Dynamics and stability of distribution networks with dispersed generation

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Dynamics and stability of distribution networks with dispersed

generation

Citation for published version (APA):

Ishchenko, A. (2008). Dynamics and stability of distribution networks with dispersed generation. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR632007

DOI:

10.6100/IR632007

Document status and date: Published: 01/01/2008

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Dynamics and Stability of

Distribution Networks with Dispersed

Generation

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op woensdag 16 januari 2008 om 16.00 uur

door

Anton Ishchenko

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Dit proefschrift is goedgekeurd door de promotor:

prof.ir. W.L. Kling

Copromotor:

dr.ir. J.M.A. Myrzik

The research was performed at the faculty of Electrical Engineering of the Eindho-ven University of Technology and was supported financially by Senter Novem in the framework of the IOP-EMVT research program (Innovatiegericht Onderzoeks-Programma ElektroMagnetische VermogensTechniek).

Printed by Printservice Technische Universiteit Eindhoven, the Netherlands. Cover design by CreanzaPrint

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Ishchenko, Anton

Dynamics and stability of distribution networks with dispersed generation / by Anton Ishchenko. – Eindhoven : Technische Universiteit Eindhoven, 2008.

Proefschrift. – ISBN 978-90-386-1714-5 NUR 959

Trefw.: elektrische energiesystemen ; stabiliteit / elektriciteitsnetten ; beveiliging / elektrische energiesystemen ; regeling / elektrische energie ; distributienetten. Subject headings: power system stability / power system protection / power system control / distribution networks.

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First promotor: prof.ir. W.L. Kling

Copromotor: dr.ir. J.M.A. Myrzik

Core committee:

prof.dr.ir. R. Belmans

prof.dr.ir. P.P.J. van den Bosch prof.dr.ir. N. Hatziargyriou

Other members:

prof.ir. M. Antal prof.dr.ir. J.H. Blom

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Summary

Dynamics and Stability of Distribution Networks

with Dispersed Generation

There is a major paradigm shift going on in the design and operation of power sys-tems. Electrical networks of the past included a relatively small number of large power plants. However, due to environmental and economical concerns this situation has started to change. The number of generators that use renewable energy sources (such as wind and photovoltaics) and have rather low rated power, is increasing. Be-sides that, also small generators that use conventional energy sources, but in a more efficient way than previously designed generators, appear nowadays. Most of them are connected to the distribution network level, and referred to as distributed or dis-persed generation (DG).

These changes will greatly influence the power system dynamics. Distribution networks can no longer be considered as passive. In future it will not be possible to use simple equivalents of distribution networks for power system dynamic modeling. The large variety of DG types adds to the complexity of the problem. In Chapter 2 of this thesis aspects related to dynamic modeling of DG as such and to power systems with DG as a whole, are considered.

In dynamic studies the complete power system cannot be represented in a de-tailed way due to the huge system dimension. Therefore, special techniques have to be applied for model aggregation and reduction. Most of the reduction techniques, known in the power engineering world, are heuristic (experience-based) or semi-heu-ristic. They have been developed for transmission networks with large directly-con-nected synchronous generators. However, DG is not bounded to synchronous gen-erators: some generators are based on induction machines; others might use power electronics as an interface, which have quite different control systems and can hardly be aggregated. Some of them even do not have rotating parts (for example, fuel cells and photovoltaics). This makes the application of traditional methods of power sys-tem model reduction nearly impossible. Therefore strict mathematical techniques, which originate from control theory, but which are rather unknown in the power en-gineering world, are selected in order to retain the necessary precision during dy-namic reduction and, at the same time, limiting the simulation time. Most of the mathematical methods of model reduction focus on linear systems. Linear or lin-earized models in many cases provide an accurate description of the physical sys-tems. This is illustrated in Chapter 3, where aspects of linearization of DG models

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are considered. Special care has to be taken, when essential nonlinearities are re-moved or linearized. Chapter 4 describes the major principles of the mathematical methods for model reduction, and in Chapter 5 the application of these methods is demonstrated by several examples.

