2000
0
2000
4000
switching action [-]
TTC
[MW
]
sequence 1sequence 2sequence 3sequence 4
Figure 3.12: Evolution of the TTC for the different transition sequences.
The problem can be described as a directed graph D = (V,A) with vertices V andarcs A [82]. Each vertex vi is represented by a state vector:
xi = [x1i , x
2i , . . . , x
ni ]T (3.15)
where x1...ni are the settings of the n phase shifters in state i. For the PST settings,
only integer numbers (degrees) are used.
52 Chapter 3
An arc is represented by aij , designating that it goes from vertex vi to vj . Thegraph is constructed in such a way that vertices which differ by only 1 in exactlyone phase shifter setting are connected by an arc:
aij ∈ A ⇐⇒ ∃!xkj ∈ xj : xkj = xki ± 1 k = 1 . . . n (3.16)
This is of course only valid for phase shifter settings within the limits of the devices.Every arc aij has an associated cost cij . This cost can for example be defined as:
cij = TTCj − TTCi + w w ≥ 0 (3.17)
where the best TTC values are the most negative ones and where w is a penalty factorthat is strictly positive when the TTC deteriorates from vertex i to vertex j. Thevalue of this parameter can be tuned depending on how severe a TTC deteriorationshould be penalised. In most cases, the cost of an arc is a negative figure. The totalcost, calculated as the sum of all costs of the arcs constituting the path, must beminimised. The problem of avoiding unfavourable transitional states reduces to thedetermination of the shortest path in the graph described above, as these unfavourablestates are heavily penalised.
3.6.2 Path strategies
Shortest path algorithms
Shortest path problems are very well-known in optimisation theory [17]. The mostbasic method of solving them is Dijkstra’s algorithm [28]. The computation time ofthis method is O(|V |2) (see appendix B for more information on this notation). Inthe original version of this method, no negative arc lengths are allowed, making it un-suitable for the particular problem discussed here. The Bellman-Ford algorithm [10]is designed to deal with negative arc lengths, with a computation time of O(|V ||A|).Next to both classic methods, a whole array of other algorithms has been developed,but the aim here is not to give an extensive overview. Even with modern methods,the calculation time can become very large, as the graph considered here is immense:there are over 5 billion vertices and even more arcs.
Greedy algorithms
The requirement of the shortest path can be relaxed to a requirement of a good path.This enables the use of a greedy algorithm [20]. This method only looks in the directneighbourhood of the current state and picks the apparent best solution at that time.For the implementation of the so-called Simple Greedy Algorithm (SGA) for theproblem considered here, a few conventions are made.
- The cost of an arc is equal to the difference in TTC between both vertices itconnects. No extra penalty factor is used.
Search Space Exploration and Path Determination for PST Settings 53
- Only arcs that result in a decreasing distance to the target vertex (in termsof the Manhattan distance2 from the current state vector to the target statevector) can be selected for the path. For example, if the target for 1 PST settingis 20, and the current setting for that PST is 10, only the setting of 11 canbe accepted, and not the arc resulting in a setting of 9. Consequently, onlyhalf of the arcs leaving a vertex are available for selection.
- From the candidate arcs, the one that results in the biggest improvement isselected for the path.
This proposed algorithm has the advantage of simplicity and limited calculationtime, but it results in excessive switching between different PSTs, which is undesirablein practice. This problem can be tackled by using the Penalised Greedy Algorithm(PGA). In this approach a penalty factor k is added to the cost function:
cij = TTCj − TTCi + λ · k (3.18)
with:
λ ∈ 0, 1 k ≥ 0 (3.19)
where the best TTC values are the most negative ones by convention. The cost of anarc is increased by a constant k if it leads to a change in a setting of another PSTthan the current one. This makes the control curves more smooth, but allows for asmall deterioration in TTC. If the penalty factor is not too large, this should not bea problem.
3.6.3 Simulations
For the simulations, the model described in 3.3.2 is used once again. For testingpurposes, 10 sets of random starting combinations of PST settings are generated,and the greedy algorithms are tested for each set (with the same target combinationof PST settings). The PGA method is tested with penalty factors of 100, 200, 300and 400 MW.
Fig. 3.13 shows the evolution of the TTC and the PST settings when the SGA isapplied in one of the example cases. The TTC improves in a monotonic way. However,the PST that alters its tap position is continuously changing. This behaviour is alsoobserved for all the other sets of random starting points.
Fig. 3.14a and 3.14b present the results of a PGA calculation with a penalty factorof 100 MW. Clearly, a large improvement is established with regard to the “random”switching of the various PSTs, but the TTC does no longer improve in a monotonic
2In an n-dimensional space, the Manhattan distance between two points (x11, x2
1, . . . , xn1 ) and
(x12, x2
2, . . . , xn2 ) equals |x1
1 − x12|+ |x2
1 − x22|+ . . . + |xn
1 − xn2 |
54 Chapter 3
0 20 40 60 80 100 120-8000
-7500
-7000
-6500
-6000
-5500
-5000
Switching action [-]
TTC
[MW
]
(a) TTC
0 20 40 60 80 100 120-30
-20
-10
0
10
20
30
Switching action [-]
PS
T se
tting
[deg
rees
]
MeedenGronauZandvlietVan Eyck 1Van Eyck 2Monceau
(b) Settings
Figure 3.13: Evolution of the TTC and the PST settings for the Simple GreedyAlgorithm.
way. However, this non-monotonic behaviour is very limited, and it does not poseany problems.
Fig. 3.14c and 3.14d show the results of the PGA calculation with a penalty factorof 400 MW. The temporary deteriorations of the TTC become more pronounced, butthe switching between PSTs reduces even further.
The simulations show that both SGA and PGA are useful, but the choice betweenboth methods depends on the requirements of the user.
3.7 Summary
In this chapter, a first step towards PST coordination is taken. The goal is to max-imise the Total Transfer Capacity, an available transfer capacity indicator. The searchspace is a multidimensional function that connects the TTC to the different PST set-tings. In the specific case of the Netherlands and Belgium, this is a six-dimensionalfunction.
Monte Carlo Simulation (MCS) can serve as a tool to explore the search space,provided that the PST settings are taken as uniform distributions in order to coverthe whole space uniformly. The higher the amount of samples, the higher the accu-racy, but also the longer the calculation time needed. The application of MCS forcalculating the TTC of the Dutch-Belgian system gives, amongst others, insight ina worst- and a best-case scenario. Further study of these scenarios shows that poorcoordination can result in internal loopflows, causing severe overloading. Alterna-tively, well-coordinated PSTs avoid such loopflows and establish an even loading ofthe interconnectors at the borders.
In order to gain more information on the optimal region of the search space in
Search Space Exploration and Path Determination for PST Settings 55
0 20 40 60 80 100 120-8000
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]
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30
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ST
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ng [d
egre
es]
MeedenGronauZandvlietVan Eyck 1Van Eyck 2Monceau
(b) Settings penalty 100 MW
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]
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0 20 40 60 80 100 120-30
-20
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T se
tting
[deg
rees
]
MeedenGronauZandvlietVan Eyck 1Van Eyck 2Monceau
(d) Settings penalty 400 MW
Figure 3.14: Evolution of the TTC and the PST settings for the Penalised GreedyAlgorithm with a penalty factor of 100 (a and b) and 400 MW (c and d).
less computation time, Multistage Monte Carlo Simulation (MMCS) is adopted. Thismethod consists of performing several subsequent MCS runs in order to zoom in onthe optimal region. The sensitivity of the TTC around this optimal point differs forevery PST, but proves to be high for the Meeden and Gronau PSTs and low for theMonceau PST.
Finally, the problem of finding a sound transition path between two sets of PSTsettings is addressed. The problem can be redefined as a shortest path problem.However, classical solution algorithms show unacceptable calculation times. Greedyalgorithms are adopted and yield a good solution. Two variants are developed: SimpleGreedy Algorithm and Penalised Greedy Algorithm. The first one is a very simpleimplementation, but causes excessive switching between different PSTs. The latteruses a penalty function to eliminate this problem.
CHAPTER 4
Metaheuristic Optimisation Methods
In the previous chapter, the search space of the TTC as a function of the PST settingswas explored with MCS and a first attempt to optimise the TTC was performed byapplying an MMCS. In this chapter, a special group of optimisation algorithms isdiscussed and applied, being metaheuristics.
In section 4.1, an introduction to this kind of methods is given. Their character-istics are discussed, including advantages and drawbacks. A general classification isgiven in section 4.2, and some algorithms are discussed in depth, as a full descriptionof all existing methods is beyond the scope of this work.
In section 4.3, two population-based metaheuristic algorithms (Evolutionary Com-putation methods in particular) are applied to the TTC optimisation problem intro-duced in the previous chapter. An important aspect is to find suitable algorithmparameters for this particular problem, since this has a major impact on convergenceand computational effort. Section 4.4 introduces Particle Swarm Optimisation andapplies it to the TTC optimisation problem. Again, parameter tuning is found to bethe key issue.
It is expected that the TTC is highly influenced by transit flows. This effect andthe consequences for the optimal settings of the PSTs is studied in section 4.5. Finally,the system losses are included in the objective function, resulting in a multiobjectiveoptimisation formulation in section 4.6. The mathematical principles are discussed
58 Chapter 4
and the results of the optimisation are shown.
4.1 Introduction to metaheuristics
4.1.1 General description
Heuristics and metaheuristics are both members of the group of Black-Box Optimisa-tion Algorithms. The main characteristic of this family of algorithms is that they relyonly on repeated evaluation of the objective function. No mathematical operationslike for example derivation are performed. This is a major advantage for problemswhere the objective function is very complex, and where a mathematical manipulationwould require an unacceptable calculation effort. Furthermore, in some problems, ananalytical expression for the objective function is not available and only a simulationmodel can be used. In these cases, it is possible to perform evaluations through simu-lation, and solve the problem with Black-Box Optimisation. Monte Carlo Simulationcan also be regarded as a black-box technique, but it is not a metaheuristic as it isin fact not an optimisation algorithm.
A heuristic1 is an iterative technique that tries to find a good solution at a rea-sonable computational effort [114]. The result is in most cases not optimal, and thereis even no guarantee that the solution does not become arbitrarily bad.
Heuristics are simple techniques used to make decisions in order to find a solution.For instance, assume you are packing odd-shaped items into the trunk of your car.Finding a perfect solution is a hard problem - there is essentially no way to do itwithout trying every possible way of packing them. What most people do then, is“put the largest items in first, then fit the smaller items into the spaces left aroundthem”. This does not necessarily give you the perfect packing, but it is usually prettygood. This is a simple example of a heuristic solution [2].
Clearly, heuristics are not very well suited as optimisation algorithms, as theytend to converge to suboptimal solutions. Metaheuristics have a two-layer structure.The bottom layer is a heuristic that performs a local search, possibly resulting in asuboptimum. The top layer is a strategy that guides the heuristic in order to cometo the global optimum. It is a form of an intelligent search.
4.1.2 Advocates and sceptics
The use of metaheuristics is a point of intense discussion between advocates andsceptics. Although the author is not in the position to judge them, it is interestingto look at the arguments from both sides.
A first point of criticism is that some researchers use metaheuristics because theydo not require further knowledge of the internal mechanisms of the system studied. Noanalytical relation between decision variables and variables to be optimised is needed,
1The word heuristic comes from the greek heuristikein: to find, to discover
Metaheuristic Optimisation Methods 59
since the simulation model provides all necessary function evaluations. Furthermore,all kinds of exotic variants of algorithms are devised, often inspired by processes foundin nature. However, a badly-tuned sophisticated metaheuristic is heavily surpassedby a well-tuned simple one. Hence, the application of adequate method parameters isa very important aspect that must be studied individually for every class of problems.
Finally, critics point out that an arbitrary black-box function can not be optimisedefficiently by any metaheuristic, since for any metaheuristic M one can easily builda function f forcing M to enumerate the whole search space (or worse) [3]. Theso-called No-free-lunch theorem proves that every optimisation algorithm performsequally well on average as any other [118]. This means that some perform very wellfor one particular problem, but very poor for all other problems, and that somealgorithms show an average performance for all problems.
Although some of these criticisms are undeniably correct, metaheuristics havebeen applied multiple times with great success. A nice example is the design of anantenna for a NASA mission [55]. The result was a structure “that expert antennadesigners would likely not produce”, showing outstanding performances.
4.2 Classification
Metaheuristics can be classified in a number of ways, but most commonly a distinctionis made between methods based on a unique solution and those based on a populationof solutions. Algorithms of the former group only work with one point in the searchspace. This single solution is improved by moving in a certain direction in the searchspace. Methods based on a population work with a group of solutions, providing moreinformation on the search space, but at the expense of increased calculation time.
An exhaustive description of all metaheuristics is beyond the scope of this thesis.Instead, a few methods of both classes mentioned above are presented.
4.2.1 Metaheuristics based on a unique solution
Simulated Annealing
In metallurgy, annealing is the process of cooling a material in a very slow way inorder to end up in the lowest energy state. All the crystal defects are removed in thisway and metastable states are avoided.
An optimisation (minimisation) algorithm called Simulated Annealing (SA) wasinvented independently by Kirkpatrick (1983) and Cerny (1985). If a solution s isused as a starting point, a next solution candidate s′ is selected in the neighbourhoodof s. The way this next candidate is obtained is defined by the user and depends onthe problem considered. The equivalent to the energy in the metallurgic process isthe value of the objective function f . If f(s′) < f(s), the new solution is accepted.In the classic algorithm, the chance of acceptance in this case is 1, but in some more
60 Chapter 4
modern variants, a smaller probability is used. If f(s′) > f(s), the new solution isaccepted with a probability p, described by:
p = e−f(s′)−f(s)
T (4.1)
with T the current temperature of the process. This temperature decreases slowlywith each step of the algorithm, i.e. the chance of acceptance of a higher energy statebecomes lower in time. The pace at which the temperature decreases in time is calledthe annealing schedule and is defined by the user.
In the beginning of the optimisation, the chance that a worse solution is acceptedis substantial. In this part of the process, the search area is explored and local optimaare avoided. Further in the process, worse solutions are unlikely to be accepted andthe focus is on improvement. If T = 0, the method reduces to a greedy algorithm,which means that only better solutions are accepted.
The advantage of this method is that it avoids local optima and it is easy toimplement. The disadvantages are the relatively slow speed (adiabatic cooling isslow by definition) and the fact that convergence to the unique optimal solution onlyhappens when time becomes infinite.
Tabu Search
The concept of Tabu Search (TS) was proposed by Fred Glover in 1986 [41]. Sinceits introduction, the algorithm has been used in many applications, although there isno formal explanation for the good performance of the method.
TS is a neighbourhood search method, i.e. the algorithm looks for the next solutionin the direct neighbourhood of the current one. In a classic TS, every neighbour isevaluated and the best solution is selected as the next one, even when this results ina deterioration.
A characteristic feature of TS is the use of memory. In order to avoid getting stuckin loops, moves that were made in the recent past are stored. Moves that reverse theeffect of recent moves are declared tabu, meaning that they are not accepted. Inthis way, looping behaviour is avoided and the possibility of convergence to a localoptimum is reduced.
In some cases, tabus can be too defensive and obstruct convergence to the bestsolution. This is the reason why aspiration criteria are introduced. If a move is tabu,but would result in a very good solution if allowed, the tabu is overruled.
Iterated Local Search
Iterated Local Search (ILS) is a search algorithm, not just searching in neighbouringsolutions, but also further away, in other local optima. If the current local optimalsolution is s∗, a perturbation is applied to reach the intermediate solution s′. Startingfrom this state, a local search using some heuristic is performed. The local optimum
Metaheuristic Optimisation Methods 61
is s′∗. If this optimum is better than the original s∗, it is accepted; otherwise, it isrejected. The principle of this method is illustrated in Fig. 4.1.
cost
solution space
s′
s∗
s′∗
Figure 4.1: Schematic illustration of the ILS principle.
The magnitude of the perturbation is determined by the user, and it is key forthe efficiency of the algorithm. A too small perturbation results in the unability toescape from local optima. An excessively large perturbation decreases the advantageof the method, reducing it to a random-restart heuristic method (which is a heuristicmethod that is executed several times with random initial values).
GRASP
The Greedy Randomised Adaptive Search Procedure (GRASP), is a method consistingof two steps. Assuming that every solution consists of several elements, for examplesubsequent direct routes between cities in the travelling salesman problem (TSP)2,a solution is constructed by adding elements one by one. The list of candidates forthe next element is called the restricted candidate list (RCL). In this list, the bestα-fraction (i.e. the (1−α)×100 percentile) of all possible candidates are summed up,ranked by incremental cost. From this list, a random candidate is picked and appliedas the next element.
The choice of α is very important in this algorithm. In the case where α approaches0, the method reduces to a fully deterministic approach. For α = 1, the methodbecomes completely random [76].
The construction phase does not guarantee an optimal solution. That is why thesolution-improvement phase is introduced: a local search algorithm starting at theconstructed solution.
2In a TSP, a number of cities is given as well as the cost of travelling between each of them. Thegoal is to find the cheapest route that visits each city exactly once and then returns to the startingcity.
62 Chapter 4
4.2.2 Metaheuristics based on a population of solutions
Evolutionary Algorithms
Evolutionary Algorithms or Evolutionary Computation are generic terms for population-based metaheuristic optimisation algorithms inspired by the biological mechanism ofevolution [117]. Three groups can be distinguished:
- Genetic Algorithms
- Evolutionary Programming
- Evolution Strategies
In a Genetic Algorithm (GA), every solution or individual is represented by alist of parameters, also referred to as a chromosome [91]. In the classical approach,the genes constituting this chromosome are binary values. If the parameters are notbinary, they should be mapped onto binary representations. The GA consists of threesteps.
- Selection
- Crossover or recombination
- Mutation
In the selection procedure, an evaluation of the fitness of all the individuals takesplace, followed by a ranking according to fitness. Then, pairs of individuals are se-lected for the next step, according to a certain algorithm. The selection algorithmused can vary. Two popular ones are roulette wheel selection and tournament selec-tion. In the roulette wheel approach, the fitness values of all individuals are summed:
ft =N∑i=1
fi (4.2)
Each individual is assigned a segment on the roulette wheel:
θi =2πfift
(4.3)
A random number in the interval [0, 2π[ is generated in order to select an individ-ual. In the tournament selection procedure, n individuals are picked and the best isselected.
The recombination step is carried out with a certain crossover probability (mostlybetween 0.6 and 1). Pairs of parent individuals are combined in order to generatechild individuals. There are several ways in which recombination can be performed.
Metaheuristic Optimisation Methods 63
- One point: The chromosomes of the parents are switched starting from a certaincrossover point. This point is a randomly-generated position.
- Two point: The chromosomes of the parents are switched between two crossoverpoints, which are randomly generated.
- Uniform crossover: Corresponding bits in the parent chromosomes are swappedwith a fixed probability (typically 0.5).
- Half-uniform crossover: Exactly half of the non-matching bits are swapped.
In the mutation step, a gene is altered with a small probability. In this way, newcharacteristics can be introduced in the population.