In general, any disturbance causes oscillations in the system; in the close neighborhood of the disturbance, oscillations are large. The network studied has to be represented in a detailed way by a nonlinear model. Neighboring networks are less affected by the disturbance and thus can be simulated using linear models. Fur-thermore, it is possible to reduce the order of these linear models. There are some networks further away (distant networks) that are not or almost not influenced by the disturbance. They can be represented just by constant inputs to the neighboring net-works (for example, by ideal voltage sources). Figure 1 shows this in a schematic way.

Study Network

(detailed nonlinear model)

Neighboring Networks

(reduced linear model)

Distant Networks (constants) Study Network (detailed nonlinear model) Neighboring Networks

(reduced linear model)

Distant Networks

(constants)

Figure 1. Modeling of a power system

A large increase of the penetration of DG units in distribution networks might lead both to stability problems under fault conditions (for instance, short-circuits) as well as instability under small disturbances (such as changes in loads and power flows that occur regularly during normal power system operation). The performance of distribution networks with DG under small disturbances (small-signal stability) is treated in Chapter 6. Contrary to the situation in transmission networks, a separated consideration of active and reactive powers through examining the voltage angles and the voltage magnitudes independently is impossible in distribution networks. Therefore the influence of reactance to resistance ratio of lines and cables has been analyzed. Besides that, it has been shown that asynchronous generators, such as wind turbines and microturbines, may introduce an undesired oscillatory behavior at high penetration levels.

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Problems concerning stability of distribution networks with DG under fault con-ditions (transient stability) are discussed in Chapter 7. They are related to the fact that short-circuits in distribution networks are cleared in a rather long time. Transient stability of generators connected at the distribution network level might be a problem in case of direct connection to the grid of electrical machines with a low inertia. The protection of the DG units has to be checked to satisfy transient stability require-ments. On the one hand, it must react fast enough to prevent instability, but on the other hand, it should not disconnect unnecessarily. In the present situation a fault at the transmission network level longer than 200-250 ms mostly leads to the discon-nection of large amounts of DG units over large geographic areas. This situation is not acceptable for a large penetration of DG, as it might lead to further cascading events in the system, as can be noticed in some recent incidents. Adequate fault ride-through concepts must be developed and implemented in the power system. The techniques and models presented in this thesis are useful for performing the neces-sary analysis in distribution systems.

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Samenvatting

Er vindt een belangrijke paradigma verandering plaats in het ontwerp en de bedrijfsvoering van elektrische energievoorzieningsystemen. Elektrische netten uit het verleden bevatten een betrekkelijk klein aantal grote elektriciteitscentrales. Echter vanwege milieu en economische belangen begint de situatie te veranderen. De hoeveelheid opwekkers die hernieuwbare energiebronnen gebruiken (zoals wind en photovoltaisch) en een relatief laag nominaal vermogen hebben, groeit. Bovendien verschijnen er tegenwoordig ook kleinere opwekkers die conventionele energiebronnen toepassen, maar op een meer efficiënte manier dan vroeger. De meeste daarvan worden aangesloten op het distributienet en worden ‘distributed’ of ‘dispersed generators’ (DG) genoemd.

Deze veranderingen zullen grote invloed hebben op de dynamica van het systeem. Distributienetten kunnen niet langer meer als passief worden gezien. In de toekomst zal het niet meer mogelijk zijn om eenvoudige equivalenten van distributienetten te gebruiken voor het dynamische modelleren van het systeem. De grote verscheidenheid van DG types vergroot de complexiteit van het probleem. In Hoofdstuk 2 van dit proefschrift worden de aspecten die met betrekking tot het dynamisch modelleren van DG op zich en elektriciteitsystemen met DG als geheel, beschouwd.

In dynamische studies kan het gehele electriciteitsnet niet in detail worden gerepresenteerd vanwege de grote dimensie van het systeem. Daarom moeten speciale technieken worden toegepast voor aggregatie en modelreductie. De meeste reductietechnieken, bekend in de wereld van elektrische energietechniek, zijn heuristisch (op ervaring gebaseerd) of semi-heuristisch. Deze zijn ontwikkeld voor transportnetten met grote direct-gekoppelde synchrone generatoren. Maar DG is niet alleen gebonden aan synchrone generatoren: sommige generatoren zijn op asynchrone machines gebaseerd; anderen kunnen vermogenselektronica als interface gebruiken, die heel verschillende regelsystemen hebben en nauwelijks zijn te aggregeren. Sommige hebben helemaal geen draaiende delen (bijvoorbeeld brandstofcellen en photovoltaisch). Dit maakt de toepassing van traditionele methoden voor modelreductie van electriciteitsnetten bijna onmogelijk. Daarom zijn strikt wiskundige technieken, komend uit de regeltechniek maar relatief onbekend in de wereld van energietechniek, geselecteerd om de benodigde precisie tijdens de dynamische modelreductie te behouden en tegelijkertijd de simulatietijd beperken. De meeste van deze wiskundige technieken van modelreductie richten zich op lineaire systemen. In veel gevallen verschaffen lineaire of gelineariseerde modellen een nauwkeurige beschrijvingen van de fysische systemen. Dit wordt in Hoofdstuk 3

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geïllustreerd, waar de aspecten van linearisering van DG modellen zijn beschouwd. Bijzondere zorg moet worden genomen, als essentiele niet-lineariteiten worden verwijderd of gelineariseerd. Hoofdstuk 4 beschrijft de belangrijkste principes van de wiskundige technieken voor modelreductie en in Hoofdstuk 5 wordt de toepassing van deze methoden met verschillende voorbeelden gedemonstreerd.

In het algemeen, veroorzaakt iedere verstoring oscillaties in het systeem; in de nabijheid van de verstoring zijn de oscillaties groot. Het bestudeerde net moet in detail met een niet-lineair model worden gerepresenteerd. Buurnetten worden bij de verstoring minder beïnvloed en kunnen dus met lineaire modellen gesimuleerd worden. Verder is het mogelijk de orde van deze lineaire modellen te reduceren. Er zijn verderweg gelegen randnetten die niet of vrijwel niet beïnvloedt worden door de verstoring. Die kunnen gewoon gerepresenteerd worden door constante inputs voor de buurnetten (bijvoorbeeld als ideale spaningsbronnen). Figuur 1 toont dit op een schematische manier.

Studienet

(gedetailleerd niet-lineair model)

Buurnetten

(reduceerd lineair model)

Randnetten (constanten) Studienet (gedetailleerd niet-lineair model) Buurnetten

(reduceerd lineair model)

Randnetten

(constanten)

Figuur 1. Modelering van een elektriciteitsnet

Een grote toename van de penetratie van DG eenheden in de distributienetten kan zowel tot stabiliteitsproblemen bij fouten leiden (bijvoorbeeld bij kortsluitingen) als ook tot instabiliteit bij kleine verstoringen (zoals veranderingen in de belasting en de vermogenstromen die regelmatig optreden tijdens normaal bedrijf). Het gedrag van distributienetten met DG bij kleine verstoringen (‘small-signal stability’) wordt in Hooftstuk 6 behandeld. In tegenstelling tot de situatie in transportnetten, is een gescheiden beschouwing van werkzaam en blindvermogen door het onafhankelijk bestuderen van de hoeken en de moduli van de spanning, is onmogelijk in distributienetten. Daarom is de invloed van de verhouding tussen de reactantie en de weerstand van lijnen en kabels geanalyseerd. Bovendien wordt aangetoond dat asynchrone generatoren, zoals bijvoorbeeld toegepast bij wind- en microturbines, bij een hoge penetratie ongewenst oscillerend gedrag kunnen introduceren.

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Problemen met stabiliteit van distributienetten met DG bij fouten (‘transient stability’) worden in Hoofdstuk 7 bediscussieerd. Deze hebben te maken met het feit dat kortsluitingen in distributienetten pas na relatief lange tijd opgeheven worden. De transiënte stabiliteit van generatoren gekoppeld aan het distributienet kan een probleem zijn in het geval van een directe verbinding aan het net van elektrische machines met een kleine inertie. De beveiliging van DG moet worden gecontroleerd om aan de eisen van transiënte stabiliteit te voldoen. Enerzijds moet deze snel genoeg reageren om instabiliteit te voorkomen, maar anderzijds moet niet onnodig worden afgeschakeld. In de huidige situatie leidt een storing in het transport net van langer dan 200-250 ms meestal tot de afschakeling van grote aantallen DG over een groot geografisch gebied. Deze situatie is niet aanvaardbaar bij een hoge penetratie van DG omdat het kan leiden tot verdere cascade gebeurtenissen in het systeem, zoals bij enkele recente incidenten ook is waargenomen. Dus adequate ‘fault ride-through’ concepten moeten worden ontwikkeld en geïmplementeerd. De technieken en modellen die in dit proefschrift zijn gepresenteerd, zijn bruikbaar om de daarvoor noodzakelijke analyses in distributienetten uit te voeren.