In Evolutionary Programming (EP), the chromosome is not limited to a binaryrepresentation, which makes it suitable for a wider range of problems. For the gen-eration of offspring, only a mutation mechanism is applied: each parent is copiedinto a child, which is then changed according to a distribution of mutation types,ranging from minor to extreme. This method implies that no couples of individualsare formed: a child is created by a single parent. The selection process is performedon the merged population of parents and children, typically by tournament selection.
The technique of Evolution Strategies (ES) is quite similar to EP, but there aretwo major differences.
- The selection procedure is performed in another way. In ES, this procedure isfully deterministic.
- In EP, only one child is produced per parent. There are basic variants of ESwhere this is also the case, but mostly there are several children per parent.
A more thorough theoretical explanation on EP and ES is given in section 4.3.
Particle Swarm Optimisation
Particle Swarm Optimisation was proposed in 1995 by James Kennedy and RusselEberhart [49]. It is a method based on the behaviour of a swarm of animals (forexample birds), which work together in order to obtain a favourable situation (findfood, protection,...). Each member of the population looks for the optimum andcommunicates its personal optimum to the rest of the swarm. More details aboutthis method are given in section 4.4.
Ant Colony Optimisation
The Ant Colony Optimisation (ACO) algorithm [29] is inspired by the foraging be-haviour of ants. If an ant walks from the nest to a food source, it leaves behind a trailof pheromone. The rate at which pheromone is deposited is higher for the shorter
64 Chapter 4
paths, because in a fixed time, more ants will pass through this path than throughthe longer one.
Furthermore, ants are more likely to choose the path with the highest pheromoneconcentration than any other path. In this way, the pheromone concentration in-creases even more, in a process called positive feedback.
The probability that an ant chooses a path (i, j) from node i to node j is:
p(i,j)k =
[τij ]
α·[ηij ]β∑l∈Nl
[τil]α·[ηil]β j ∈ Ni
0 j /∈ Ni(4.4)
where:
Ni neighbourhood of i
p(i,j)k probability of ant k in node i choosing arc (i, j)
τij pheromone concentration on arc (i, j)
ηij attractiveness of arc (i, j)
α, β weighting factors
The pheromone trace is updated in every iteration:
τij = (1− ρ)τij + ∆τ (4.5)
In (4.5), ρ represents the rate of evaporation of the trace. This is needed in order toavoid convergence to local minima and allows other routes to be explored. The term∆τ depends on the performance of the solution.
4.3 Application of Evolutionary Computation
4.3.1 Theoretical background
As described in section 4.2.2, the group of Evolutionary Computation algorithmsconsists of three subgroups: Genetic Algorithms (GA), Evolutionary Programming(EP) and Evolution Strategies (ES). The latter two methods are less known, althoughthey have shown outstanding performances in different applications [9]. Therefore,the focus in this section is on these two methods.
(Meta-) Evolutionary Programming
EP is a technique based on an evolving population of solutions. The generation ofoffspring is based only on a mutation mechanism.
Metaheuristic Optimisation Methods 65
Algorithm 4.1 Pseudo-code for Evolutionary Programming.parents ← Initialise()repeat
children ← Copy(parents)children ← Mutate(children)population ← Merge(parents, children)parents ← Select(population)
until stopping condition met
The pseudo-code can be seen in Algorithm 4.1 [108]. The parent individuals createchildren by mutation. The parent and child populations are merged, and a selectionprocedure is performed. In EP, there is usually one child per parent, so that after theselection procedure, half of the total population remains.
The selection procedure is performed based on a stochastic q-tournament. Eachmember of the candidates is compared with q random other candidates. The membergets a score equal to the number of candidates performing worse. The members areranked in descending order of the score values, and the best half is retained.
An important aspect of EP is how mutations are performed. Every component xiof the state vector x mutates according to:
xk+1i = xki +
(√βiF (x) + γi
)·Ni(0, 1) (4.6)
where β and γ are parameters that have to be tuned in order to make the factor underthe square root positive. Ni(0, 1) is a sample from a standard normal distribution(i.e. with expected value 0 and standard deviation σ = 1). F (x) is a fitness function,the value of which goes to zero when the optimum is reached.
The working principle of this method is straightforward: as the solution ap-proaches the optimum, the mutations become smaller and smaller. However, in orderto make this algorithm work, the minimum value has to be known beforehand, orat least an approximation has to be made. This can be a serious problem in someapplications.
To cope with this problem, Meta-EP (MEP) has been proposed [37]. In thisapproach, the dependence on the value of the fitness function is eliminated, by incor-porating mutation strengths in the evolution process:
xk+1i = xki + σki Ni(0, 1) i = 1 . . . d (4.7)(σk+1i
)2=(σki)2
+ ζσki Ni(0, 1) i = 1 . . . d (4.8)
Next to the state vector x, a strategy vector σ ∈ Rd+ is introduced, also withdimension d. σi indicates the mutation strength coupled with state vector elementxi. This is the non-isotropic implementation of the method, in which mutationswith equal probability are hyperellipses in Rd. In the isotropic case, one mutation
66 Chapter 4
(a) isotropic (b) non-isotropic
Figure 4.2: Mutation strengths in a two-dimensional example for the isotropic andnon-isotropic case. The optimum is somewhere in the middle of the picture.
strength is used for all search directions, reducing the hyperellipses to hyperspheres.A graphical representation for a two-dimensional case can be seen in Fig. 4.2. Inthese figures, the optimum is located somewhere in the middle. In the isotropic case(Fig. 4.2a), the mutation strength is equally large in both search directions, indicatedby a circle around every intermediate solution. In the non-isotropic case, the mutationstrength is larger in the direction of the largest gradient.
A tuning parameter ζ is introduced in order to keep the mutation strength positive.This strategy of co-evolving the mutation strengths makes the algorithm self-adaptive,which is important in order to be able to deal with a whole range of problems. Achallenge that remains is the choice of the initial mutation strengths.
Evolution Strategies
The technique of Evolution Strategies (ES) is also based on a mutation operator. Inan ES with µ parents and λ offspring, two possible implementations are (µ, λ)-ES and(µ + λ)-ES. In the first (the non-elitist approach), the best child individuals replacethe parent generation. In the latter implementation (the elitist approach), selectionis performed on the merged population of parents and offspring. A typical value ofλ/µ of 5 is suggested in literature [25].
The mutation of the strategy variables is comparable to the implementation inMEP, and is performed in the following way:
xk+1i = xki + σk+1
i Ni(0, 1) i = 1 . . . d (4.9)
σk+1i = σki · exp (τ ′N(0, 1) + τNi(0, 1)) i = 1 . . . d (4.10)
Metaheuristic Optimisation Methods 67
N(0, 1) denotes a normal distribution with 0 mean and standard deviation 1.Ni(0, 1) designates the same function with a different sample for every value of i. τis the coordinate-wise learning rate, τ ′ the overall learning rate. Literature indicatesthe following guidelines for the choice of these parameters:
τ ∝(
4√
4d)−1
τ ′ ∝(√
2d)−1
(4.11)
In contrast with MEP, the selection algorithm in ES is completely deterministic.The candidates are ranked according to their performance and µ of the best membersare retained.
The algorithm described here is the so-called non-isotropic self-adaptation imple-mentation. An isotropic version can also be used, where τ = 0 in (4.10). This isotropicvariant is not used in this research as not all dimensions are equally important in theproblem considered.
Differences with GAs
In the field of Evolutionary Computation, GAs are the best-known variants. Thereare some fundamental differences between GAs on one side and MEP and ES on theother:
- GAs are based on a binary representation of data, while MEP and ES usereal-valued representations.
- GAs use several operators, like mutation and recombination. Originally, MEPand ES are only based on mutations, although modern implementations of ESalso allow recombination.
- MEP and ES are controlled by self-adapting strategy parameters, which is notthe case in GAs.
4.3.2 Problem definition
The goal is to optimise the import TTC of the Netherlands and Belgium from Ger-many and France. This is performed by changing the settings of the six PSTs, re-sulting in a search space consisting of six dimensions. The borders of this space aremarked by the maximum and minimum settings of each phase shifter. The mathe-matical description is:
Max TTC = f(x) x ∈ R6 (4.12)
subject to:
xi,min 6 xi 6 xi,max i = 1 . . . 6 (4.13)
68 Chapter 4
ρ value σ0Mee () σ0
Gro / σ0Mon () σ0
ZV / σ0V E1 / σ0
V E2 ()ρ1 5 12 4.8 10ρ2 8 7.5 3 6.25ρ3 12 5 2 4.17ρ4 15 4 1.6 3.33ρ5 20 3 1.2 2.5ρ6 25 2.4 0.96 2
Table 4.1: Range factors and corresponding initial mutation strengths for the differentPSTs.
This is a constrained optimisation problem. An unconstrained problem can beobtained by incorporating the constraints in the objective function. A limit functionL is defined, with L = 0 outside the limits and L = 1 elsewhere. TTC = 0 outside thelimits, and the constraints become soft: the area outside the limits is not unfeasiblebut becomes unattractive. Mathematically, the unconstrained optimisation problemcan be written as follows:
Max TTC = f(x) · L(x) x ∈ R6 (4.14)
with:
L(x) =
0 outside limits
1 elsewhere(4.15)
An important note is that in the actual optimisation process, the import capacityis considered, being a negative value by convention, i.e. the TTC is minimised insteadof maximised.
4.3.3 Simulation setup
The simulations are performed on the model introduced in chapter 3. The optimisa-tion algorithms are programmed in Python, and all the load flow computations areperformed in PSS/E. The evaluation of the TTC is done in the same way as wasdescribed in section 3.1.2.
Both optimisation methods are tested with a population size of 30, 60, 90, 120and 150. The number of iterations is chosen to be 100 for MEP and 20 for ES. Asa λ/µ of 5 is taken for ES, the calculation time per iteration can be expected to beapproximately 5 times larger than with MEP. Therefore, with 100 iterations for MEPand 20 for ES, the calculation time for both methods can be expected to be of thesame order of magnitude.
The initial mutation strengths are determined as follows:
σ0i =
αmax,i − αmin,iρ
(4.16)
Metaheuristic Optimisation Methods 69
20 40 60 80 100 120 140 160-7750
-7700
-7650
-7600
-7550
-7500
-7450
population size
TT
C [M
W]
ρ1
ρ2
ρ3
ρ4
ρ5
ρ6
(a) TTC
20 40 60 80 100 120 140 1600
1000
2000
3000
4000
5000
6000
7000
population sizeca
lcu
latio
n tim
e [s]
ρ1
ρ2
ρ3
ρ4
ρ5
ρ6
(b) calculation time
Figure 4.3: Simulation results for MEP with six range factors.
where αmax,i and αmin,i are the maximal and minimal angle of PST i and ρ is a rangefactor. The algorithms are assessed for different values of ρ, as listed in Table 4.1.
Each of the two algorithms has been run 10 times for each parameter combination,and the average TTC and calculation time have been computed.
4.3.4 Simulation results
The simulation results for MEP can be seen in Fig. 4.3. The graphs show the resultingTTC (averaged over 10 runs per parameter combination) and the calculation time fordifferent population sizes and for different range factors. Clearly, the choice of therange factor is important for the performance of the algorithm. If the initial mutationstrength is large, there are a lot of non-feasible solutions in the beginning, having anegative influence on convergence. This explains the relatively small calculation timefor large initial mutations as well: the TTC of a non-feasible solution is not calculated,saving time. Furthermore, if a parent is outside the limits, there is a good possibilitythat its child will also be outside. In this way, this behaviour is propagated throughsubsequent iterations, leading to a decreased convergence rate. This propagation isless likely to occur when more children are produced per parent, as is the case in ES.
If the calculation time is not considered, a large value of ρ, combined with alarge population yields the best results for MEP. However, a considerable amount ofcalculation time is required to find this solution, being mostly not unacceptable inpower system operations. Accounting for the calculation time, ρ5 with 30 particlesalso yields very good results.
For ES, the results are given in Fig. 4.4. What stands out is the fact that thealgorithm is less sensitive to the initial mutation strength, as seen in the reducedspread in the TTC results and in the calculation time. This observation confirms the
70 Chapter 4
20 40 60 80 100 120 140 160-7750
-7700
-7650
-7600
-7550
-7500
-7450
population size
TT
C [M
W]
ρ1
ρ2
ρ3
ρ4
ρ5
ρ6
(a) TTC
20 40 60 80 100 120 140 1600
1000
2000
3000
4000
5000
6000
7000
population size
ca
lcu
latio
n tim
e [s]
ρ1
ρ2
ρ3
ρ4
ρ5
ρ6
(b) calculation time
Figure 4.4: Simulation results for ES with six range factors.
theory of propagation given for MEP.The results are reasonably good for every value of ρ. It is impossible to select a
certain parameter combination as the best one, as the differences in performance arenot significant.
If a choice has to be made between both algorithms, ES could have a slightadvantage over MEP as it is less dependent on the initial conditions.
4.4 Application of Particle Swarm Optimisation
4.4.1 Theoretical background
In the Particle Swarm Optimisation (PSO) approach, the particles (as members of theswarm are called in the algorithm) fly around in a multidimensional space (dimensiond) [109]. Their state can be described by their current position x and velocity v,which are d-dimensional vectors. The velocity of particle i depends on the previousvelocity due to supposed inertia, which is a controllable parameter in the algorithm:
vk+1i = Hiv
ki + . . . (4.17)
where Hi is the inertia matrix (diagonal). A large inertia means that the velocity isvery much influenced by its previous value, so rapid changes are impossible. A smallinertia results in a very nervous behaviour, which is suited for further convergencetowards the optimum if the region where this optimum is located is already found.Experiments have been carried out with a decreasing inertia over time.
The particle i is assumed to have a simple form of memory. It can remember itspersonal best and at which position pi it occurred. The particle is influenced by this
Metaheuristic Optimisation Methods 71
knowledge as it tends to stay in the vicinity. This is implemented in the algorithm inthe following way:
vk+1i = Hiv
ki +C1R1(pi − xki ) + . . . (4.18)
where C1 is a diagonal parameter matrix of the algorithm to control the influence ofpersonal memory andR1 a diagonal matrix with random numbers in the interval [0, 1].Clearly the velocity is always corrected in the direction of the personal optimum.
A very important aspect of swarm dynamics is the communication between itsmembers. Every particle knows what the best solution is so far for the whole swarm,and at which position pg this occurred. This is reflected in the final form of thealgorithm:
vk+1i = Hiv
ki +C1 R1(pi − xki ) +C2R2(pg − xki ) (4.19)
where C2 is a diagonal parameter matrix of the algorithm to control the influence ofthe group memory and R2 a diagonal matrix with random numbers in the interval[0, 1].
In each iteration, the velocity is calculated and the position updated accordingto:
xk+1i = xki + χvk+1
i (4.20)
where χ is a constriction factor, which controls the magnitude of the velocity. It isused to increase the stability of the algorithm if needed, χ = 1.
4.4.2 Simulations
PSO is applied to the import TTC optimisation problem of the Netherlands andBelgium from Germany and France [104]. In the actual simulations, the settings ofdifferent algorithm parameters are compared, while others are taken as a constant.The parameters to be determined are:
- C1, self-confidence matrix
- C2, group-confidence matrix
- Hi, inertia matrix
- χ, velocity constriction factor
- number of particles
- maximum number of iterations
72 Chapter 4
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iterations
|TT
C| [M
W]
inertia = 0.2inertia = 0.4inertia = 0.6inertia = 0.8
(a) 15 particles
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|TT
C| [M
W]
inertia = 0.2inertia = 0.4inertia = 0.6inertia = 0.8
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(c) 60 particles
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C| [M
W]
inertia = 0.2inertia = 0.4inertia = 0.6inertia = 0.8
(d) 120 particles
Figure 4.5: TTC for different inertia values.
Both the self-confidence and group-confidence matrices are chosen to be of theform kI (with I the unity matrix), meaning that the diagonal elements are equal.In [49], it is proposed to choose k = 2 for both matrices, as the mean values of thediagonal elements of C1R1 and C2R2 become 1 in that case. This mean value of 1indicates that the particles will overshoot the optimum with a probability of 50%.
The inertia matrix is taken identical for all particles. Furthermore, its non-zeroelements are chosen to be identical, as there is no reason to make a distinction betweendifferent dimensions at this point. The performance of the algorithm is tested withinertia values of 0.2, 0.4, 0.6, and 0.8.
A value of 1 is chosen for the velocity constriction factor χ. This value can bechanged if there is any problem with velocities becoming too large.
The number of particles is varied (15, 30, 60, and 120), and the behaviour of themethod is monitored for each value. The maximum number of iterations is chosen tobe 80.
Metaheuristic Optimisation Methods 73
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particles = 15particles = 30particles = 60particles = 120
(d) inertia 0.8
Figure 4.6: TTC for different numbers of particles.
In Fig. 4.5, the effect of the inertia factor can be studied. The figure showsthe evolution of the magnitude of the best TTC value in time (number of iterationsperformed), averaged over 10 runs. Clearly, an inertia of 0.8 is too large to accomplisha good and fast convergence. A value of 0.4 seems to perform well. An inertia of0.2 is also acceptable, although the performance is somewhat worse, possibly becausethere is a risk of convergence towards a local optimum.
Fig. 4.6 shows the results with different values for the number of particles within 1graph. Once again, from Fig. 4.6d it can be seen that a large inertia is not suitable atall. Fig. 4.6b clearly shows that the performance of 15 and 30 particles is comparable,but that 60 or 120 particles perform better when considering the value of the optimumfound in the end.
The calculation time for 80 iterations is proportional to the number of particles:
calculation time (s)|80 it = 14.6 · particles+ 110 (4.21)
74 Chapter 4
This equation is obtained by fitting experimental data (obtained from a P4 3.2GHz PC) with a linear trendline. The increase in calculation time from 15 to 60particles is too significant compared to the gain in TTC (about 20 MW). For thepurpose of this research, the use of 15 or 30 particles is sufficient.
The result from the best PSO run with inertia 0.4, 30 particles and 80 iterationsis shown in Table 4.2. The optimal settings are very close to the ones shown inTable 3.3, only in this case, the TTC is slightly better and the computational effortis considerably lower.
TTC Mee Gro Zv VE1 VE2 Mon-7751 -21.74 -11.99 -22.93 2.78 24.79 11.99
Table 4.2: Import TTC [MW] and PST settings [] for the best PSO run with inertia0.4, 30 particles and 80 iterations.
The result obtained by PSO is comparable to the ones obtained by MEP and ESwhen it comes to the maximum TTC found. However, the required calculation timeis much shorter in the case of PSO. Hence, PSO is preferred to MEP and ES for thisparticular problem.
4.5 Transit flow sensitivity
In section 4.4, it was demonstrated that PSO is a suitable method to find the optimalPST settings in order to maximise the import TTC of the Netherlands and Belgium.Transit flows through both countries are expected to have an important influence onthe TTC.
In order to study the sensitivity of the TTC to transit flows, the principle of powershift (PS) is used: the generation is increased in Germany and decreased in France bythe same amount, resulting in a transport from Germany to France, a part of whichtransits the Netherlands and Belgium. The opposite procedure can be used to inducea transit in the other direction.