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Chapter 1

Introduction

1.1 Background and problem definition

Electrical power systems all over the world experience substantial changes in the principles and philosophy of operation. Some of these changes are related with ad-vances in technology – for instance, breakthroughs in communication, development of power electronics, digital processors and protection devices. Other changes are linked to liberalization of electricity markets. The aim is to create a competitive mar-ket with a large number of independent participants instead of one monopoly, which fully controls generation, transmission and distribution of electrical energy.

But the most drastic change is initiated by the Kyoto Protocol. This international political agreement is an amendment to the United Nations Framework Convention on Climate Change (UNFCC). The objective of the protocol is the “stabilization of greenhouse gas concentrations in the atmosphere at a level that would prevent dan-gerous anthropogenic interference with the climate system” [UNF 05]. In 2007 post-Kyoto negotiations on greenhouse emissions have started at the meeting of the G8+5 Climate Change Dialogue. As a result of these agreements nowadays there is a sig-nificant increase in use of renewable energy sources for electric power generation (such as wind, hydro, photovoltaics, etc.).

One of the consequences of the deregulation of electricity markets and the ratifi-cation of the Kyoto Protocol is a fast development and increase of the amount of de-centralized or distributed generation (DG), which is revolutionary for the design and operation of distribution grids. For sure large conventional or renewable power plants (such as large wind parks) will be connected to the transmission network, but also small power producers are stimulated to come to the market and some types of generators are small by their nature.

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A large number of combined heat and power plants (CHP), microturbines, wind turbines and PV-systems are being connected to the distribution networks. Such gen-erators are defined as distributed gengen-erators [Bar 00]. Even at the present time the penetration level of DG can be up to 30% in certain regions. In future this number will continue to grow, and for sure it will cause impacts on the power system opera-tion. One of the major impacts is the change of the distribution network nature from passive (containing only loads) to active (containing a mixture of loads and small to medium-scale generators) [Jen 00]. The nature of this change is illustrated in Figure 1.1. The power system of the future is characterized by the possibility of a reversed direction of power flows (from the distribution to the transmission grid), a significant growth in the amount of generators and an increased generation uncertainty.

Loads Loads Loads

Distributed Generators Distributed Generators Distributed Generators Generators Transmission Network Distribution Network Generators Distribution Network Distribution Network

Loads Loads Loads

Generators Transmission Network Distribution Network Generators Generators Distribution Network Distribution Network Future Past

Loads Loads Loads

Generators Transmission Network Distribution Network Generators Generators Distribution Network Distribution Network

Loads Loads Loads

Generators Transmission Network Distribution Network Generators Generators Distribution Network Distribution Network Future Past Future Past

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These profound alterations have a considerable effect on power system dynam-ics. The larger the penetration level of DG in a power system, the more difficult it becomes to predict, to model, to analyze and to control the behavior of such a sys-tem. The model of a power system as a whole is becoming a model with exponen-tially increasing complexity. Lack of knowledge about system dynamics may result in improper system operation, local technical problems (instability of a specific gen-erator, wrong settings for generator protection leading to unnecessary tripping or damage of equipment, etc.) as well as global problems (such as inter-area oscilla-tions, cascading events and consequent blackouts over larger geographical areas). Although blackouts have a global nature, they are usually initiated by a sequence of local events. Taking into account the growing penetration levels of DG and the in-creasing role of distribution networks in the power system, consideration of distribu-tion networks dynamics and stability becomes a quesdistribu-tion of significant importance. Deployment of DG in the existing passive distribution networks is reaching a critical point whereby it can no longer be installed in the typical “fit and forget” fashion without impacting network operation and stability [Dja 07].

A power system is complex. Its dynamics includes phenomena with quite differ-ent time scales. This fact is illustrated on Figure 1.2. In this thesis the emphasis is on the slow part of the electromagnetic transients (linked with the operation of the pro-tections) and the fast part of the electromechanical transients (related with the local oscillation modes). The time scale of interest is in the range from tens of millisec-onds to several secmillisec-onds.

Wave Phenomena Electromagnetic Transients Electromechanical Transients 1 mmmms 1 ms 1 s 10 s Switchings of power electronics, lightnings, traveling waves Short-circuit behavior, operation of protection Electromechanical oscillations, transient and

small-signal stability

Voltage collapse, automatic generation

control, spot market Long term dynamics Wave Phenomena Electromagnetic Transients Electromechanical Transients 1 mmmms 1 ms 1 s 10 s Switchings of power electronics, lightnings, traveling waves Short-circuit behavior, operation of protection Electromechanical oscillations, transient and

small-signal stability

Voltage collapse, automatic generation

control, spot market Long term dynamics

Figure 1.2. Different time scales in power system dynamics

Various problems with stability of individual generators and stability of the sys-tem are expected to happen in the future. These problems depend to a large extent on the strength of the external (transmission) system and type of DG units that dominate inside a specific distribution network. The strength of the external system on a fast time scale (order of tens to hundreds milliseconds) is related to the short-circuit

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power at the point of common coupling (PCC) between the transmission system op-erator (TSO) and the distribution network opop-erator (DNO). For a more slow time ho-rizon (order of several seconds) it can be characterized by the amount of energy, which the system is able to deliver during a disturbance, and the time during which this energy can be delivered. In traditional power systems it is related to the system inertia. However, in the future a more general characteristic will be required, because of a significant presence of inverter-connected DG units and an increasing role of power electronics in the system. Even today, for instance in case of a connection of wind turbines to long distribution feeders, large voltage variations and, consequently, the possibility of unstable operation might appear [Kan 02]. The analysis of stability and dynamics of a distribution network coupled to a weak or very weak transmission system requires in general a quite detailed representation of the external system. A system is regarded as strong when the short circuit power at the point of coupling is at least a factor 3 higher then the load connected to it. A weak system cannot be seen as a reference or a source of infinite power, its response to a disturbance in the distri-bution system depends on many factors and becomes difficult to predict. This is why in such a case a quite detailed dynamic model is required. At the same time, it is im-portant to keep the model size (or model order) reasonable since the amount of time spent on simulation grows exponentially with the system order. Therefore the devel-opment of reduced order models is essential. The area of application for such re-duced models is quite wide, and is not limited to studies of a distribution network connected to a weaker transmission system, but for strong systems as well. In present research there is a major effort going on in the development of reduced order models. There are two possible solutions. The first one is in the development of reduced or-der models of components (usually generators) based on engineering experience or physical insight, and then the formulation of an overall system model using these models. One of the typical examples of such an approach can be found, for instance, in [Soe 05], where a low order transfer function for a variable-speed wind turbine is derived. The second solution is formulating the overall system model in full detail, and then reducing the model order mathematically exploiting the model structure. Such an approach can not be found so often in the literature. Although there are sev-eral examples of its application [Kan 03], [Cha 05], mathematical methods originat-ing from control and systems theory (such as balanced truncation and Hankel norm approximation) are still rather unknown in the power engineering world. The re-search presented in this thesis is intended to bridge this gap.

Various DG units might bring different problems related to stability, protection and system operation. There exist many types of DG, and each of them has its own limitations. For example, directly-connected asynchronous generators tend to con-sume significant amounts of reactive power during short-circuits, which might create dynamic voltage instability. Power electronics-interfaced DG units are very sensitive to voltage dips, and are almost immediately be disconnected by the protection. This

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is not desirable in the future, because a fault in the transmission network might lead to the disconnection of large amounts of DG. Development of fault ride-through re-quirements for DG is necessary. These rere-quirements must be based on transient sta-bility studies since the DG protection has to prevent generator instasta-bility and, at the same time, to avoid unnecessary disconnection.