In Fig. 4.7a, the import TTC can be seen as a function of the PS. By convention,a positive PS means that the total generation is increased in Germany. In the casewithout PSTs, a positive PS is beneficial for the TTC. This can be explained bythe fact that the bottleneck is the Belgian-French border, where the interconnectorsare relatively heavily loaded due to imports to Belgium. A transit from Germany toFrance relieves this bottleneck, while a transit from France to Germany makes theproblem worse.
It can also be seen that an optimal coordination of the Meeden and Gronau PSTsresults in a better TTC, but the shape of the curve is still the same, as the bottleneckis still at the Belgian-French border.
Metaheuristic Optimisation Methods 75
-4000 -3000 -2000 -1000 0 1000 2000 3000 4000-8000
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impo
rt TT
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30
power shift [MW]
PS
T se
tting
[deg
rees
]
Meeden Gronau Zandvliet Van Eyck 1 Van Eyck 2 Monceau
(b) settings
Figure 4.7: Maximum import TTC and corresponding PST settings as a function ofpower shift.
76 Chapter 4
If the settings of all PSTs are optimised for each level of PS, the TTC is muchhigher and the curve is also much flatter, indicating that the sensitivity to transitflows is drastically reduced. Clearly, at a PS lower than -2000 MW, one of the PSTsreaches its limit, resulting in a less flat curve. Fig. 4.7b shows that it is the Van Eyck1 PST that reaches its maximum positive setting at this point. For the region wherethis PST is not at its maximum, the device is very important for the reduction of thesensitivity to PS changes.
4.6 Multiobjective optimisation including losses
It is likely that the active power losses in the system show a significant increasewhen PSTs are used in the grid due to the fact that the “natural” flow patternis disturbed. While the TTC is optimised, it is very well possible that the lossesincrease dramatically. This is why they are accounted for in this section by means ofa multiobjective optimisation formulation [107].
4.6.1 Definitions
(Strict) Pareto dominance: In the context of a minimisation problem3, vector u(strictly) dominates vector v if all components of u are equal to or smaller than thecorresponding components of v. Furthermore, at least one of the components of u isstrictly smaller than the corresponding component of v.
u ≺ v ⇔ ∀i : ui ≤ vi ∧ ∃i : ui < vi (4.22)
Pareto optimality: A solution x is Pareto optimal if its objective vector f(x) is notdominated by any other objective vector. All the Pareto optimal solutions togetherare called the Pareto optimal set P∗.
x ∈ P∗ ⇔ ¬∃x′ : f(x′) ≺ f(x) (4.23)
Pareto front: A Pareto front PF∗ is the curve formed by the group of objectivevectors corresponding to the Pareto optimal set P∗.
PF∗ = f(x) = (f1(x), . . . , fnf (x))|x ∈ P∗ (4.24)
Convex Pareto front: A Pareto front PF∗ is convex if:
∀x1,x2 ∈ PF∗,∀λ ∈ [0, 1],∃x3 ∈ PF∗ : λ|x1| + (1 − λ)|x2| ≥ |x3| (4.25)
In other words: if two random points of the Pareto front are selected, the front isconvex if the whole line segment connecting these points belongs to the feasible region.If the sense of the inequality in (4.25) is reversed, the Pareto front is concave.
3The definition of Pareto dominance can be altered for a maximisation problem by changing allthe inequality signs.
Metaheuristic Optimisation Methods 77
f2 =
−w1w
2 f1 +
Kw2
f1
f2
optimum
(a) CWA for convex Pareto front
f2 =
−w1w
2 f1 +
Kw2
f1
f2
A
B
C
(b) CWA for concave Pareto front
Figure 4.8: Graphical representation of the Conventional Weighted Aggregation ap-proach, for a two-objective problem.
4.6.2 Conventional Weighted Aggregation
A method often used to obtain the Pareto front is the Conventional Weighted Ag-gregation (CWA) [69]. In this approach, all objectives are weighted and added up inorder to come to a new objective function:
F =nf∑i=1
wifi(x) (4.26)
This reduces the problem to a single-objective optimisation problem, which canbe solved with the methods available in that domain. If CWA is used, it must be keptin mind that a different optimum is found for every combination of weight factors.The choice of these factors is crucial in the optimisation process.
If a two-objective problem is considered, the CWA results in the following objectivefunction:
w1f1 + w2f2 = K ⇔ f2 = −w1
w2f1 +
K
w2(4.27)
In a graph, this represents a straight line (the cost line), with the slope dependingon the ratio of the weight factors and the intercept depending on the cost K and w2.
In Fig. 4.8a, it can be seen how the optimum is reached for one set of weightfactors for a convex Pareto front. This optimum is one point on the front. A changein the ratio of the weight factors results in a different slope of the cost function, whichresults in a different optimum. A method for obtaining the Pareto front is thereforeto repeat the optimisation with different weight factors.
The situation is different for a concave front, however. As shown in Fig. 4.8b, theoptimum value obtained is either A or B. This means that a concave Pareto front cannot be found by repeating the optimisation for different values of the weights.
78 Chapter 4
f2
f1
(a) 0
f1
f2
(b) 45
f1
f2
(c) 90
Figure 4.9: Rotated convex Pareto front.
f2
f1
(a) 0
f1
f2
(b) 45
f1
f2
(c) 90
Figure 4.10: Rotated concave Pareto front.
Instead of rotating the cost line when the weight factors change, it is also possibleto keep this line horizontal and to rotate the Pareto front [48]. The relation betweenthe rotation angle and the weights is:
φ = arctanw1
w2(4.28)
This is illustrated in Fig. 4.9 for a convex Pareto front and in Fig. 4.10 for a concaveone. The solid circles indicate the stable optima, whereas the dashed circles indicatethe unstable ones.
4.6.3 Simulations
The multiobjective cost function is now a weighted sum of the import TTC of theNetherlands and Belgium from Germany and France (which is a negative number inthe simulations, resulting in a minimisation problem), and the system losses of theNetherlands and Belgium together. If the PST limits are converted to soft constraints,the problem formulation becomes:
Metaheuristic Optimisation Methods 79
Min F = [w1f1(x) + w2f2(x)] · L(x) (4.29)
with:
L(x) =
0 outside limits
1 elsewhere(4.30)
In order to solve this problem, PSO is adopted. The parameters are chosen ac-cording to the study performed in section 4.4:
- C1 = C2 = 2 · I, with I the unity matrix
- Hi = 0.4 · I- χ = 1
- 30 particles
- maximum 80 iterations
Algorithm 4.2 Pseudo-Code for PSO optimising TTC and losses.Initialize()while iteration < max iterations do
while particle < number particles dov ←CalculateNewVelocity(v, bp, bg)x ←MoveParticle(x, v)ttc ←CalcTTC(x)losses ←CalcLosses(x)obj ←CalcObjective(ttc, losses)bp ←UpdatePersBest(obj)bg ←UpdateGroupBest(obj)
end whileend while
procedure CalcTTC(x)ApplySettings(x)DCLoadflow()ACLoadflow()CalcDistributionFactors()ttc ←ExtractTTC()return ttc
The implementation of the algorithm is described in pseudo-code in Algorithm 4.2.It is ran several times, with different values for w1 and w2. The sum of the weightfactors is kept constant and equal to 1. Each run results in one Pareto optimal point,and 15 of these points are calculated to obtain the Pareto front. The final result is
80 Chapter 4
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(d) settings Van Eyck 2 and Monceau
Figure 4.11: Pareto front of the import TTC versus the active power losses of Belgiumand the Netherlands and the corresponding PST settings.
shown in Fig. 4.11a. The front is a typical convex Pareto front, and that is why itcan be obtained rather easily by CWA.
The system losses without PSTs sum up to a total of 220 MW. In that case, theimport TTC is about 4600 MW. The Pareto front shows that optimal coordinationof the PSTs can boost the TTC to a level of about 7100 MW with the same systemlosses of 220 MW.
It can be seen from the curve that an import TTC of 7 to 7.5 GW is a good target.At higher TTC values,
∣∣∂losses∂TTC
∣∣ becomes too high, resulting in a significant increasein losses for only a small gain in TTC. On the other hand, at low TTC values, thederivative becomes too small, resulting in a large deterioration of the TTC for onlya small amelioration of the losses.
Metaheuristic Optimisation Methods 81
Next to the Pareto front, Fig. 4.11 also shows the corresponding optimal settingsof the different PSTs. The figure shows that the optimal settings are less extremewhen loss minimisation is more important than TTC maximisation.
4.7 Summary
In this chapter, the family of metaheuristics has been discussed and applied to theTTC optimisation problem. Metaheuristics are algorithms that only rely on evalua-tions of the objective function, which makes them very suitable for simulation-basedoptimisation. They consist of a heuristic local search algorithm guided by a top-levelstrategy, in order to avoid convergence to local optima.
A widespread classification scheme is based on the number of intermediate solu-tions used. Some variants use a single solution that is improved in every iterationby moving in a certain direction in the search space. Others use a population ofsolutions, providing more information on the search space, but at the expense ofcalculation time.
The class of Evolutionary Computation algorithms consists of three members.Two of those, being Meta-Evolutionary Programming and Evolution Strategies areused for the TTC optimisation. The results show that with appropriate parametertuning, good convergence can be obtained, but with a fairly high computational effort.Furthermore, Particle Swarm Optimisation has been applied to the TTC optimisationand this method yields very good results at a limited computational cost, making itthe optimisation method of choice for further research.
Transit flows have a large impact on the import TTC of the Netherlands andBelgium from Germany and France. It is shown that without PSTs, a transit flowfrom Germany to France has a positive effect on the import TTC, while a flow in theother direction worsens the bottleneck at the Belgian-French border. The optimal co-ordination of PSTs results in an improved TTC value and in a much lower sensitivityto transits within the control capabilities of the devices.
Finally, system losses can be incorporated in the cost function, resulting in amultiobjective optimisation formulation. The weight factors assigned to the TTCand the losses determine what the optimal point is; for every combination of weightfactors, a different optimum exists. The concept of the Pareto front offers a solutionfor this problem, allowing for a trade-off between TTC on the one hand, and losses onthe other. The simulation results show that the losses increase steeply if the absolutebest TTC value is to be obtained. If this is not acceptable, a lower TTC value canbe targeted for.
CHAPTER 5
PSTs in Linearised Equations
Metaheuristic optimisation offers a fast and accurate means to obtain the maximumTTC or the optimal value of any other goal function and it is therefore of greatuse for the coordination of several PST devices. However, the black-box approachrelies on simulations and it does not give any analytical information on the impactand operation of a PST in a meshed grid whatsoever. Obviously, the TTC can becalculated for a certain set of PST settings by running load flow calculations, but noanalytically closed equations of this relation are available at this point. In this chapter,some basic assumptions are adopted in order to simplify the analytical equations ofthe network and the PSTs. By doing so, a more profound insight can be obtained inthe impact that these devices have on a meshed grid.
The assumptions that are made are associated with the approximations used inDC load flow calculations. By adopting these approximations, the inherent nonlinearnature of the load flow calculation disappears, and the grid can be analysed by a set oflinear equations. If PSTs are integrated in these calculations, closed-form expressionsresult with the settings of the PSTs as variables and the overall impact of the devicesbecomes clear. This is discussed in section 5.1.
With these expressions, the relation between the PST settings and the line flowscan be derived. Since the TTC is based on line flows, an analytical expression forthis quantity can be developed also, as is shown in section 5.2. Such an expression
84 Chapter 5
is not only important for understanding the effects of a PST, but it also allowsfor optimisation with Linear Programming, as discussed in section 5.3. Also, thesensitivity of the TTC to the PST settings can be determined (section 5.4).
The theoretical principles developed in this chapter lead to new possibilities formodelling PSTs; in section 5.5 the device is replaced by an equivalent reactance.
5.1 Linearised power flow equations
5.1.1 DC load flow
Performing a DC load flow calculation is a technique that uses approximations inorder to avoid the iterative procedures that are needed in AC load flow methods [62].It results in a linear set of equations that can be solved easily by algebraic methods.The assumptions that are made are the following [74]:
- The voltage profile is assumed to be flat, i.e. all voltages are close to 1 pu.
- The X/R ratio is assumed to be large (larger than 3 [23] or 4 [75]), so thatthe resistances can be neglected. This means that no losses are present inthe simplified system. The method can be applied for transmission systems,where this particular assumption is reasonable, but in general it is not validfor distribution systems, where the X/R ratio is substantially lower. This isillustrated in Table 5.1, where resistance and reactance data for the Belgiangrid are listed [75].
- All the bus phase-angle differences are assumed to be small, so that sin δ ≈ δ,where δ is expressed in radians. Particular care has to be taken when thismethod is applied to heavily-loaded systems, where phase differences can besubstantial. Also, when PSTs are present in the system, increasing phase shiftangles can give rise to some inaccuracy [100].
If these assumptions are adopted, a linear relation can be written between the businjections PB and the bus angles δ:
PB = Y δ ⇐⇒ δ = Y −1PB (5.1)
where Y is the admittance matrix [42]. Furthermore, the line flow PFij over a line ijcan be written as:
PFij =δi − δjXij
=δijXij
= Bijδij (5.2)
If (5.1) is combined with (5.2), a linear relation between the power flow on line ijand the bus injections PBk can be written1 (where the elements of Y −1 are indicated
1This is true when a virtual slack bus is assumed to be bus n, for which P B = 0. If this is notthe case, the indexing of the c-elements is not correct anymore. The most easy way to deal with thisis to assume that cij is the element (i, j) of the augmented matrix of Y −1 where a dummy row andcolumn are added at the position of the slack bus.
PSTs in Linearised Equations 85
Voltage level [kV] 380 220 150 70Rmin [Ω/km] 0.025 0.038 0.018 0.034Rmax [Ω/km] 0.038 0.088 0.292 0.425Ravg [Ω/km] 0.031 0.067 0.090 0.174Xmin [Ω/km] 0.278 0.184 0.071 0.034Xmax [Ω/km] 0.353 0.429 1.458 0.756Xavg [Ω/km] 0.325 0.364 0.374 0.360(X/R)min [-] 8.4 3.5 1.0 0.8(X/R)max [-] 12.5 8.0 12.0 9.0(X/R)avg [-] 10.5 5.5 4.2 2.1
Table 5.1: Resistance and reactance data for the Belgian high-voltage grid.
as c):
PFij =n−1∑k=1
Bij (cik − cjk)PBk =n−1∑k=1
H(ij)kPBk (5.3)
or in matrix form:P F = HPB (5.4)
The matrix H is called the distribution factor matrix (DFM) and represents theparticipation of every bus injection in every line flow [6].
5.1.2 PSTs in a DC load flow
Impact of a single PST
If a single PST with phase shift αij is connected in series to the line ij, the powerthrough that line can be written as:
PFij = Bij(δij + αij) (5.5)
Taking the phase shift into account, (5.1) becomes [112]:
PB = Y δ + Yααij (5.6)
where Yα is a vector with value Bij at position i, −Bij at position j, and zeroelsewhere. By solving this matrix equation, the following results:
δ = Y −1(PB − Yααij
)(5.7)
We refer to the element (i, j) of the matrix Y −1 as cij ; the angles δi and δj can thenbe written as follows:
δi = ci1PB1 + . . .+cii(PBi −Bijαij)+ . . .+cij(PBj +Bijαij)+ . . .+ci(n−1)P
Bn−1 (5.8)
86 Chapter 5
δj = cj1PB1 + . . .+cji(PBi −Bijαij)+ . . .+cjj(PBj +Bijαij)+ . . .+cj(n−1)P
Bn−1 (5.9)
Using these two equations and taking into account that Y −1 is a symmetrical matrix2,the angle difference can be calculated:
δij = δi − δj =n−1∑k=1
PBk (cik − cjk) +Bijαij(2cij − cii − cjj) (5.10)
By combining (5.5) and (5.10), the active power through line ij as a consequence ofthe PST in that line, becomes:
PFij = Bij
(n−1∑k=1
PBk (cik − cjk) + αij(Bij(2cij − cii − cjj) + 1)
)(5.11)
Thus, the power flow in the line with a PST ij corresponds to a weighted sum of thesystem power injections PBk plus a linear term in the phase shift angle αij .
Similarly, the power flow in a line pq can be written as a function of the settingof the PST in line ij:
PFpq = Bpq
(n−1∑k=1
PBk (cpk − cqk) + αijBij(cpj − cpi + cqi − cqj))
(5.12)
Since Y is a highly sparse matrix, Y −1 is very dense [15]. Thus, the linear term inαij in (5.12) is nonzero for every line. This illustrates the fact that the PST influencesthe entire network. Obviously, the influence can be very small for distant lines.
The derivative of the power to αij is referred to as the phase shifter distributionfactor (PSDF), and can be expressed as follows [44]:
ξijαij =∂PFij∂αij
= Bij(1 +Bij(2cij − cii − cjj)) (5.13)
ξpqαij =∂PFpq∂αij
= BpqBij(cpj − cpi + cqi − cqj) (5.14)
These factors, and consequently the impact of the PSTs on system flows, depend onlyon the network configuration and not on the system power injections (generations andloads).
The power flows in the system lines can then be approximated as follows:
PFij = PFij,0 + αijξijαij (5.15)
PFpq = PFpq,0 + αijξpqαij (5.16)
where PFij,0 and PFpq,0 indicate the line flows at zero phase shift; they represent theDC load flow with all PSTs set to zero.
2If a matrix A is symmetric, then A = AT. From this it can be derived that A−1 = (AT)−1 =(A−1)T, which means that the inverse of a symmetric matrix is also symmetric.
PSTs in Linearised Equations 87
Impact of multiple PSTs
If multiple PSTs are installed in the system, the equations for the line power flowsmust be adapted accordingly. In case of presence of PSTs in the lines kl too, thepower flow in the PST-line ij is calculated as:
PFij = PFij,0 + αijξijαij +
∑(k,l)
(k,l) 6=(i,j)
αklξijαkl
(5.17)
Hence, every PST contributes an extra term to the equation and the resulting powerflow in the line ij can be expressed as a linear function of the settings of the differentphase shifters.
Similarly, the power flow in the line pq lacking a PST but with PSTs in the lineskl, is:
PFpq = PFpq,0 +∑(k,l)
αklξpqαkl
(5.18)
Again, every PST contributes an extra term to the power flow in the line.We may thus conclude that under DC load flow assumptions, the active power
flow in a system line is a linear function of the PST settings. The contribution ofeach PST to the power flow in each line depends on the PSDFs and the phase shiftangles. Hence, it is the configuration of the system that is the key element and notat all the power injections. Therefore, if the matrix of PSDFs is written as Ξ, thechoice of a particular set of phase shift angles α leads to the addition of a constantterm ∆PF = Ξα (positive or negative) to the line flows at zero phase shift.
PSDFs for the Dutch and Belgian grid
The PSDFs related to the different PSTs installed in the system have been calculatedfor all the lines of the 380 kV grid of the Netherlands and Belgium. The resultsare shown in Fig. 5.1. A large value of the PSDF for a certain transmission line isindicated by a thick dark line, while small PSDF values are symbolised by thin lightlines.