The disturbance (nearly blackout) over the large area of Europe in November 2006 is an example, which demonstrated that nuisance tripping of DG units can sig-nificantly aggravate the situation under emergency conditions. During the distur-bance, a significant amount of generation units tripped due to the frequency drop in the Western area of the UCTE system. This contributed to the deterioration of system conditions and to the delay for restoring secure normal conditions. In addition, most of the TSOs do not have access to the real-time data of the power units connected to the distribution grids. This did not allow them to perform a good evaluation of the system state during and after the disturbance. For instance, in the North-Eastern area, the uncontrolled reconnection of generation units induced unexpected severe condi-tions and the need for additional time to recover a normal secure operation.

Although some of the above mentioned problems related to protection and sta-bility of distribution networks with DG are already treated in literature (for example, in [Can 03], [Chi 02], [Mil 02]), a detailed and systematic analysis of these problems in distribution networks and its consequences for the transmission system is hardly performed. This thesis tries to give insight to this. It proposes necessary tools and methodologies for stability analysis at the distribution network level. Application of these tools is illustrated by examples.

1.2 Objective and research questions

Based on the problem definition given above the following research objective has been defined:

Analyze the influence of DG on the stability of a distribution network and develop methods to incorporate the dynamics of a distribution network with DG in stabil-ity studies of the whole power system.

Several research questions have been posed:

1. How to model a distribution network with DG in such a way that the dynam-ics is represented adequately, while the model order is significantly

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re-duced? And how to integrate these reduced order models in performing the stability analysis of the large-scale power system?

2. How does DG affect the stability of a distribution network under normal conditions (following small disturbances)?

3. What is the influence of DG on the stability of a distribution network under emergency conditions (during and after a fault)?

4. Are the protection settings of DG properly adjusted to be able to prevent un-necessary DG tripping in case of distant faults and, at the same time, to guarantee that a generator does not become unstable?

5. What are the limits of a typical Dutch distribution network for incorporating DG from the stability point of view?

1.3 Methods

As a first step, dynamic models of different types of DG are created using Mat-lab/Simulink. The overall model of a power system is formulated in a global rotating

dq0

reference frame. Such a formulation is essential for the following steps.

Linearization of DG is done analytically. State-space averaging is done for generators with power electronics interfaces.

Reduced order models suitable for stability analysis of a large-scale power sys-tem are obtained by means of model reduction. Model reduction techniques used in-clude balanced truncation, Hankel norm approximation and Krylov-based methods. These methods need a linear or linearized system as an input.

The stability of a distribution network with DG under normal conditions (small-signal stability) is assessed by modal analysis of the linearized power system model. The Lelystad distribution network, which network structure is regarded as typical for the Netherlands, has been used.

The stability under emergency conditions (transient stability) is analyzed by means of a time-domain simulation. Again this analysis was performed using the Le-lystad distribution network.

A detailed description of the methods used in this thesis can be found in the fur-ther chapters.

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1.4 Outline

This section presents an outline of the thesis.

Chapter 2 – The general principles of dynamic modeling and simulation of power systems are explained. A method for the formulation of an electrical network model is proposed. It assumes that the overall state-space model of the electrical elements (which takes into account the network topology) is created in an

abc

refer-ence frame. Then state-space matrices of the overall system model (instead of models of the separate components) are transformed from the

abc

to the

dq0

reference frame. After that, dynamic models of different types of wind turbines and a microtur-bine are presented in a transparent and easy-to-follow way.

Chapter 3 – Different linearization methods are described. An analytical linearization is performed for a squirrel-cage induction generator wind turbine. Af-terwards, linearization of the model of power electronic converters by means of state-space averaging is illustrated. These elements are often present in DG systems. The linearization is a necessary prerequisite for further small-signal stability analysis and model reduction.

Chapter 4 – An overview of various model reduction techniques originating from the systems and control theory is given. These techniques are rather unknown in the power engineering world. The described methods include balanced truncation, Hankel norm approximation and Krylov-based model reduction.

Chapter 5 – The model reduction techniques described in Chapter 4 are ap-plied on distribution networks with DG. The results show that it is possible to sig-nificantly reduce the order of a system, while retaining the desired precision. In the end, a method for the integration of reduced order models in the power system simulation software is proposed.

Chapter 6 – Firstly, an algorithm for the formulation of the overall system state-space model is derived. After that, the small-signal stability limits of a MV ca-ble network and a MV overhead line network are compared. Modal analysis is per-formed for various typical distribution networks and different penetration levels of DG.

Chapter 7 – The transient stability of a typical Dutch distribution network is evaluated under three-phase faults at different locations. Cable lengths are made up to two and three times longer to analyze the influence. Furthermore, the cables are

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substituted by overhead lines having quite different parameters, and similar analysis is performed. It is shown that fault ride-through capability included in the DG pro-tection allows to prevent cascading events and, at the same time, to guarantee that a generator does not become unstable. In the end, the use of reduced order models for transient stability analysis of a distribution system is illustrated.

Chapter 8 – General conclusions and recommendations for further research are given.

1.5 Intelligent power systems research project

The research presented in this thesis has been performed within the framework of the ‘Intelligent Power Systems’ project. The project is part of the IOP-EMVT pro-gram (Innovation Oriented research Propro-gram – Electro-Magnetic Power Technol-ogy), which is financially supported by SenterNovem, an agency of the Dutch Min-istry of Economical Affairs. The ‘Intelligent Power Systems’ project is initiated by the Electrical Power Systems and Electrical Power Processing groups of the Delft University of Technology and the Electrical Power Systems and Control Systems groups of the Eindhoven University of Technology. In total 10 Ph.D. students, who work closely together, are involved in the project.

The research focuses on the effects of the structural changes in generation and consumption which are taking place, like for instance the large-scale introduction of distributed (renewable) generators.

Such a large-scale implementation of distributed generators leads to a gradual transition from the current ‘vertically-operated power system’, which is supported mainly by several big centralized generators, into a future ‘horizontally-operated power system’, having also a large number of small to medium-sized distributed (re-newable) generators. The project consists of four parts (as illustrated in Figure 1.3).

The first part investigates the influence of uncontrolled decentralized generation on the stability and dynamic behavior of the transmission network. As a consequence of the transition in the generation, less centralized plants will be connected to the transmission network as more generation takes place in the distribution networks, whereas the rest is possibly generated further away in neighboring systems. Solutions that are investigated include the control of centralized and decentralized generation, the application of power electronic interfaces and monitoring of the stability of the system.

The second part focuses on the distribution network, which becomes ‘active’. Technologies and strategies have to be developed that can operate the distribution network in different modes and support the operation and robustness of the network.

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

Figure 1.3. The four parts of intelligent power systems research project The project investigates how the power electronic interfaces of decentralized gen-erators or between network parts can be used to support the grid. Also the stability of the distribution network and the effect of the stochastic behavior of decentralized generators on the voltage level are investigated. The research presented in this thesis belongs to this part of the project.

In the third part autonomous networks are considered. When the amount of power generated in a part of the distribution network is sufficient to supply the local loads, the network can be operated autonomously but as a matter of fact remains connected to the rest of the grid for security reasons. The project investigates the control functions needed to operate the autonomous networks optimal and secure.

The interaction between the grid and the connected appliances has a large influ-ence on the power quality. The last part of the project analyses all aspects of the power quality. The goal is to support the discussion between the polluter and the grid operator who has to take measures to comply with the standards. The realization of a power quality test lab is an integral part of the project.

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Chapter 2

Dynamic Modeling of Power Systems

with Dispersed Generation

2.1 Introduction

Modeling is the first step towards the analysis of real physical phenomena, for consequent design of devices and for making adjustments in already existing schemes or equipment (if necessary). In most cases, if more precise models are used, more accurate results are obtained, which allow to predict better the behavior of the system analyzed in different situations and to improve the system performance. Us-ing too simple models may lead to completely wrong results and erroneous conclu-sions. On the other hand, too detailed models may be very complex and difficult to understand and will contain information not really necessary for the analysis. Besides that, it becomes much more difficult to obtain or to identify parameters of these models, and to understand the impact of a specific parameter on the overall behavior.

In this chapter dynamic models of DG of different types are given in such a de-tail, sufficient for the analysis of the behavior of the overall system rather than the behavior of a specific component. The emphasis is on the analysis of the slow part of the electromagnetic and the fast electromechanical transients in power systems and the stability issues associated with them.