The plots offer an instant overview of the influence of the different devices. Itcan for example be observed that the Meeden PSTs have a large impact on the twomost northern interconnectors on the Dutch-German border, while the impact on theBelgian grid is small. The Gronau PST is located more central at the Dutch-Germanborder, which is reflected by a considerable effect on all interconnectors between theNetherlands and Germany.
The Van Eyck 1 PST has an important impact on the loop Zandvliet - Geertrui-denberg - Eindhoven - Maasbracht - Van Eyck 1 - Zandvliet. The Zandvliet PST hasa large influence on the power flows in this loop as well, but also on the interconnec-tor Avelin-Avelgem at the Belgian-French border. The Van Eyck 2 PST has a largeimpact on the Achene-Lonny interconnector at the Belgian-French border.
88 Chapter 5
(a) Meeden (b) Gronau
(c) Zandvliet (d) Van Eyck 1
(e) Van Eyck 2
Figure 5.1: PSDFs related to different PSTs for the Dutch and Belgian grid.
PSTs in Linearised Equations 89
Line 7-8 Line 8-9 Line 7-5 Line 5-4 Line 4-6 Line 6-9AC -2.21 -2.19 2.31 2.20 2.25 2.23DC -2.24 -2.24 2.24 2.24 2.24 2.24
Table 5.2: Slopes of the line flow as a function of the PST setting [MW/].
5.1.3 Case study
As a case study, the 9-bus system that was already discussed in section 2.6 is consid-ered. Fig. 2.19 showed the power flows in the system lines as a function of the settingof the PST which is installed between the busses 6 and 9 (calculated with an ACload flow). The curves are almost linear, which can be explained by the theory de-veloped in the current chapter. The slope of the linear fit of the curves is determinedand listed in Table 5.2. The calculated PSDFs (calculated with the DC load flowapproach) are also listed in this table and it can be seen that the difference betweenthe AC and DC values is very small. The difference in line flow (DC flow minus ACflow) is plotted in Fig. 5.2. The figure shows that the absolute deviations are verysmall, and that the effect of the non-linearity of the AC curves is limited.
-20 -15 -10 -5 0 5 10 15 20-4
-3
-2
-1
0
1
2
3
4
5
6
PST setting [degrees]
Lin
e f
low
diffe
ren
ce
[M
W]
Line 7-8Line 8-9Line 7-5Line 5-4Line 4-6Line 6-9
Figure 5.2: Difference between AC and DC line flows as a function of the PST setting.
90 Chapter 5
5.2 Analytical expression for TTC
In this section, an analytical expression for the TTC is derived. In order not tocomplicate the problem, the N secure TTC is considered initially, and a generalisationto the N − 1 case will be performed afterwards.
In a two-area system, the base case exchange (BCE), as discussed in section 3.1,is the sum of the power flows through the Ni interconnectors between the systemswithout any power shift. The N secure TTC can then be written as:
TTCN sec
=Ni∑i=1
PFi + PSmax (5.19)
The relation between the line flows and the PST angles can be written as a linearfunction under DC load flow assumptions, as was described in the previous section.The variation of the maximum power shift (PS) from one area to another one as afunction of the PST settings must be investigated.
As discussed in section 3.1.2, the change in active line flow is related to the PSthrough the sensitivity factor sl. The sensitivity factors are calculated once by ap-plying a certain power shift. The linear curves are then extrapolated until one linereaches its maximum rated power (Fig. 3.2).
sl is taken as a constant, which can be validated by considering (5.11) or (5.12).If a power shift is applied, these equations are influenced, as the injected powers atthe different busses change when taking part in the PS:
PBk → PBk + fk · PS (5.20)
where fk is a participation factor for bus k. If this equation is combined with (5.11)or (5.12), and the derivative with respect to the power shift is taken, the followingexpression can be found for an arbitrary interconnector gh (with or without PST andassuming that the PST settings do not change):
sgh =∂PFgh∂PS
= Bgh
n−1∑k=1
fk(cgk − chk) (5.21)
This shows that the sensitivity is constant under the given approximations, as isassumed in the linear projection method.
The maximum power shift PSmax is determined by the line that is overloadedfirst. For every line, the constraint that the maximum line capacity may not beexceeded must be fulfilled:
PFl + sl · PS ≤ sgn(sl) · PFl,max ∀l (5.22)
where sgn(.) is the sign function. It is assumed that the magnitudes of the positiveand negative line ratings are identical.
PSTs in Linearised Equations 91
Equation (5.22) can be rewritten in order to come to an expression for the maxi-mum power shift:
PSmax = minl
(sgn(sl) · PFl,max − PFl
sl
)(5.23)
where l are all the lines that are relevant for the calculation of the TTC3.The TTC for a system with Ni interconnectors to the adjacent area can now be
written as a function of Np PST settings4:
TTCN sec
=Ni∑i=1
PFi,0 +Np∑p=1
αpξiαp
+ minl
sgn(sl)sl
PFl,max −1sl
PFl,0 +Np∑p=1
αpξlαp
(5.24)
The BCE is a linear function of the different PST settings5. Furthermore, PSmaxis a piecewise linear function of these settings, as in different setting ranges, differentlines can be limiting. This means that the N secure TTC is also a piecewise linearfunction of the PST settings.
The extension to the N − 1 secure TTC is fairly straightforward:
TTCN−1 sec
= mincont
(TTC|cont) (5.25)
Every contingency results in a new system situation for which the TTC can becalculated as a function of α. The N − 1 secure TTC is the minimum of all TTCsunder the contingency conditions. Obviously, the N−1 secure TTC is also a piecewiselinear function of every α.
5.3 TTC optimisation
5.3.1 N secure TTC
Suppose we have a set of linear functions f0, f1, f2, . . . and that these functions dependon a vector x. Given these functions, it is easy to see that in general the followingholds for a certain value of x:
f0(x) + min (f1(x), f2(x), . . .) = min (f0(x) + f1(x), f0(x) + f2(x), . . .) (5.26)
3In a TTC calculation, a number of lines must be monitored for overloading. This list of linesis not necessarily limited to the interconnectors only; there might be reasons for monitoring otherlines.
4From this point on, the same notation is used for PSDFs with regard to a PST in the same lineor in another line. The difference should be clear from the context.
5This is a consequence of the convention that the BCE is calculated as the sum of the intercon-nector flows between both areas.
92 Chapter 5
By using this principle, (5.24) can be rewritten in the following form:
TTCN sec
= minl
Ni∑i=1
PFi,0 +sgn(sl) · PFl,max − PFl,0
sl+
Np∑p=1
(Ni∑i=1
ξiαp −1slξlαp
)αp
(5.27)
This equation expresses the N secure TTC as a piecewise linear function in thePST settings. The next challenge is to maximise this expression as a function of thesesettings.
Let F be defined as the minimum of a set of Nl functions:
F (x) = min (f1(x), f2(x), . . . , fNl(x)) (5.28)
If these functions are linear in x, where x is a Np-dimensional vector, they can bewritten in the following general form:
fi(x) = ai1x1 + ai2x2 + . . .+ aiNpxNp + ki (5.29)
If the maximum of F (x) over x is designated as u, an expression for u can be derivedby introducing an auxiliary variable t:
u = max (t : t− fi(x) ≤ 0, i = 1 . . . Nl) (5.30)
This principle is illustrated for one dimension in Fig. 5.3. The variable t is limitedto the hatched area (t can theoretically also be negative, but this is not shown in thefigure).
xmin xmax
fi(x)
x
f3f1
f2
tmax
Figure 5.3: Maximum of a piecewise linear function.
An auxiliary function is introduced to simplify the problem:
f∗i (x) = fi(x)− ki (5.31)
The problem can now be written as:
u = max (t : t− f∗i (x) ≤ ki, i = 1 . . . Nl) (5.32)
PSTs in Linearised Equations 93
variable dimensionc (Np + 1)× 1z (Np + 1)× 1b Nl × 1A Nl × (Np + 1)
Table 5.3: Dimensions of variables.
In a more general form, this becomes:
max(cTz : Az ≤ b
)(5.33)
which is a Linear Programming (LP) problem (Appendix B) [61], where:
c =
100...
z =
t
x1
x2
...
b =
k1
k2
k3
...
A =
1 −a11 −a12 · · ·1 −a21 −a22 · · ·...
......
1 −an1 −an2 · · ·
(5.34)
The dimensions of these matrices are listed in Table 5.3. Np designates the numberof variables (the number of PSTs in this particular problem), and Nl is the numberof constrained lines in the N secure case, which is equal to the number of lines thatis monitored.
If (5.27) is to be maximised, the elements of the A-matrix and b-vector become:
kl =Ni∑i=1
PFi,0 +sgn(sl) · PFl,max − PFl,0
sl(5.35)
−alp = −Ni∑i=1
ξiαp +1slξlαp (5.36)
The vector z consists of the TTC t (which is maximal in the optimum) and thesettings of the different PSTs x1, x2, . . .
5.3.2 N − 1 secure TTC
As indicated in (5.27), the N secure TTC can be written as the minimum of aset of linear functions. Every function stands for one line that is monitored in thecalculation process. If the theory is to be generalised to N − 1 security, a new set oflinear functions must be written for each contingency. These functions are differentas the initial admittance matrix has to be changed due to one line being taken out ofservice. The N − 1 secure TTC is the minimum of all these functions:
TTCN−1 sec
= min(f c11 , f c12 , . . . , f c21 , fc22 , . . .
)(5.37)
94 Chapter 5
If this is to be maximised, it can be incorporated in (5.33) and (5.34). Everycontingency adds a number of rows in the A-matrix and b-vector. In general, A is ablock matrix and b is a block vector, where every block corresponds to one particularcontingency (and one block corresponds to the case without contingencies).
Contingency on a non-monitored line
If the contingency occurs on a non-monitored line, no particular attention needs tobe paid to singularities in the calculation. If the outaged line contains a PST, thePSDFs of all lines related to that PST become zero, resulting in a column of zeros inthe A-submatrix for this particular contingency.
Contingency on a monitored line
For a contingency on a monitored line, attention has to be paid to singularities. ThePSDFs of the outaged line become zero, as is the case for a non-monitored line. Alsothe sensitivity of the line (to power shifts) becomes zero; this leads to a division byzero in the maximum power shift calculation. Hence, the power shift can becomeinfinitely large, while the power on the outaged line remains zero. This means thatthis line, under the specified contingency, will never be the limiting factor and willtherefore not interfere with the TTC calculation.
The row of elements in the A-submatrix belonging to this particular combinationof monitored line and contingency should be removed. Of course, the matching b-vector element should be removed as well.
Dimensions of the matrices under contingency conditions
Suppose that Nl lines are monitored for the TTC calculation. Every contingency on amonitored line adds (Nl−1) rows to theA-matrix. A contingency on a non-monitoredline adds Nl rows.
If the number of contingencies on monitored and non-monitored lines is designatedas Cm and Cn respectively, the dimensions of the A-matrix and b-vector become:
A→ (Cm(Nl − 1) + CnNl)× (Np + 1)b → (Cm(Nl − 1) + CnNl)× 1
(5.38)
These equations are valid if the case without contingencies is not considered. If itis considered, both matrices have Nl extra rows.
5.3.3 Case study
As a case study, the 39-bus New England test system is adopted [63]. This systemconsists of 46 transmission lines, 10 generators and 39 busses. The system is modifiedin order to fit the needs of this study. Two PSTs are added: one is located between
PSTs in Linearised Equations 95
busses 4 and 14 and the other one between busses 17 and 18, as shown in Fig. 5.4.Both are modelled with the parameters of the Meeden PSTs.
CG
1
1
30
2
25
1
3
1
CG
1
1
1
37
28 29
CG
1
1
18
1
1
1
1
21
1
CG
CG
35
22
23
36
1
CG
CG
5
1
CG
1
1
CG
1
1
CG
7
39
10
6
8
9
12
31
38
13
11
32
4
27
17
24
16
15
33
20
34
19
14
26
Area EastArea West
Figure 5.4: Single-line diagram of the 39-bus New England test system.
The system is split up into two interconnected areas, named area West and areaEast respectively. The two areas are connected by three lines: 14-15, 17-18 and 25-26.These three lines are the ones that are monitored during the TTC calculations.
The N secure import TTC of area West opposed to area East is shown in Fig. 5.5.In each of the plots, one PST setting is kept at zero degrees while the TTC is calcu-lated as a function of the other. The calculation is compared with an AC load flowcalculation in PSS/E. Clearly, the deviations are small and can be attributed to theapproximations made in the DC load flow approach.
The A-matrix is a 3 × 3 matrix with the following elements (referring to thenaming conventions of equation (5.34)):
96 Chapter 5
-30 -20 -10 0 10 20 301400
1500
1600
1700
1800
1900
2000
PST 4-14 setting (degrees)
Imp
ort
TT
C f
or
Are
a W
est
(MW
)
DC calculation
AC calculation
(a) PST 4-14
-30 -20 -10 0 10 20 30800
1000
1200
1400
1600
1800
2000
2200
PST 17-18 setting (degrees)
Imp
ort
TT
C f
or
Are
a W
est
(MW
)
DC calculation
AC calculation
(b) PST 17-18
Figure 5.5: Absolute value of the N secure import TTC of area West as a functionof the PST settings. In each plot, one PST setting is kept at zero degrees, while theother one varies.
PSTs in Linearised Equations 97
−a11 = −(ξ26,25α4,14
+ ξ17,18α4,14
+ ξ15,14α4,14
)+ 1
s26,25ξ26,25α4,14
−a21 = −(ξ26,25α4,14
+ ξ17,18α4,14
+ ξ15,14α4,14
)+ 1
s17,18ξ17,18α4,14
−a31 = −(ξ26,25α4,14
+ ξ17,18α4,14
+ ξ15,14α4,14
)+ 1
s15,14ξ15,14α4,14
−a12 = −(ξ26,25α17,18
+ ξ17,18α17,18
+ ξ15,14α17,18
)+ 1
s26,25ξ26,25α17,18
−a22 = −(ξ26,25α17,18
+ ξ17,18α17,18
+ ξ15,14α17,18
)+ 1
s17,18ξ17,18α17,18
−a32 = −(ξ26,25α17,18
+ ξ17,18α17,18
+ ξ15,14α17,18
)+ 1
s15,14ξ15,14α17,18
(5.39)
The b-vector has 3 elements:
k1 = P26,25,0 + P17,18,0 + P15,14,0 +sgn(s26,25) · P26,25,max − P26,25,0
s26,25
k2 = P26,25,0 + P17,18,0 + P15,14,0 +sgn(s17,18) · P17,18,max − P17,18,0
s17,18
k3 = P26,25,0 + P17,18,0 + P15,14,0 +sgn(s15,14) · P15,14,max − P15,14,0
s15,14
(5.40)
When all these values are calculated, the following optimisation case results:
max
100
T
·
t
α4,14
α17,18
s.t.
1 3.89 −10.471 6.04 41.071 −6.32 −9.90
· t
α4,14
α17,18
≤ 19.8
30.017.8
(5.41)
This linear optimisation problem is solved by using the MOSEK toolbox for Mat-lab (www.mosek.com). The maximum value for t, which is the maximum TTC, is2100 MW. The optimal PST settings are 12.1 degrees for α4,14 and -10.8 degrees forα17,18. From the contour plot of Fig. 5.6 (obtained through a repetitive evaluation of(5.24)) it is clear that these settings are indeed optimal.
The next step is to calculate the maximum N − 1 secure TTC. The contingenciesare applied to the interconnectors only. A preliminary study in PSS/E showed thatthe optimum should be somewhere around 1400 MW for optimal settings of about+9 and -4 degrees for PST 4-14 and 17-18 respectively. There are 3 interconnectorsand 3 monitored lines, so Ni = 3, Nl = 3, Cm = 3 and Cn = 0. By using (5.38), itcan be calculated that the dimensions of A and b are 6 × 3 and 6 × 1 respectively(the case without contingencies is not considered).
These matrices are constructed by applying one contingency at a time, and calcu-lating the elements of the rows corresponding to the specific contingency. The finalresult is:
98 Chapter 5
PST 17-18 setting [degrees]
PS
T 4-
14 s
ettin
g [d
egre
es]
-30 -20 -10 0 10 20 30-30
-20
-10
0
10
20
30
600
800
1000
1200
1400
1600
1800
2000
Figure 5.6: Contour plot of the calculated TTC as a function of the two PST settingsfor the 39-bus New England test system.
A =
1 4.88 16.041 −3.07 −10.081 4.33 01 −3.93 01 1.59 −10.261 −2.26 14.59
b =
16.1212.6714.7513.3213.2415.07
(5.42)
The matrices are partitioned in order to clearly show what the contribution is ofevery separate contingency.
The result is a maximum TTC of 1392 MW with settings of 8.82 and -5.16 degreesfor PST 4-14 and 17-18 respectively. This result was already more or less predictedby the PSS/E calculations. The optimisation itself is very fast (seconds), but the cal-culation of the matrices takes a bit longer, mainly due to the fact that some elementsof the inverse admittance matrix are required. The complete calculation takes a fewminutes for a large system, which is a lot faster than the methods developed in theprevious chapters.
PSTs in Linearised Equations 99
5.4 TTC sensitivity
In the previous sections, an analytical expression for the TTC between two areaswas found and it is possible to optimise it by using LP. However, it would be veryinteresting to gain some insight in its sensitivity, i.e. the way in which the TTC varieswith the PST settings.
As discussed before, the TTC is the sum of the Base Case Exchange (BCE) andthe maximum allowable power shift PSmax. The partial derivative of the TTC to acertain PST setting αp is:
∂TTCN sec
∂αp=∂BCE
∂αp+∂PSmax∂αp
(5.43)
5.4.1 Sensitivity of the BCE
The BCE is equal to the sum of the line flows through all interconnectors without anypower shift. If every line power is expressed as a linear function of the PST settings,the BCE can be written as:
BCE =Ni∑i=1
PFi,0 +Ni∑i=1
Np∑p=1
αpξiαp (5.44)
with Np the number of PSTs, and Ni the number of interconnectors. The sensitivityof the BCE to PST setting αp is:
∂BCE
∂αp=
Ni∑i=1
ξiαp (5.45)
This result is easy to understand: the total flow over a border is only influenced bythe PSDFs of the interconnectors regarding the PST that is studied.
In a two-area system, the PSDFs of the interconnectors sum up in such a way thatthe sensitivity of the BCE becomes zero. The PSTs do not change the BCE in thiscase; only the distribution of the flows over the different interconnectors is altered.
5.4.2 Sensitivity of the maximum power shift
If it is supposed that line j is limiting, the maximum power shift can be written as:
PSmax =1sj
sgn(sj) · PFj,max −
PFj,0 +Np∑p=1
αpξjαp
(5.46)
The sensitivity of PSmax to PST setting αp can be calculated as:
∂PSmax∂αp
= − 1sjξjαp (5.47)
100 Chapter 5
P Fmax
∆(PSmax)1 ∆(PSmax)2
P F
PS
ξαp∆αp
P F ′
P F
Figure 5.7: PSmax for different PST settings and different sensitivities.
In Fig. 5.7, it is illustrated what happens to PSmax if a PST setting is changed.The PSDF is an important factor, as it determines the intercept of the linear curvewith the Y-axis. An increasing PSDF results in a decreasing PSmax, which explainsthe minus sign in (5.47). Furthermore, the figure shows that for a high power shiftsensitivity (steep curve), PSmax does not vary much with a changing PST setting.For a low power shift sensitivity, a small change in PST setting can have a big impacton the maximum power shift. These considerations explain the mathematical relationin (5.47).
5.4.3 Total sensitivity
By adding (5.45) and (5.47), an expression of the total sensitivity of the N secureTTC with regard to PST setting αp is obtained:
∂TTCN sec
∂αp=
Ni∑i=1
ξiαp −1sjξjαp (5.48)
where j is the limiting line in the TTC calculation. For a two-area system, the firstterm becomes zero, as explained before.
For an area in the Np-dimensional PST setting space where j is the limiting line,expression (5.48) is a constant number. If the limiting line changes, the sensitivitychanges as well. This means that the sensitivity is a piecewise constant function inthe PST setting space.
As shown in (5.25), the N − 1 secure TTC is the minimum value over all contin-gencies. Equation (5.48) still holds for this case, but it is important to realise thats and ξ are different for every contingency, as a topology change takes place everytime.
PSTs in Linearised Equations 101
5.5 Equivalent reactance
For modelling purposes, it would be practical to represent a PST by a single reactanceinstead of an ideal phase shift in combination with a series reactance, as was donebefore. In this section, it is shown that it is possible to model the PST together withthe transmission line as a variable reactance X ′T .
XP + XL
Vs 6 δ′ Vr 6 0X ′T
Vs 6 δ′′ Vr 6 0α
Vs 6 δ′′ + α
Figure 5.8: Representation of a PST in a transmission line.
By using DC load flow approximations and the notation shown in Fig 5.8, thepower through the equivalent reactance can be written as:
PF′
=δ′
X ′T(5.49)
The power flow through a line with a PST can be written as:
PF′′
=δ′′ + α
Xl +Xpst(5.50)
If both cases are equivalent, the line powers must be equal (PF′
= PF′′). Fur-
thermore, the phase difference between the sending and receiving end must be equal(δ′ = δ′′ = δ). By doing so (and by using Xl +Xpst = XT ), the equivalent reactancebecomes:
X ′T =δ
δ + αXT (5.51)
In paragraph 5.1.2, it was demonstrated that the power through a line with a PSTcan be written as (assuming that no other PSTs are present in the system):
PF =δ + α
XT= PF0 + αξ ⇐⇒ δ = XTP
F0 + α (XT ξ − 1) (5.52)
where PF0 is the line flow at zero phase shift, XT is the total reactance in the line,and ξ is the PSDF of the line with regard to the PST in the line. By combining (5.51)and (5.52), an expression for the equivalent reactance is found:
102 Chapter 5
X ′T = XT −(PF0α
+ ξ
)−1
(5.53)
If other PSTs are present in the system, this expression can be generalised. Theequivalent reactance for PST k becomes:
X ′T,k = XT,k −
PF0 +
Np∑p=1,p6=k
ξkαpαp
αk+ ξkαk
−1
(5.54)
In order to demonstrate this modelling technique, the New England test systemis equipped with one PST in the line 4-14. The equivalent reactance of this PST iscalculated and plotted in Fig 5.9.
-30 -20 -10 0 10 20 30-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
PST setting [degrees]
Xe
q [
pu
]
self-compensation point
zero-power point
Figure 5.9: Equivalent reactance of the PST in the New England test system.
By looking at the curve, some characteristics can be observed. First, the zero-power point is reached at an angle of about -8 degrees. At this point, the equivalentreactance becomes infinity, resulting in a zero line flow. Also, the self-compensationpoint can be defined. At that point, the equivalent reactance becomes equal to thereactance of the line without the PST. This particular point is reached at a settingof about +10 degrees.
PSTs in Linearised Equations 103
The line flow through the PST is monitored and compared to the case where thedevice is replaced by the equivalent reactance. The result can be seen in Fig. 5.10.
-30 -20 -10 0 10 20 30-400
-200
0
200
400
600
800
PST setting [degrees]
Plin
e [
MW
]
PST
Variable X
Figure 5.10: Power flow through line 14-4 with a PST and with its calculated equiv-alent reactance.
The deviation between the two methods is small and can be attributed to theassumptions that are made in the DC load flow.
5.6 Summary
Due to the nonlinear nature of the AC load flow equations, difficulties arise for defin-ing analytically-closed equations that describe the impact of PST operation. Theassumptions that are adopted for a DC load flow calculation (small angle differences,no line resistance, flat voltage profile) allow for a linearised formulation of the powerflow equations, making iterative techniques unnecessary.
The DC load flow equations can incorporate the behaviour of PSTs. The resultingexpressions show that the active line flows are a linear function of the PST settings.The sensitivity factors that link the PST settings to the power flows are the so-called phase shifter distribution factors or PSDFs. These factors only depend on thenetwork topology and not on the injection pattern, and give valuable information onthe impact of an installed control device.
Based on the linear equations that were derived for the line flows, a formulation ofthe object function (TTC) can be developed. It is shown that the TTC is a piecewise
104 Chapter 5
linear function of the different PST settings. This structure allows to treat it asa Linear Programming or LP problem. The fact that an analytical expression isavailable for the TTC means that also the sensitivity of the TTC to the PST settingscan be determined by taking the partial derivative.
The linearisation as proposed in this chapter introduces inaccuracy, in particularwhen the DC load flow assumptions are far from reality. This can for example be thecase in heavily loaded systems or in cases where line resistances are relatively highcompared to the line reactances. However, the method is very useful when consideringlarge systems, allowing the TTC to be optimised in a limited amount of calculationtime.
By using DC load flow approximations, an equivalent reactance can be calculatedfor the PST. This reactance is a function of the PST setting, but depends also on thenetwork topology. It can be used as an alternative method to model PSTs for loadflow studies.
CHAPTER 6
Application of Optimisation Methods
In the previous chapters, several search and optimisation methods were derived forthe coordination of PSTs in a power system. These methods are applied on a varietyof problems.
In the unit commitment task, the goal is to distribute the total power demandamongst the generating facilities in the most cost-effective way. However, severalconstraints have to be taken into account, such as congestion in the network. Insection 6.1, a linearised optimal power flow is introduced to handle this problem. Itis shown that with the equations developed in the previous chapter, PSTs can beintegrated in this formulation. By doing so, the coordination of the settings of thePSTs can help to relieve congestions.
Another application is described in section 6.2, where equal relative tie-line load-ings are the target. Since analytical expressions for the line flows are available, therequired settings for an equal power distribution can be calculated. Depending onthe number of PSTs compared to the number of tie-lines, the set of equations caneither be solved exactly, or by means of a Linear Least Squares approach.
In the third example application, a stochastic approach is adopted to quantifythe risk of congestion. In section 6.3 it is proven that with a PST, this risk can bealtered in a straightforward way for a single line. However, in a meshed system, theminimisation of the global congestion risk requires a global optimisation algorithm
106 Chapter 6
for the calculation of the optimal settings.As a last application, a reinforcement study of the interconnectors on the Dutch-
German border is performed in section 6.4. Several possibilities are taken into accountand for each of them the maximum attainable TTC (with optimal PST coordination)is calculated with the methods developed in the previous chapters.
6.1 Generation unit dispatch with PSTs
6.1.1 DC OPF without PSTs
As discussed in chapter 5, line flows may be regarded as linear functions of the businjections. They can be split into positive contributions by generators and negativeones by loads. Two separate bus injection vectors can be written and the distributionfactor matrix (DFM) H is split accordingly. The line flow vector introduced inequation (5.4) becomes:
P F = HGPBG +HLPBL (6.1)
These generation and load DFMs are derived from the general DFM by selecting theappropriate columns.
In the classical unit commitment task1, the goal is to distribute the requiredgeneration over the generating facilities so that the total operational cost is minimal.An optimal power flow (OPF) formulation can be developed, accounting for additionalconstraints such as the prevention of congestion in the network [26; 119]. It is assumedthat the marginal costs of all plants can be regarded as a constant vector CG. Thenthe objective function can be formulated as:
min (CG)TPBG (6.2)
The first constraint is that the total load must be matched by the total generation:
Ng∑r=1
PBGr = −Nl∑s=1
PBLs (6.3)
Furthermore, the line flows are not allowed to exceed the limits of the lines (indicatedby P FR):
HGPBG ≤ P FR −HLPBL (6.4)
−HGPBG ≤ P FR +HLPBL (6.5)
Finally, every generator can only generate power within a certain range:
PBGm ≤ PBG ≤ PBGM (6.6)
1The classical unit commitment task and the method that is developed in this section are notapplicable in the current liberalised framework in Europe. However, a number of useful insights canbe obtained by taking them into consideration.
Application of Optimisation Methods 107
It is also possible to add ramp rate constraints if successive optimisations areperformed within a certain time frame. These constraints reflect the limited speedwith which the power output of plants can be adjusted.
The optimisation formulation developed above is a Linear Programming (LP)problem and can be tackled by a standard LP solver [40].
6.1.2 DC OPF with PSTs
If PSTs are installed in the system, (5.3) must be adapted according to (5.17) and(5.18):
PFij =n−1∑k=1
H(ij)kPBk +
Np∑p=1
ξijαpαp (6.7)
In (block) matrix form, this becomes:
P F = HPB + Ξα = [H |Ξ] ·[PB
α
](6.8)
where Ξ contains all PSDF factors. The PSDFs and angles are expressed relative tothe maximum positive angle setting of the PST. It can be seen that (6.8) is math-ematically equivalent to (5.4). This means the PST can be regarded as a combinedgenerator and load, with a power injection equal to the angle in pu and with distri-bution factors given by the PSDFs in pu2.
If a cost penalty factor is to be assigned to a PST in order to prevent unnecessarycontrol actions, one has to take care not to disturb the LP formulation. The costfactor should be positive for positive settings of the PST and negative otherwise. Inorder to model this, the PST is split in two subdevices: one for positive and one fornegative angles. Accordingly, one subdevice has a positive cost factor and the other anegative one with the same absolute value. The constraints for these subdevices arethat their angle (in pu) must be between -1 and 0 for the positive one, and between0 and +1 for the other. Their PSDFs are obviously the same. The actual setting ofthe PST is the sum of the settings of its two subdevices.
If the settings of the subdevices are indicated by α+ and α−, and the (positive)marginal cost is written as c, the total cost related to the PST is:
Cpst = cα+ − cα− = c(α+ − α−) (6.9)
As this cost constant is minimised, a positive α is always accomplished with onlythe positive component and not as a combination of a positive and a negative term.In the same way, a negative angle is accomplished with only a negative component.So, when solving the LP problem, only one of the subdevices per PST is active.
2It must be noted that H is influenced by the presence of PSTs, as the reactance of these devicesmust be accounted for in the admittance matrix.
108 Chapter 6
type # units operating cost Pmax Pmin ramp rate(e/MWh) (MW) (MW) (MW/quarter)
I coal 3 21-23 1200 300 150II gas 4 36-39 800 200 250III turbojet 2 80-81 200 0 300
Table 6.1: Assumed generator data for the New England test system.
6.1.3 Application to the New England system
In order to illustrate the method developed, the New England 39-bus test system isused (appendix C). The system is adapted for this particular study.
A first modification is the addition of two PSTs, as discussed in section 5.3.3. Fur-thermore, the loads are not fixed to the values given in the original system. Instead,a load curve for one week (15-minute values) is taken from a real system and is scaledrelative to the peak load value of this time series. These relative values (between0 and 1) are multiplied by the load values specified for the original New Englandsystem.
Also, the data for the generators is altered. Three types of generators are distin-guished (Table 6.1). Type I plants have a low operating cost and a low ramp rate.Type III plants are very fast but also very expensive. Type II plants are in betweenboth types.
0 1 2 3 4 5 6 70
1000
2000
3000
4000
5000
6000
7000
day
gene
rato
r com
mitm
ent [
MW
]
Figure 6.1: Unit commitment without PSTs for 1 week.
Application of Optimisation Methods 109
Type I generators are located at busses 30, 32 and 33, type II at busses 34 to 37,and type III at busses 38 and 39. The ratings of all lines are set to 900 MW.
The optimal unit commitment without PSTs is performed for a whole week. Theresults are shown in Fig. 6.1. It can be seen that the cheapest (coal) plant is alwaysrunning at rated power. The second cheapest has to be ramped down during mostof the day in order not to violate line constraints. During peak loads on weekdays,expensive type III generators are switched on, resulting in increased costs.
The unit commitment with PSTs is performed for the same period of 1 week. Thecost of the PSTs is taken as a very small value (0.1 e/30). By doing so, the PSTstend towards 0 if they are not needed (e.g. during nighttime), while their influenceon the total cost can be neglected. It may be a possibility to assign a higher cost tomodel increased losses, but this is not done here.
The ramp rate of the PSTs is chosen to be approximately 1 per 15 minutes. Thisdoes not necessarily correspond to the technical limitations of the device, but it ischosen this way to avoid excessive switching, which may cause a lot of wear and tearon the tap changer.
The results of the optimisation can be seen in Fig. 6.2. The PSTs enable the secondcheapest coal plant to be used at maximum output for a longer time, resulting in alower overall operational cost. Furthermore, it can be seen that the expensive typeIII peak units are not used at all, which also lowers the cost. Integrated over the spanof a week, the total operational cost is 21.3 Me without and 20.7 Me with PSTs,resulting in a gain of 0.6 Me. Fig. 6.2b shows that the PST settings follow a cyclicpattern, with a different pattern for the weekend days.
The cost reduction resulting from using PSTs is depicted in Fig. 6.3. Duringnighttime, when the flows are limited, the PSTs are not really needed, and theirbenefit is negligible. This is clearly visible where the curves coincide. On the otherhand, during daytime, the benefits are clear. In these periods, the PSTs allow a moreefficient unit dispatch given the line flow constraints.
The optimisation method presented here does not take into account system losses.The gain in fuel cost obtained by optimal PST use may be counteracted by increasedlosses. To show this, the obtained unit commitment data (including PST settings)is used as input for AC load flow simulations in PSS/E and the resulting systemlosses are evaluated, with and without PSTs. The difference in losses is shown in thehistogram of Fig. 6.4. Frequently, the difference is more or less zero, correspondingto nighttime operation when the PSTs are at 0. There are some cases where thelosses are actually decreased by the PST operation, but in most (daytime) cases,they increase. It can be seen that this increase is about 10 MW in the worst case(where the maximum losses without PSTs are about 120 MW). The total losses areintegrated over 1 week, resulting in 13089 MWh with PSTs and 13019 MWh without,i.e. a difference of 70 MWh, or a mere 0.5%. If this figure is multiplied by an averageoperational cost of 40 e/MWh, the extra losses in terms of money come to 2800 e,
110 Chapter 6
0 1 2 3 4 5 6 70
1000
2000
3000
4000
5000
6000
7000
day
gene
rato
r com
mitm
ent [
MW
]
(a) Commitment of the generators
0 1 2 3 4 5 6 7
-30
-20
-10
0
10
20
30
day
PS
T se
tting
[deg
rees
]
PST 1PST 2
(b) PST settings
Figure 6.2: Unit commitment with PSTs for 1 week.
Application of Optimisation Methods 111
0 1 2 3 4 5 6 71.5
2
2.5
3
3.5
4
4.5
5x 10
4
day
co
st/
15
min
[E
uro
]
PSTs zero
PSTs optimal
Figure 6.3: Comparison of the operational cost with and without PSTs.
which is a very small number compared to the 0.6 Me fuel cost reduction obtainedwithout accounting for losses.
6.2 Border-flow control
Under DC load flow assumptions, the power flow in a line can be calculated as alinear function of PST settings, as was discussed in chapter 5. The flows on theinterconnectors on the border of two systems can therefore also be written as a set oflinear equations, as shown in equations (5.17) and (5.18). In this section, instead ofaiming for an optimal TTC, the PSTs are coordinated in a different way. The goal isto make the relative loadings of the tie-lines at a border equal. The desired values ofthe tie-line flows can not always be exactly achieved, but they can be approximatedby the Linear Least Squares method [110].
6.2.1 Linear Least Squares
Consider the following overdetermined3 system of linear equations:
Ax ≈ b (6.10)
3A system of linear equations is considered overdetermined if the number of equations exceedsthe number of unknowns.
112 Chapter 6
-20 -15 -10 -5 0 50
50
100
150
200
250
∆ losses [MW]
frequ
ency
Figure 6.4: Difference in system losses with and without PSTs.
In a Linear Least Squares (LLS) approach [40], the aim is to find the x for which||Ax− b||2 is minimal (hence the name):
minx
J = (Ax− b)T (Ax− b) (6.11)
It is fairly straightforward to verify that this minimisation problem has the fol-lowing solution:
x0 = (ATA)−1AT b (6.12)
If a border of two systems with all its interconnectors is considered, the activepower flows can be described:
P F = P F0 + Ξ ·∆α ≈ P F
ref (6.13)
where P F0 is the vector of power flows with the PSTs set to their preference position
(for example: all at 0), Ξ is a matrix of PSDFs and P Fref is a reference power flow
distribution to be approximated.In the application considered here, the relative loading of the interconnectors
should be made equal. So, the elements of P Fref are:
Application of Optimisation Methods 113
PFi =
n∑i=1
PFi
n∑i=1
PFi,r
· PFi,r (6.14)
where PFi is the power flow through one of the interconnectors and PFi,r the ratedpower of that line. If (6.13) is rearranged, it can be written as:
Ξ ·∆α ≈ P Fref − P F
0 (6.15)
which can be identified with (6.10). The change in phase shifter settings resulting inthe best approximation of an equal loading scenario can be found by using (6.12).
6.2.2 Single border control
Border types
In meshed systems, borders can be classified as one of the following types:
- Type 1 borders have less than l− 1 PSTs for l interconnectors. This means theflow distribution on the border can not be fully controlled.
- Type 2 borders have l− 1 PSTs for l interconnectors. In this case, the flow dis-tribution can be controlled within the limits of the PSTs, but the total transfercan not be fixed.
- Type 3 borders have a PST in every interconnector. In this way, the flowdistribution can be controlled and the total transfer over the border can be set,as long as the limits of the PST settings are not reached.
Dutch-German border
The Dutch-German border is of type 1 (see section 2.5). The LLS method is appliedstarting from four base cases, in which the settings of the PSTs on the Belgian-Dutchborder are chosen differently and the PSTs on the Dutch-German border are at 0 inthe base case. Both devices in Meeden are modelled as one, as they are operated inthat way. Furthermore, the PST of Gronau in Germany is taken into account, andthere are two other interconnectors without PSTs. For every case, the optimal ∆α iscalculated for Meeden and Gronau (Table 6.2 - “estimated optimum”).