Models of DG, tested and experimentally validated by numerous authors ([Slo 03], [Ack 02], [Mil 03], [Gud 06], etc.) have been used for simulation. Here they are slightly modified and represented in an unified form, which is easy to read, to under-stand and to implement. These models are utilized in further chapters for lineariza-tion, model reduction and stability analysis.

Squirrel-cage and doubly-fed induction generator wind turbines are considered in this chapter in detail, since nowadays they are the most widely used wind turbine

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systems connected to the medium voltage (MV) distribution level. The split-shaft microturbine model is described since it is a good representative for gas turbines with comparatively small rated power but significant enough for taking into account at MV level. Such microturbines can be used for small-scale combined heat and power (CHP) plants.

2.2 Dynamic modeling and simulation of power

systems

In general, dynamic modeling of a power system can be performed using two different approaches: system-oriented or object-oriented.

The algorithm of the system-oriented approach is shown in Figure 2.1.

2. Create a description of the global equivalent circuit

Difference equations Differential

equations

3. Introduce control systems and formulate the overall dynamic system model

1. Substitute each power system component by an equivalent circuit

4. Calculate the initial conditions

5. Run the numerical integration

2. Create a description of the global equivalent circuit 2. Create a description of the global equivalent circuit

Difference equations Differential

equations

3. Introduce control systems and formulate the overall dynamic system model 3. Introduce control systems and formulate

the overall dynamic system model

1. Substitute each power system component by an equivalent circuit 1. Substitute each power system component by an equivalent circuit

4. Calculate the initial conditions 4. Calculate the initial conditions

5. Run the numerical integration 5. Run the numerical integration

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Firstly, each power system component (transformer, line, electrical machine, etc.) is substituted by an equivalent circuit consisting of inductances, capacitors, re-sistors, voltage and current sources. Taking into account the topology of the network, the global equivalent circuit, which characterizes a power system, is built and the system of differential or difference equations describing this circuit is formed.

For the representation using differential equations (typically in the form of a state-space) state variables are usually selected to be currents through inductances and volt-ages across capacitors. This selection is based on the physical law that the stored en-ergy does not change abruptly. However, in case of nonlinearities in inductances or capacitors, the fluxes in inductances and the charges in capacitors have to be selected as state variables. This process is illustrated using one phase of a three phase trans-former (without saturation) as an example. The equivalent circuit is shown in Figure 2.2. All quantities are referred to the primary side of the transformer.

v₁ v′_m v_R1 R₁ L₁ v_L1 i₁ v′_L2 L′_m L′₂ R′₂ v′_R2 v′₂ i′₂ i′_m + + v₁ v′_m v_R1 v_R1 R₁ L₁ v_L1 v_L1 i₁ v′_L2 v′_L2 L′_m L′₂ R′₂ v′_R2 v′_R2 v′₂ i′₂ i′_m + +

Figure 2.2. Equivalent circuit of a two winding transformer (one phase, neglecting iron losses, linearized)

Its elements are described by:

1 2 1 2 1 1 2 2 1 2 1 2 , , , m R R m m L L L v R i v R i di di di v L v L v L dt dt d t = ′ = ′ ′ ′ ′ = ′ = ′ ′ = ′ (2.1)

The Kirchhoff current law (KCL) equation for the circuit is:

1 2 m 0

i +i′ ′− =i (2.2)

While the Kirchhoff voltage law (KVL) equations are:

1 1 2 2 1 2 0 0 m R L m R L v v v v v v v v + + + + + + − ′ = − ′ ′ ′ ′ = (2.3)

Substituting (2.1) in the KVL (2.3) and getting rid of the magnetizing current im

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(

)

(

)

1 2 1 1 1 1 1 2 2 2 2 2 0 0 m m m m di d i v R i L L L dt dt di d i v R i L L L dt dt + + + + + + ′ − + ′ ⋅ ′ = ′ − ′ ′ ′ ′ + ′ ⋅ = (2.4)

Solving them with respect to the current derivatives, the state-space equations in an explicit form (with inductor currents as states and terminal voltages as inputs) can be written as follows: 2 2 1 2 1 1 1 2 1 2 1 2 1 2 m m m m m m m m L L L L L L R R i L L i L L v d dt i L L L i L L L v R R L L L L σ σ σ σ σ σ σ σ           _ _   _ _     _ _   _