The optimum PST settings and the corresponding line loadings L are calculatedby LLS and (5.17) and (5.18). The optimum settings are used in an AC calculationin PSS/E in order to verify the line loadings (“simulated optimum” in the table).
From the results, the following conclusions can be drawn:
114 Chapter 6
L1 L2 L3 L4
base 1 0.15 0.53 0.48 0.57estimated optimum 1 0.43 0.43 0.39 0.49simulated optimum 1 0.47 0.42 0.38 0.48
base 2 0.20 0.58 0.44 0.54estimated optimum 2 0.43 0.43 0.39 0.48simulated optimum 2 0.47 0.41 0.38 0.48
base 3 0.20 0.54 0.24 0.29estimated optimum 3 0.30 0.30 0.27 0.32simulated optimum 3 0.31 0.26 0.28 0.32
base 4 0.10 0.38 0.16 0.17estimated optimum 4 0.19 0.19 0.19 0.19simulated optimum 4 0.20 0.16 0.19 0.20
Table 6.2: Estimated and simulated line loadings at the Dutch-German border forfour base cases (relative values per interconnector).
- The LLS method is able to balance the border flows to a certain extent. How-ever, perfect balancing is not possible as all four line flows can not be fullycontrolled by only two PSTs.
- The estimated and simulated values differ slightly due to the limitations of theDC load flow approach.
Dutch-Belgian border
The Dutch-Belgian border is of type 3 when accounting for the new PSTs in Belgium(as discussed in section 2.5). An LLS approach is not strictly necessary as the desiredline flows can be obtained exactly, since there are 3 flows to control with 3 PSTs(i.e a square system of equations). In this case, it suffices to solve the set of linearequations. If the aim is an equal loading, the desired line powers are set to the desiredpower transfer divided by three, as the rated powers of the lines at the Dutch-Belgianborder are all equal.
A calculation is performed for four different cases, which differ by the settings ofthe PSTs at the Dutch-German border. In the base cases, the PSTs at the Dutch-Belgian border are at 0. In every optimised case the desired power transfer fromBelgium to the Netherlands is set to 300, 600 and 900 MW successively. For thesethree transfers, the line loadings should be 0.07, 0.14 and 0.21 respectively. Theestimated PST settings to obtain an equal loading as well as to establish the desiredtransfer are calculated and used in the simulation model. The resulting line loadingsfrom the simulation can be seen in Table 6.3. The results from the simulations can
Application of Optimisation Methods 115
case L1 L2 L3
Base case
1 0.12 -0.11 0.482 0.10 -0.10 0.483 0.15 -0.11 0.484 0.23 -0.09 0.52
Simulated optimum
1/0.07 0.07 0.09 0.021/0.14 0.15 0.17 0.101/0.21 0.22 0.24 0.182/0.07 0.07 0.09 0.022/0.14 0.15 0.16 0.102/0.21 0.23 0.24 0.193/0.07 0.06 0.09 0.023/0.14 0.14 0.17 0.103/0.21 0.22 0.24 0.184/0.07 0.05 0.09 0.024/0.14 0.13 0.16 0.104/0.21 0.21 0.24 0.18
Table 6.3: Estimated and simulated line loadings at the Dutch-Belgian border forfour base cases (relative values per interconnector).
differ sometimes considerably from the desired values, especially for line 3. The maincause is inaccuracy introduced by the limitations of the DC load flow approach.
6.2.3 Combined border control
Σ
Pset
∆α
∆α
Control PSTs
∆Psystem
P F
0
sign
signLLSK
s
LLSK
s
Figure 6.5: Block diagram for the combined border control.
In case of a system having two borders to other systems (such as the Dutch sys-tem), both border controllers can be combined in a single scheme (Fig. 6.5). The LLSblocks solve (6.13); in the specific case of the Dutch system, the controller for theDutch-Belgian border has an extra input for the total power exchange (Pset). Theinitial line powers are applied to the LLS functions, resulting in a ∆α vector for eachborder. The PST is modelled as a combination of a sign function, a gain and an inte-
116 Chapter 6
0 50 100 150 200-20
-15
-10
-5
0
5
10
15
time [s]
PS
T se
tting
[deg
rees
]
Meeden
Van Eyck 2
Van Eyck 1
Gronau
Zandvliet
(a) PST settings
0 50 100 150 200-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time [s]
line
lo
ad
ing
[p
u]
Van Eyck 2
Van Eyck 1
Zandvliet
Maasbracht-RommerskirchenGronau
Maasbracht-SiersdorfMeeden
(b) Line loadings
Figure 6.6: PST settings and relative line loadings for the Dutch case. The referencepower exchange between Belgium and the Netherlands is 300 MW at first, and stepsto 900 MW at t = 100 s.
grator. The integrator and sign function result in a linear ramping behaviour, whichapproximates the behaviour of the mechanical tap changer. The gain K determinesthe slope of the ramp (i.e. the speed of the tap changer).
The settings of the PSTs are fed into a system block. In this block, a multiplicationwith the PSDF matrix is performed.
Fig. 6.6 shows a simulation example for the Dutch system. At t = 0, all PSTsare at 0. The initial reference transfer value between Belgium and the Netherlandsis 300 MW. As the Belgian-Dutch interconnectors all have equal ratings, the flowson this border converge towards 100 MW (or 0.07 pu). At t = 100 s, the referencetransfer power is changed to 900 MW. Although the LLS calculations are virtuallyinstantaneous, a delay is introduced, caused by the mechanical tap changers of thePSTs.
6.2.4 Influence of grid topology changes
The LLS controller relies on the PSDF matrix. If the system topology changes, forinstance due to a line outage, this matrix must be updated. If the PSDF matrix in theLLS blocks is not updated, errors will occur. In the following sections, a contingency issimulated by changing the PSDF matrix in the system block in Fig. 6.5 and updatingthe PF0 vector to new values.
Internal contingencies
As an example of an internal contingency, a 380 kV line between Hengelo and Doet-inchem is taken out of service at t = 100 s. The PSDF matrix in the LLS blocks isinstantaneously updated (Fig. 6.7). After the contingency, the controller adapts the
Application of Optimisation Methods 117
0 50 100 150 200-20
-15
-10
-5
0
5
10
time [s]
PS
T se
tting
[deg
rees
]
Zandvliet
Gronau
Van Eyck 1
Van Eyck 2
Meeden
(a) PST settings
0 50 100 150 200-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time [s]
line
lo
ad
ing
[p
u]
Van Eyck 2
Van Eyck 1
Zandvliet
Maasbracht-Siersdorf
Maasbracht-Rommerskirchen
Gronau
Meeden
(b) Line loadings
Figure 6.7: PST settings and relative line loadings for the Dutch case. At t = 100 s,an internal line from Hengelo to Doetinchem is tripped and the LLS controllers areupdated instantaneously.
0 50 100 150 200-25
-20
-15
-10
-5
0
5
10
15
time [s]
PS
T se
tting
[deg
rees
]
Zandvliet
Gronau
Van Eyck 1
Van Eyck 2
Meeden
(a) PST settings
0 50 100 150 200-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time [s]
line
lo
ad
ing
[p
u]
Van Eyck 2
Zandvliet
Van Eyck 1
Gronau
Meeden
Maasbracht-Siersdorf
Maasbracht-Rommerskirchen
(b) Line loadings
Figure 6.8: PST settings and relative line loadings for the Dutch case. At t = 100 s,the line from Maasbracht to Rommerskirchen is tripped, and the LLS controllers areupdated instantaneously.
PST settings in order to return to the optimal line loadings. If the PSDF matrixin the LLS blocks is not updated in this case, there is no significant difference withthe situation when updated. The reason for this is the minor impact of an internalcontingency on the PSDF matrix.
Interconnector contingencies
If a contingency occurs on an interconnector, the change in the PSDF matrix is muchlarger. If the PSDF matrix in the LLS blocks is not updated when this kind of outageoccurs, the errors can be large. As an example, a simulation is performed in which
118 Chapter 6
0 50 100 150 200-20
-15
-10
-5
0
5
10
time [s]
PS
T se
tting
[deg
rees
]
Zandvliet
Gronau
Van Eyck 1
Van Eyck 2
Meeden
(a) PST settings
0 50 100 150 200-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time [s]
line
lo
ad
ing
[p
u]
Van Eyck 2
Van Eyck 1
Zandvliet
Gronau
Maasbracht-RommerskirchenMeeden
Maasbracht-Siersdorf
(b) Line loadings
Figure 6.9: PST settings and relative line loadings for the Dutch case. At t = 100 s,the line from Maasbracht to Rommerskirchen is tripped, but the LLS controllers arenot updated.
the line Maasbracht-Rommerskirchen is tripped at t = 100 s. From the simulationresults in Fig. 6.8, it can be seen that the outage is counteracted by PST controlactions. The Dutch-German border becomes a type 2 border after the contingency,so that the loading of the remaining lines can be made equal.
If the PSDF matrix in the LLS blocks is not updated, a large error is introduced(Fig. 6.9). The same setpoint for every line power is maintained, so the Meeden andGronau lines carry the same power as before. As the line to Rommerkirchen is outof service, a large part of its power flow is taken over by the line to Siersdorf.
6.2.5 Modelling of discrete behaviour
In the previous sections, it was assumed that a PST setting can be changed in acontinuous way. This is however not the case in practice due to the mechanical tapchanger. Its discrete behaviour can be simulated by adding a quantiser block afterthe integrator in the PST model. Some consequences are:
- As the PSTs can only be set to discrete positions, the power flows through thelines are also controlled in a discrete way and the loadings of the interconnectorscan not be made exactly equal in general.
- Small disturbances and noise can result in excessive switching of a PST betweentwo states. In order to avoid this problem, a dead band block can be insertedafter the LLS control block.
Simulation results can be seen in Fig. 6.10.
Application of Optimisation Methods 119
0 50 100 150 200-20
-15
-10
-5
0
5
10
15
time [s]
PS
T se
tting
[deg
rees
]
Zandvliet
Gronau
Van Eyck 1
Van Eyck 2
Meeden
(a) PST settings
0 50 100 150 200-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time [s]
line
lo
ad
ing
[p
u]
Van Eyck 2
Zandvliet
Maasbracht-Rommerskirchen
Maasbracht-Siersdorf
Gronau
Van Eyck 1
Meeden
(b) Line loadings
Figure 6.10: PST settings and relative line loadings for a test case with discretemodelling. The reference power exchange between Belgium and the Netherlands is300 MW at first, and steps to 900 MW at t = 100 s.
6.3 Congestion risk minimisation
In this example, the PST settings are optimised taking into account that the flows inthe system change stochastically in the operational period for which the settings arefixed.
6.3.1 Quantification of uncertainty in power system flows
The stochastic nature of the load and of some prime movers can lead to difficultiesto foresee or to forecast the system state [67]. It can, for example, be very hard topredict whether a certain transmission line will be congested, as not all conditionsare perfectly known. Uncertainty techniques offer a tool to deal with these problems[53].
In order to quantify the uncertainty of the forecasted power flows in the system,one should first assess the uncertainty in the system inputs and then use the powersystem steady-state model to estimate how this uncertainty is translated into variabil-ity of the power flows, represented by their probability density functions (PDFs) [14].Monte Carlo Simulation (MCS) offers a straightforward solution to this problem. Thegeneral formulation of the problem involves two modelling steps: first, one should as-sess the distributions of the system stochastic inputs (loads and stochastic generation)for the specific period of operation and then sample these distributions based on theprevailing stochastic dependence structure. For sampling routines appropriate for thepower system modelling, one can refer to [65].
The degree of uncertainty depends on the operational time frame taken into ac-count for the optimisation. This period may span from some hours to one day,depending on the operational strategy of the operator. The uncertainty stems from
120 Chapter 6
the uncertainty in forecasting of the system load and generation (for the case of windpower, one can refer to [73]). The settings of the PSTs may be actively used toalleviate the system congestion risk.
6.3.2 Impact of PSTs on system power flows
In chapter 5, it is shown that every line flow can be written as a linear functionof PST settings under DC load flow assumptions. Generalisation to all possiblesystem inputs means that the entire line power flow distribution is shifted by anamount ∆PF = Ξ∆α. Under DC load flow assumptions, this shifting causes nodistortion of the shape of the PDF. In Fig. 6.11, the shifting effect of the PSTson the line power flow PDF is illustrated. This principle can be used to reducesystem congestions. Congestion management is a very important topic nowadays,as international transports are increasing, and uncertainty in forecasts makes theproblem even harder to tackle. In this context, a method is proposed to find the PSTsettings minimising the overall congestion risk in the system [102].
p(C)
P F
P Fr
fP
fP0fP
P Fmax,0P F
max
∆P F =Ξ∆α
Figure 6.11: Active power flow PDF for zero phase shift (fP0) and for a phase shift∆α (fP ).
6.3.3 Minimisation of the overall system congestion risk
Suppose a line has a certain rating PFr . When the line power flow exceeds this rating,congestion occurs. The congestion risk p(C) corresponds to the probability that theline power flow exceeds PFr :
p(C) = p(|PF | ≥ PFr ) (6.16)
=∫ −PFrPFmin
fP dPF +
∫ PFmax
PFr
fP dPF (6.17)
Application of Optimisation Methods 121
where fP is the line power flow PDF and PFmax and PFmin are the maximum andminimum line ratings respectively. This probability is indicated in Fig. 6.11 as ashaded area (for the case where only the positive rated power is limiting).
Obviously, shifting the distribution has an impact on p(C). The relation betweenthe PST settings and p(C) is given by (6.17) where PF = PF0 + Ξ∆α. Substitutionyields:
p(C) =
−(PFr +Ξ∆α)∫PFmin,0
fP0dPF +
PFmax,0∫PFr −Ξ∆α
fP0dPF (6.18)
Thus, p(C) can also be calculated by using fP0 , which is the power flow distributionfor 0 phase shift, and altering the rated power.
Minimising p(C) for a single line is straightforward. It corresponds to the min-imisation of the area presented in Fig. 6.11. However, in a meshed system the powerflows are coupled. Therefore, a decrease in p(C) in one line (i.e. reduction of thepower flows of the line) can lead to an increase in another. An optimisation methodhas to be applied in order to obtain the PST settings minimising the risk p(C) forthe whole system. The objective function should therefore be proportional to p(C)of each line separately. The most basic implementation is thus to minimise the sumof all contingency risks:
minα
Nl∑i=1
pi(C) (6.19)
In order to evaluate equation (6.18), the PDF of the line power flow should beknown analytically. This is impossible as it follows from stochastic simulations (MCS)or other uncertainty analysis techniques. Therefore, conventional optimisation meth-ods can not be used. Instead, a simulation-based optimisation algorithm should beapplied. In this research, PSO has been chosen.
6.3.4 Case study
In this study case, the modified version of the 39-bus New England test system withtwo PSTs is used. The rated power PFr of all power system lines is 400 MW, whilethe rated values of the system loads can be found in appendix C.
The modelling procedure involves three steps: first the MCS input sampling dataare generated in Matlab. These data are written to a text file and read by a Pythonscript. An MCS is performed based on these input data. The MCS results are usedin a PSO, resulting in optimal settings for the PSTs. A verification is performed inPSS/E.
In this case study, the uncertainty in the system is related to the stochastic be-haviour of the system loads for the operational period under consideration. For eachsystem load, a forecast is available, corresponding to point predictions for specific
122 Chapter 6
150 200 250 300 350 400 450 500 550 6000
50
100
150
200
250
300
load bus 4 [MW]
frequ
ency
Figure 6.12: Load distribution for 4-TF segmentation (10000-sample MCS).
time frames (TFs) spanning this period [66]. The forecast error may then be mod-elled by superimposing random noise variables to the point forecast [16]. Thus, a loadin a specific TF is modelled as a normal random variable (r.v.) with a specific meanand standard deviation given by the point prediction and assessment of uncertainty.In each TF, the forecast error between different loads in the system is considered tobe uncorrelated. Thus, in each TF, all system loads are modelled as independent r.v.The distribution for the whole period under consideration is obtained as a mixtureof these normals. This mixture can be sampled using an independent uniform r.v.UTF as TF-indicator. In particular, based on the relative duration of each TF, eachsample drawn from UTF is matched to a specific TF. According to the indicator, asample is drawn from the respective normal distribution belonging to the specific TFfor the specific load [65].
In this specific study case, 4 TFs are used. The mean value for each TF corre-sponds to a percentage of the nominal value of each load PLn , while the standarddeviation is obtained as a percentage of the load point forecast. In Table 6.4 themean values/standard deviation settings for this TF segmentation is presented, to-gether with the time-ratio for the relative duration of each TF. According to thistime-ratio, for UTF > 0.8, TF4 is chosen (high-load TF), thus a sample is drawnfrom a normal r.v. with mean value µTF4 = PLn and standard deviation 0.03 · µTF4,for 0.5 < UTF < 0.8, TF3 is chosen, thus a sample is drawn from a normal r.v. withmean value µTF3 = 0.85 × PLn and standard deviation 0.1 · µTF3, etc. In Fig. 6.12,
Application of Optimisation Methods 123
the resulting distribution for the load on bus 4 based on a 10000-sample MCS and a4-TF segmentation is presented.
As mentioned, the same TF indicator is used for all system loads. This imposes ahigh stochastic dependence between the load distributions. In Fig. 6.13, the scatterdiagram for a 10000-sample MCS for the modelling of two loads in the system ispresented. Although in each TF the r.v. are independent, the resulting distributionsare highly correlated. In this particular case, the rank correlation obtained at theoutput samples is 0.89.
150 200 250 300 350 400 450 500 550 600200
250
300
350
400
450
500
550
600
650
700
load bus 4 [MW]
load
bus
20
[MW
]
Figure 6.13: Scatter diagram between two system loads for 4-TF segmentation (10000-sample MCS).
The dispatch of the generating units in the system should be performed for eachsystem state, defined by the MCS sampling of the system loads. The rated capacityof the units is presented in appendix C. In this specific study case, the units in the
Time Ratio Mean load value St. Deviation(% PLn) (% mean load)
TF1 0.2 50 6TF2 0.3 65 10TF3 0.3 85 10TF4 0.2 100 3
Table 6.4: TF settings for 4-TF load modelling for the New England test system.
124 Chapter 6
-400 -300 -200 -100 0 1000
50
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150
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350
P [MW]
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(a) line 14-15 (PSDF=-7.3 MW/degree)
-100 -50 0 50 100 150 200 2500
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P [MW]
frequ
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zero setting10 degrees
(b) line 17-27 (PSDF=6.9 MW/degree)
Figure 6.14: Histograms of two lines with zero phase shift and with 10 degrees phaseshift for PST1.
system are considered to be thermal units of the same type. For each sample, thepower generation of the plants (except for the slack bus) is pro-rata.
In order to demonstrate the impact of a PST on the line power flow distribution,these distributions are plotted for two lines (14-15 and 17-27), with PST2 (in line17-18) at 0 and PST1 (in line 4-14) once at 0 and once at 10. The PSDFs of lines14-15 and 17-27 regarding PST1 are -7.3 MW/ and +6.9 MW/ respectively. Thismeans that at a setting of 10, the line power flow distribution of line 14-15 shouldbe shifted 73 MW to the left and that of line 17-27 69 MW to the right. This isconfirmed by the simulation results obtained with an AC load flow MCS. The resultsare presented in Fig. 6.14.
By performing an MCS, the overloading probability (i.e. congestion risk) for allthe system lines can be calculated. The system operation involves actions to reducethe overall risk for the period under consideration. Traditional actions would be toperform generation redispatch in order to alleviate the loading of critical lines. Inthis case study it is shown how PSTs can be used to redistribute the power flows inorder to come to a more favourable overall situation and to avoid or limit redispatch.
In Fig. 6.15, the congestion risks for the different system lines are presented withand without optimal PST actions.
For the “zero settings” case, the MCS yields a high congestion risk for several lines.This risk reaches almost 80% for some lines (Fig. 6.15). In Fig. 6.16, the boxplot4
for the power flows in the lines is presented. It can be seen that although in severallines the power flows exceed the line limits, other system lines are underloaded. The
4In descriptive statistics, a boxplot is a convenient way of graphically depicting groups of nu-merical data through their five-number summaries (the smallest observation, lower quartile (Q1),median, upper quartile (Q3), and largest observation). The box itself is bounded by Q1 and Q3(with a line inside to indicate the median), and on both sides whiskers extend to 1.5 times theinter-quartile distance. All data points beyond the whiskers are considered to be outliers.
Application of Optimisation Methods 125
0
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1p(
C) [
pu]
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Zero SettingsBoth PSTs Optimal
Figure 6.15: Overloading probabilities in the New England system for different lines.
proposed methodology actually allows to use this spare capacity by shifting the loadbetween lines.
The methodology to alleviate the overall system congestion risk is applied usingboth PSTs. The settings are optimised by PSO (α1 = −17.6, α2 = −13.6), anda new MCS is performed in order to calculate the new overloading probabilities.From Fig. 6.15, the reduction of the system congestion risk can be seen. In Fig. 6.16and 6.17, the boxplot for the system power flows are presented. It can be seen fromthe shape and positions of the boxplots that the PSTs lead to a shift of the systempower flow distributions, without changing their shape, in accordance to the theorypresented before. Special cases of improvement are lines 2-3, 5-6, 6-11 and 10-11.However, the overload in lines 16-19 and 21-22 remains the same. A glance at thePSDF values for these lines shows that the PSTs have no influence on them. Thereason for this is the fact that they are part of a subgrid with a single connectionpoint at bus 16. Therefore, the power flows inside the subgrid can not be altered bythe installed PSTs.
6.4 Reinforcement study on the Dutch-German border
The Dutch-German border is a typical example of a border with congestion problemsas the capacity requested by the market is often higher than available. This led to thediscussion on a new overhead line on the Dutch-German border, between the TenneTTSO and RWE control areas. A joint study has been performed by TenneT and RWETransportnetz Strom, considering multiple options under different scenarios [93]. Thescenarios can be split into two categories: the so-called “West” scenarios incorporate
126 Chapter 6
−800
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Figure 6.16: Boxplot of the line power flows for zero phase shifter settings. The linelimits are 400 and -400MW and indicated by the horizontal lines.
−800
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er [M
W]
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Figure 6.17: Boxplot of the line power flows for optimal phase shifter settings. Theline limits are 400 and -400MW and indicated by the horizontal lines.
Application of Optimisation Methods 127
NL D
HGL
DTC
DOD
BMR
MBT DUE
NIE
KU
GRO
HA
R4R2
R1
R3
Figure 6.18: Considered reinforcements at the Dutch-German border.
assumptions resulting in large flows from the north to the south in the western partof UCTE. The “East” scenarios are characterised by large flows from the north tothe south in the more eastern zones of UCTE. The outcome of the study showedthat the best techo-economical option would be to build a new overhead line fromNiederrhein (Germany) to Doetinchem (the Netherlands), with an estimated cost ofabout 70 Me [95]. A memorandum of understanding was signed by both TSOs inDecember 2006 [94].
In this section, TTC calculations are performed for the different reinforcements onthe Dutch-German border considered by TenneT and RWE [103]. These alternativesare compared here as an application of the TTC optimisation methods developed inthe previous chapters. The reinforcements taken into account are:
R1 A double 400 kV line from Niederrhein to Boxmeer. Each of these lines has arating of 1790 MVA.
R2 A double 400 kV line from Niederrhein to Doetinchem (1790 MVA each).
R3 A double 400 kV line from Duelken to Maasbracht (1790 MVA each).
R4 A PST inserted in the line Kusenhorst-Gronau, as well as an upgrade of theexisting PST at Gronau. The rating both PSTs is set to 1425 MVA and themaximum setting is 30.
R5 A combination of R4 and R2.
The different options are shown schematically in Fig. 6.18.
128 Chapter 6
R1 R2 R3 R4 R50
500
1000
1500
2000
2500
3000
3500
Case
TTC
[MW
]
PSOAnalytical
Figure 6.19: Relative (to the base case) TTC increase for the different reinforcementoptions.
Case Method Mee 1 Mee 2 Gro Zv VE1 VE2 Gro2BC PSO -12,6 -12,6 -7,8 23,2 25,0 25,0
Analytical -12,6 -12,0 -7,7 19,0 25,0 22,2R1 PSO -18,2 -18,2 -9,2 8,6 25,0 19,1
Analytical -17,8 -17,2 -8,8 5,7 25,0 16,9R2 PSO -16,1 -16,1 -4,4 5,7 25,0 17,0
Analytical -15,5 -15,0 -3,9 2,6 25,0 14,9R3 PSO -14,2 -14,2 -8,1 18,9 25,0 24,4
Analytical -14,0 -13,3 -7,9 15,3 25,0 21,7R4 PSO -14,8 -14,8 -5,6 21,0 25,0 24,4 -0,4
Analytical -15,0 -14,4 -5,6 16,8 25,0 21,0 -0,7R5 PSO -15,8 -15,8 0,4 5,1 25,0 16,7 6,7
Analytical -15,3 -14,7 0,1 2,0 25,0 14,6 6,8
Table 6.5: Optimal settings for the different reinforcement options.
For each of the 5 reinforcements, the maximum TTC is calculated for the Nether-lands opposed to Belgium and Germany for one of the scenarios of the study performedby the TSOs. In order to maximise the TTC, all PSTs in the Benelux as well as theGronau PST are used. For R4 and R5, the second PST in Gronau is also included inthe optimisation. The calculation is performed by using Particle Swarm Optimisation(PSO) and by analytical calculations (with DC load flow approximation).
The results from the optimisation are shown in the bar graph of Fig. 6.19, wherethe relative increase in TTC is given compared to the base case without reinforcement.The deviation between both calculation methods is very small. It can be caused by
Application of Optimisation Methods 129
a non-perfect convergence of the PSO algorithm and/or by shortcomings of the DCload flow approximation.
The figure shows that the line from Duelken to Maasbracht (R3) does not improvethe TTC much. The line Niederrhein-Doetinchem would be a much better investment,since a gain of almost 3 GW can be obtained. An extra PST in Gronau does notcontribute a lot as well, but in combination with a new line from Niederrhein toDoetinchem, a significant gain is obtained. However, the cost of such a device ismuch higher than the benefits in this case. The optimal settings found by PSO andthe analytical method are listed in Table 6.5.
6.5 Summary
In this chapter, the methodologies for optimal coordination of phase shifters, devel-oped in this work, are applied on a variety of problems.
In a classical unit commitment task, the goal is to distribute the demand amongstthe generating facilities so that the total operational cost is minimal. In case ofcongestion in the network, more expensive peak units may need to be dispatched,resulting in an increased overall operational cost. In a DC load flow approach, PSTscan be easily included in the OPF formulation, accounting for the congestion handling.Optimal use of the PSTs can result in a lower total operational cost, as they allow aredirection of the power flows, possibly avoiding the dispatch of more expensive peakunits.
Under DC load flow assumptions, the power flow on a line can be calculated as alinear function of PST settings. The flows on the interconnectors of a border of twosystems can be written as a set of linear equations. Border flow control is the secondexample of using PSTs in the system. Depending on the number of PSTs relative tothe number of interconnectors, the desired power flows on the borders can be attainedexactly or approximately. In the latter case, the Linear Least Squares (LLS) methodis applied to calculate the optimal PST settings; in the former case the set of linearequations can be solved analytically. A case study shows that an LLS controller canbe implemented at each border, and that each border can be optimised separately.
In a third example, the PST settings are optimised taking into account that theflows in the system change stochastically in the operational period for which thesettings are fixed. In order to quantify the uncertainty of the power flows in thesystem, one should first assess the uncertainty in the system inputs and then estimatehow this uncertainty is translated into variability in the power flows. MCS offers astraightforward solution to this problem, resulting in a PDF of the power flow forevery line. The risk of congestion can be deduced from these distributions for everysingle line. If the concept of the PSDFs is generalised to all possible system states,it can be seen that a PST shifts the entire PDF of a line flow by a certain amount,depending on its setting and the PSDF. A single PST definitely leads to a reduction
130 Chapter 6
of the power flow and de-congestion of the line where it is installed. However, it maylead to overloading of another line. In a meshed grid the problem becomes complexand the final objective of the optimisation should be to minimise the overall risk ofcongestion in all the lines. This concept is shown for a test system with PSO as theoptimisation algorithm.
In 2006, the Dutch TSO (TenneT) and one of the German TSOs (RWE) decidedto reinforce the border between both countries; the decision was taken to build aninterconnector between Niederrhein and Doetinchem. As a last application, TTCcalculations were performed for the different reinforcement options. For every option,both Particle Swarm Optimisation (PSO) and the analytical DC method were appliedto find the maximum attainable TTC, and the values were compared. The case-studyindicated that the deviation between PSO and the analytical method is small, andthat the results provide valuable information for the reinforcement strategy.
CHAPTER 7
Conclusions and Recommendations
7.1 Conclusions
In the recent past, power systems have undergone a major transformation as a numberof developments have taken place, leading to a change in the way grids are operated.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), are installed at an increasing number of locations.
It has been shown in this thesis that a lack of coordination between multiplePSTs can lead to inefficient use of the existing infrastructure, and also to operationalproblems in the grid, such as loopflows and congestion. On the other hand, goodcoordination results in a balanced flow distribution over the different interconnectorsbetween areas. As an indicator of how well PSTs are coordinated, the Total TransferCapacity (TTC) has been evaluated.
The number of possible combinations of PST settings can grow very large. Anexhaustive enumeration of all possible TTC values is highly impractical as a meansto find the optimum, as every additional PST adds an extra dimension to the searchspace containing the optimum TTC. Monte Carlo Simulation offers a way to estimatethe best- and worst-case scenarios in terms of TTC values with a limited amount ofcalculation time. Multistage Monte Carlo Simulation offers the possibility to zoom
132 Chapter 7
into an area of interest, thereby providing more detailed information about e.g. thebest-case scenario regarding the TTC, at the cost of a relatively large computationaleffort.
If the TTC is adopted as the indicator for PST coordination, it is very hard towrite the objective function of the optimisation problem as a function of the PSTsettings, also due to the required contingency analysis. Metaheuristic optimisation,which is based on evaluations of the objective function through simulations, offers asolution for this problem. There are several algorithms to choose from, but not allwork equally well. It has been shown that Particle Swarm Optimisation performs well,while Evolutionary Programming and Evolution Strategies are considerably slower.
The problem of PST coordination can be simplified by adopting DC load flowassumptions, leading to linearised equations, containing phase shifter distributionfactors (PSDFs). In this way the TTC can be expressed as a piecewise linear function,and the PST settings that result in the maximum TTC can be found by solving aLinear Programming problem. The method offers a very fast optimisation, at thecost of inaccuracy caused by the approximations.
It is shown that the concept of PSDFs is very useful in some applications. Thelinearised power flow equations allow the use of a Linear Least Squares methodologyto balance border flows and also enable the incorporation of PSTs in a unit dispatchfunction. Furthermore, it has been proven that in stochastic simulations, the powerflow histogram can be shifted by a predictable amount, based on PSDFs. Finally,both the linearised and metaheuristic optimisation can be adopted for the assessmentof possible grid reinforcements, as they indicate the maximum attainable TTC valuefor every option.
7.2 Recommendations
7.2.1 Economic Aspect
This thesis aims at a purely technical optimisation, while economic aspects are nottaken into account. An important research question for future work is how to incor-porate PSTs in a market environment. It might be possible to consider a power flowcontroller as a separate economic entity that can generate cash flow. This option isdiscussed in [81].
In the framework of flow based market coupling (FMC), where coupling of mar-kets is combined with the consideration of physical flow paths, the application ofPSTs becomes very attractive. FMC maximises social welfare, while respecting theconstraints of the grid. PSTs offer extra control variables to realise a higher socialwelfare in a region [57].
Conclusions and Recommendations 133
7.2.2 Multi- or Interarea Coordination
In this thesis, the focus is on the optimisation for one or two areas. Other neighbouringareas can experience negative consequences due to this local optimisation, e.g. in theform of congestions. A challenge for future research is to find a way to obtain a globaloptimisation or to find a way to make local optimisations co-exist. Promising stepsin this direction are described in [8] and [56].
7.2.3 Dynamic Behaviour
In section 2.6.2 it was shown that, even for a small test system, it is very difficultto formulate general conclusions about the influence of PSTs on transient stability.Simulations can be performed in the time domain or by applying energy functions,as shown in [38], but such observations do not give insight in the underlying mech-anisms. The same holds for small-signal stability: eigenvalue analysis does not offerfundamental knowledge about the relation between PST settings and stability. Aprofound study is needed to clarify these matters, ideally by looking for analyticalexpressions that describe the fundamental relationships.
APPENDIX A
Abbreviations, Symbols and Operators
A.1 List of abbreviations
ACO - Ant Colony OptimisationBCE - base case exchangeCDF - cumulative density functionDACF - day-ahead congestion forecastDFM - distribution factor matrixEP - Evolutionary ProgrammingES - Evolution StrategiesETSO - organisation of European Transmission System OperatorsFACTS - flexible AC transmission systemsFMC - flow based market couplingGA - Genetic AlgorithmGRASP - Greedy Randomised Adaptive Search ProcedureHVDC - high voltage direct currentIGBT - insulated gate bipolar transistorILS - Iterated Local SearchLLS - Linear Least Squares
136 Appendix A
LP - Linear ProgrammingMCS - Monte Carlo SimulationMEP - Meta-Evolutionary ProgrammingMMCS - Multistage Monte Carlo SimulationMRSD - maximum rotor speed deviationNTC - Net Transfer CapacityPDF - probability density functionPGA - Penalised Greedy AlgorithmPSAT - Power Systems Analysis ToolboxPSDF - phase shifter distribution factorPS - power shiftPSO - Particle Swarm OptimisationPST - phase shifting transformerPTDF - power transfer distribution factorPWM - pulse width modulationR - reinforcementRCL - restricted candidate listr.v. - random variableSA - Simulated AnnealingSGA - Simple Greedy AlgorithmSSSC - solid state series compensatorSV - search volumeTCR - thyristor controlled reactorTCSC - thyristor controlled series capacitorTF - time frameTRM - Transmission Reliability MarginTS - Tabu SearchTSO - transmission system operatorTSP - travelling salesman problemTTC - Total Transfer CapacityUPFC - unified power flow controllerVSC - voltage-source converterZL - zoomlevel
A.2 List of symbols
a - coefficient of linear equation [MW/]A - set of arcsA - coefficient matrix in Linear Programmingb - coefficient vector in Linear ProgrammingB - susceptance [S]
Abbreviations, Symbols and Operators 137
c - arc cost [MW], inverse admittance matrix element, or marginal cost [e]c - coefficient vector in Linear ProgrammingC - capacitance [F], number of contingencies, total cost [e], or congestionC - confidence matrix or cost vectorcon - outaged lined - dimensionD - directed graphf - function, fitness value, or participation factorF - fitness function or objective functionH - inertia matrix or distribution factor matrixI - current [A] or integralI - unity matrixk - penalty factor [MW] or constant term of linear equation [MW]K - total cost [MW] or gain coefficientl - number of interconnectorsL - inductance [H], input terminal, constraint selector, or line loadinglim - limiting linemax - maximum valuemin - minimum valueN - number of samples or normal distributionp - probabilityp - optimal position vectorP - active power [W]q - number of contestants in tournament selection processQ - reactive power [VAr] or quartileR - resistance [Ω]R - random matrixs - sensitivity [MW/MW] or solutionS - complex power [VA], secondary terminal, or “hit” region in MCSSV - search volumet - winding ratio, auxiliary variable, or time [s]T - temperature [C]u - maximum of piecewise linear functionU - time frame selectorv - vertex or velocity [/iteration]V - voltage [V] or set of verticesw - penalty factor [MW] or weighting factorx - state or position []X - reactance [Ω] or output variableY - admittance [S]Y - admittance matrix
138 Appendix A
Yα - admittance correction vectorz - vector of variables in Linear ProgrammingZ - impedance [Ω]Ni - neighbourhoodP∗ - Pareto optimal setPF∗ - Pareto frontα - phase shift angle [ or rad], fraction of RCL, or weighting factorβ - weighting factor or tuning parameter in EPγ - tuning parameter in EPδ - voltage angle [ or rad]ε - fraction of power exchangeη - attractiveness of an arcθ - estimator in Monte Carlo integration or section of roulette wheel [rad]λ - penalty factor selector or number of child individualsµ - number of parent individuals or mean valueξ - phase shifter distribution factor [MW/]Ξ - phase shifter distribution factor matrixρ - evaporation of a trace or range factorσ - standard deviation or mutation strength []τ - pheromone concentration on an arc or learning rateϕ - load angle [ or rad]χ - velocity constriction factorω - angular speed [rad/s]Ω - rectangle in Monte Carlo integration
A.3 List of subscripts
0 - no-load, base case, initial, zero phase shift, or preference positionavg - averagebase - base valueG - generationH - hiti - interconnectorinj - injectedj - limiting linel - under loading conditions or related to a certain lineL - line, input terminal, or loadm - monitored linemax - maximumn - non-monitored line or nominalp - phase shifter index
Abbreviations, Symbols and Operators 139
pst - phase shifting transformerpu - per unitP - power flowr - rated or receiving endref - references - sending endS - secondarySC - short circuitT - totalTF - time framez - zoomlevelα - related to a certain phase shifterθ - regarding estimator in Monte Carlo integration
A.4 List of superscripts
B - bus injectionc - for a certain contingencyF - line flowG - generationL - loadm - lower limitM - upper limitR - rated+ - positive direction− - negative directionˆ - approximated
A.5 List of operators
∗ - complex conjugate∀ - for all|.| - absolute value, magnitude, or norm∃ - there exists∃! - there exists exactly one∝ - proportional to≺ - Pareto dominance∧ - and¬ - notsgn(.) - signum function
APPENDIX B
Mathematical Background
B.1 Big O notation
In mathematics, the big O notation, which is an example of a so-called Landausymbol, describes an asymptotic upper bound for the magnitude of a function interms of another, often simpler function. The notation is as follows:
f(x) = O(g(x)) (B.1)
meaning that f(x) is of the order g(x). However, this does not mean that f(x) isalways smaller than g(x). Mathematically, f(x) is of the order g(x) if for a positiveconstant c and for large values of x, c · g(x) is an upper bound for f(x) [86]:
∃c, k > 0 : 0 ≤ f(x) ≤ c · g(x),∀x ≥ k (B.2)
As an example, consider the following polynomial function f(x):
f(x) =12x3 + 4x2 + 5 (B.3)
The question is of which order this function is.
- f(x) = O(x4) as x4 > 12x
3 + 4x2 + 5 for large x
142 Appendix B
- f(x) = O(x3) as 10x3 > 12x
3 + 4x2 + 5 for large x
- f(x) 6= O(x2) as c · x2 < 12x
3 + 4x2 + 5 for large x
Although it is true that f(x) is both of order x4 and x3, the latter statement is morestrict [51].
The big O notation is used very frequently in algorithm analysis and especially ingraph theory, as it indicates an upper bound for the worst-case computation time interms of the size of the problem. Sets of problems with the same order of complexitycan be grouped in complexity classes [64]. For example, problems related to breakingan n-digit decimal code have a computation time of O(10n), i.e. they are a memberof the exponential time complexity class.
B.2 Linear Programming
Linear Programming (LP) is the optimisation of a linear objective function, subjectto linear constraints [11]. Linear problems can be written in a number a ways, butthe most common one is the so-called standard form, consisting of three parts:
- The linear function to be maximised is of the form
cTx (B.4)
- All constraints can be written as:
Ax ≤ b (B.5)
- The variables are all non-negative:
x ≥ 0 (B.6)
This formulation can be adapted to deal with minimisation problems, negative vari-ables and other non-standard cases.
As an example, suppose that a farmer has an area of land L, and he can choose toplant it with wheat, barley, or a combination of both. Limited amounts of fertiliserF and insecticide P are available, and each of the crops requires different amountsper unit area, (F1,P1) and (F2,P2). If S1 and S2 are the selling prices per unit areaof wheat and barley respectively, the goal is to maximise the total revenue R fromselling the crops. This problem can be written as an LP problem, where x1 and x2
are the areas planted with wheat and barley respectively [4]:
max R = S1x1 + S2x2 (B.7)
subject to:x1 + x2 ≤ L (B.8)
Mathematical Background 143
F1x1 + F2x2 ≤ F (B.9)
P1x1 + P2x2 ≤ P (B.10)
x1 ≥ 0 x2 ≥ 0 (B.11)
These equations are graphically represented in the x1−x2 plane in Fig. B.1. Theconstraints determine a polygon marking the feasible region (indicated as the shadedregion in the figure) and the objective function can be drawn as a group of parallellines, one for each value of R. In the example drawing, the optimum is indicated asa bold dot where constraints 1 and 3 meet.
4
3R
4
123
x1
x2
S1x1 + S2x2 = RP1x1 + P2x2 = PF1x1 + F2x2 = Fx1 + x2 = L
12
Figure B.1: Visual representation of the example problem.
In general, the constraints of an LP problem constitute a polytope, and the ob-jective function reaches its optimum in a vertex of this polytope. This principle isused in the so-called simplex algorithm, where an admissible solution is constructedin a vertex of the polytope and the optimum is reached by walking along the edgesto vertices with successively higher values of the objective function. As linearityimplies convexity, a local optimum is automatically a global optimum, due to theKarush-Kuhn-Tucker conditions [40]. A more modern class of solution algorithms arethe interior-point methods, which move through the interior of the feasible region,instead of following the edges [120].
APPENDIX C
New England Test System Data
The data in this appendix are adopted from [63].
Table C.1: Bus Data of the New England 39 Bus Test System.
Bus Volts Load Load Gen(nr.) (pu) (MW) (MVAr) (MW)
1 - 0.0 0.0 -2 - 0.0 0.0 -3 - 322.0 2.4 -4 - 500.0 184.0 -5 - 0.0 0.0 -6 - 0.0 0.0 -7 - 233.8 84.0 -8 - 522.0 176.0 -9 - 0.0 0.0 -10 - 0.0 0.0 -11 - 0.0 0.0 -12 - 7.5 88.0 -
Continued on next page
146 Appendix C
Table C.1 – continued from previous page
Bus Volts Load Load Gen(nr.) (pu) (MW) (MVAr) (MW)13 - 0.0 0.0 -14 - 0.0 0.0 -15 - 320.0 153.0 -16 - 329.0 32.3 -17 - 0.0 0.0 -18 - 158.0 30.0 -19 - 0.0 0.0 -20 - 628.0 103.0 -21 - 274.0 115.0 -22 - 0.0 0.0 -23 - 247.5 84.6 -24 - 308.6 -92.2 -25 - 224.0 47.2 -26 - 139.0 17.0 -27 - 281.0 75.5 -28 - 206.0 27.6 -29 - 283.5 26.9 -30 1.0475 0.0 0.0 25031 0.982 9.2 4.6 -32 0.9831 0.0 0.0 65033 0.9972 0.0 0.0 63234 1.0123 0.0 0.0 50835 1.0493 0.0 0.0 65036 1.0635 0.0 0.0 56037 1.0278 0.0 0.0 54038 1.0265 0.0 0.0 83039 1.03 1104.0 250.0 1000
Table C.2: Line Data of the New England 39 Bus Test System.
Line Data Resistance Reactance Susceptance Transformer TapBus Bus (pu) (pu) (pu) Magnitude Angle
1 2 0.0035 0.0411 0.6987 0 01 39 0.0010 0.0250 0.7500 0 02 3 0.0013 0.0151 0.2572 0 0
Continued on next page
New England Test System Data 147
Table C.2 – continued from previous page
Line Data Resistance Reactance Susceptance Transformer TapBus Bus (pu) (pu) (pu) Magnitude Angle
2 25 0.0070 0.0086 0.1460 0 03 4 0.0013 0.0213 0.2214 0 03 18 0.0011 0.0133 0.2138 0 04 5 0.0008 0.0128 0.1342 0 04 14 0.0008 0.0129 0.1382 0 05 6 0.0002 0.0026 0.0434 0 05 8 0.0008 0.0112 0.1476 0 06 7 0.0006 0.0092 0.1130 0 06 11 0.0007 0.0082 0.1389 0 07 8 0.0004 0.0046 0.0780 0 08 9 0.0023 0.0363 0.3804 0 09 39 0.0010 0.0250 1.2000 0 010 11 0.0004 0.0043 0.0729 0 010 13 0.0004 0.0043 0.0729 0 013 14 0.0009 0.0101 0.1723 0 014 15 0.0018 0.0217 0.3660 0 015 16 0.0009 0.0094 0.1710 0 016 17 0.0007 0.0089 0.1342 0 016 19 0.0016 0.0195 0.3040 0 016 21 0.0008 0.0135 0.2548 0 016 24 0.0003 0.0059 0.0680 0 017 18 0.0007 0.0082 0.1319 0 017 27 0.0013 0.0173 0.3216 0 021 22 0.0008 0.0140 0.2565 0 022 23 0.0006 0.0096 0.1846 0 023 24 0.0022 0.0350 0.3610 0 025 26 0.0032 0.0323 0.5130 0 026 27 0.0014 0.0147 0.2396 0 026 28 0.0043 0.0474 0.7802 0 026 29 0.0057 0.0625 1.0290 0 028 29 0.0014 0.0151 0.2490 0 02 30 0.0000 0.0181 0.0000 1.025 06 31 0.0000 0.0250 0.0000 1.07 010 32 0.0000 0.0200 0.0000 1.07 012 11 0.0016 0.0435 0.0000 1.006 012 13 0.0016 0.0435 0.0000 1.006 019 20 0.0007 0.0138 0.0000 1.06 0
Continued on next page
148 Appendix C
Table C.2 – continued from previous page
Line Data Resistance Reactance Susceptance Transformer TapBus Bus (pu) (pu) (pu) Magnitude Angle19 33 0.0007 0.0142 0.0000 1.07 020 34 0.0009 0.0180 0.0000 1.009 022 35 0.0000 0.0143 0.0000 1.025 023 36 0.0005 0.0272 0.0000 1 025 37 0.0006 0.0232 0.0000 1.025 029 38 0.0008 0.0156 0.0000 1.025 0
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Publications
Journal publications
1. J. Verboomen, D. Van Hertem, P. H. Schavemaker, F.J.C.M.Spaan, J.-M. Delince,W. L. Kling, and R. Belmans, “Phase shifter coordination for optimal trans-mission capacity using particle swarm optimization,” Electric Power SystemsResearch (EPSR), vol. 78, no. 9, pp. 1648–1653, September 2008.
2. J. Verboomen, D. Van Hertem, P. H. Schavemaker, W. L. Kling, and R. Bel-mans, “Analytical approach to grid operation with phase shifting transformers,”IEEE Transactions on Power Systems, vol. 23, no. 1, pp. 41–46, February 2008.
3. J. Verboomen, D. Van Hertem, P. H. Schavemaker, F.J.C.M.Spaan, J.-M. Delince,W. L. Kling, and R. Belmans, “Monte carlo simulation techniques for optimi-sation of phase shifter settings,” European Transactions on Electrical Power(ETEP), vol. 17, no. 3, pp. 285–296, November 2006.
Conference publications
1. J. Verboomen, F. J. C. M. Spaan, P. H. Schavemaker, and W. L. Kling, “Methodfor calculating maximum total transfer capacity by optimising phase shiftingtransformer settings,” in CIGRE Session 2008 - C1-111, Paris, France, 2008, 7pages.
2. J. Verboomen, G. Papaefthymiou, W. L. Kling, and L. van der Sluis, “Useof phase shifting transformers for minimising congestion risk,” in ProbabilisticMethods applied to Power Systems (PMAPS) 2008, Rincon, Puerto Rico, 2008,6 pages.
3. J. Verboomen, D. Van Hertem, P. H. Schavemaker, W. L. Kling, and R. Bel-mans, “Indication of safe transition paths of phase shifter settings by greedy al-
156 Publications
gorithms,” in Universities Power Engineering Conference (UPEC) 2007, Brighton,UK, September 2007, 6 pages.
4. A. M. van Voorden, L. M. Ramirez Elizondo, G. C. Paap, J. Verboomen, andL. van der Sluis, “The application of super capacitors to relieve battery-storagesystems in autonomous renewable energy systems,” in Powertech 2007, Lau-sanne, Switzerland, July 2007, 6 pages.
5. G. Papaefthymiou, J. Verboomen, P. H. Schavemaker, and L. van der Sluis,“Estimation of power system variability due to wind power,” in Powertech 2007,Lausanne, Switzerland, July 2007, 6 pages.
6. J. Verboomen, D. Van Hertem, P. H. Schavemaker, W. L. Kling, and R. Bel-mans, “Border flow control by means of phase shifting transformers,” in Pow-ertech 2007, Lausanne, Switzerland, July 2007, 6 pages.
7. D. Van Hertem, J. Verboomen, S. Cole, W. L. Kling, and R. Belmans, “Influenceof phase shifting transformers and hvdc on power system losses,” in PowerEngineering Society General Meeting 2007, Tampa, USA, June 2007, 8 pages.
8. J. Verboomen, D. Van Hertem, P. H. Schavemaker, W. L. Kling, and R. Bel-mans, “Coordination of phase shifters by means of multi-objective optimisa-tion,” in Universities Power Engineering Conference (UPEC) 2006, Newcastleupon Tyne, UK, September 2006, pp. 432–436.
9. J. Verboomen, D. Van Hertem, P. H. Schavemaker, W. L. Kling, and R. Bel-mans, “Optimal phase shifter coordination: a multidimensional problem,” inIASTED Artificial Intelligence and Soft Computing (ASC) 2006, Palma de Mal-lorca, Spain, August 2006, 6 pages.
10. G. Papaefthymiou, J. Verboomen, P. H. Schavemaker, and L. van der Sluis,“Impact of stochastic generation in power systems contingency analysis,” inProbabilistic Methods applied to Power Systems (PMAPS) 2006, Stockholm,Sweden, June 2006, 6 pages.
11. J. Verboomen, D. Van Hertem, P. H. Schavemaker, W. L. Kling, and R. Bel-mans, “Optimal coordinated phase shifter control by using meta-evolutionaryprogramming and evolution strategies,” in IEEE Young Researcher Symposiumin Power Engineering, Ghent, Belgium, April 2006, 6 pages.
12. D. Van Hertem, J. Verboomen, K. Purchala, R. Belmans, and W. L. Kling,“Usefulness of dc power flow for active power flow analysis with flow controllingdevices,” in IEE International Conference on AC and DC Power Transmission2006, London, United Kingdom, March 2006, pp. 58–62.
Publications 157
13. N. Saraf, J. Verboomen, P. Schavemaker, and L. van der Sluis, “A model forthe static synchronous series compensator for the real time digital simulator,”in Future Power Systems (FPS) 2005, Amsterdam, the Netherlands, November2005, 6 pages.
14. D. Van Hertem, J. Verboomen, R. Belmans, and W. L. Kling, “Power flowcontrol devices: An overview of their working principles and their applicationrange,” in Future Power Systems (FPS) 2005, Amsterdam, the Netherlands,November 2005, 6 pages.
15. J. Verboomen, D. Van Hertem, P. H. Schavemaker, W. L. Kling, and R. Bel-mans, “Phase shifting transformers: Principles and applications,” in FuturePower Systems (FPS) 2005, Amsterdam, the Netherlands, November 2005,6 pages.
16. J. Verboomen, D. Van Hertem, P. H. Schavemaker, W. L. Kling, and R. Bel-mans, “The influence of phase shifting transformers on transient stability,”in Universities Power Engineering Conference (UPEC) 2005, Cork, Ireland,September 2005, 5 pages.
Other
1. J. Verboomen, “FACTS: an overview,” in IEEE Student Branch Symposium:Evolutions in Power Engineering, Leuven, Belgium, March 2006.
Acknowledgement
Although only my own name is printed on the cover of this thesis, it is actually muchmore of a combined effort of several people, both in academics and outside. I am verygrateful to all of them and their contribution will be remembered.
I would like to thank my promotor prof.ir. W.L. Kling (Wil) for making this researchpossible in the first place, for giving me the freedom to find my own way in thisscientific adventure and for supporting me in my decisions. It was a pleasure to workwith him and I am sure we will meet again in our future professional careers.
Also many thanks to my supervisor dr.ir. P.H. Schavemaker (Pieter) for always be-ing open to new, unconventional ideas. Although he left the TU Delft during theresearch, we still had fruitful conversations at regular times after that.
Many thanks go out to prof.ir. L. van der Sluis (Lou) for giving me the opportunity toperform research in the Power Systems Lab and for being an excellent travel partneron several occasions. Furthermore, the help of prof.dr. C. Roos in solving variousoptimisation problems is highly appreciated.
I would like to thank the doctoral examination committee for taking the time to gothrough my work. Their comments have raised the quality of this thesis, for which Iam grateful. A special thanks goes out to prof.dr.ir. R. Belmans (Ronnie) for showingme the first steps in the power engineering world and for the continuing support, evenafter I left the KU Leuven.
As this research has been performed within the IOP-EMVT framework, I wouldlike to acknowledge prof.ir. M. Antal (chairman of the program committee) andir. G.W. Boltje (program coordinator) for enabling this initiative, and for their at-tention given to my work.
160 Acknowledgement
There is an old African proverb that says: “If you want to go fast, go alone. If youwant to go far, go together.” I think this applies to my direct colleagues at TU Delft.Many thanks to Freek Baalbergen, Deborah Dongor, Nima Farkhondeh Jahromi,Madeleine Gibescu, Ralph Hendriks, Zongyu Liu, Bob Paap, George Papaefthymiou,Marjan Popov, Laura Ramirez Elizondo, Muhammad Reza, Barbara Slagter, BartUmmels, Ezra van Lanen, Arjan van Voorden, Else Veldman, Johan Vijftigschild,Ioanna Xyngi and Boukje Ypma. A special thanks goes out to Bart Ummels forbeing an excellent roommate, in spite of the fact that we are two opposites in manyways, and to George Papaefthymiou, for being a great friend and for teaching me theGreek way of life.
In the framework of IOP-EMVT, I have been working together with many inspiringpeople. I would like to thank Sjoerd de Haan and Johan Morren of the EPP groupat TU Delft and some people at TU Eindhoven: prof.dr.ir. J.H. Blom, Sjef Cobben,Edward Coster, Roald de Graaff, Anton Ishchenko, Andrej Jokic, Panagiotis Karalio-lios, Johanna Myrzik, Frans Provoost, Cai Rong and prof.dr.ir. P.P.J. van den Bosch.Our regular meetings were always a pleasure.
I would like to thank Dirk Van Hertem of the KU Leuven for our fruitful cooperation.Furthermore, the help of Frank Spaan of TenneT is greatly appreciated, not only forproviding me with valuable data, but also for showing me the practical relevance ofmy work.
Of course there is also a non-academic side to life, which is not to be neglected. Thesupport of my family and friends throughout the years has proven invaluable. Thereare too many people too mention, but they know who they are.
A special thanks goes out to my parents, who have offered me a happy childhood,which is the best present one can get. Their continuing support has enabled me toset my own course in life, for which I am endlessly grateful.
Finally, I would like to thank my wife Annechien, for giving me happiness in life.
Biography
Jody Verboomen was born on January 21 1979, in Leuven, Belgium. He attendedsecondary school at the Don Bosco institute in Haacht (Belgium), where he graduatedon Latin-Mathematics. He enrolled at Group T in Leuven, Belgium in 1997 andobtained the Master of Industrial Sciences in Electronical Engineering degree (magnacum laude) in 2001. For his master thesis about heart rate variability analysis atthe Academic Hospital of Gasthuisberg (Leuven), he was nominated “laureate of thefellowship of Biomedical Technology and Health Care of the Technological Institute ofthe Royal Flemish Engineering Society”. In 2004, he obtained the degree of Master ofScience in Electrotechnical-Mechanical Engineering (magna cum laude) from the KULeuven, Belgium. During this education, he joined Electrabel (Brussels, Belgium) foran internship concerning the coordination of renewable energy projects, and spent ashort time at the Universidad Politecnica de Madrid (Spain) for an intensive courseon nuclear radiation. After graduation from the KU Leuven, he joined the ElectricalPower Systems group of the Delft University of Technology as a PhD researcher withinthe framework of the “Intelligent Power Systems” research program. In October 2008,he joined Siemens AG in Erlangen, Germany as a consultant.
This book was typeset by the author in LATEX 2ε(pdfLATEX more in particular) using the “memoir”document class. The main text was set using a 10 ptComputer Modern Roman font. The cover was de-signed in Inkscape. Most of the graphs were plottedby using Matlab and included in the main documentas a PDF file. Most of the graphics were drawn inXfig.
![2 CTPT [Transformers]](https://img.pdfslide.tips/doc/110x75/56d6c00d1a28ab301698bbc3/2-ctpt-transformers.jpg)
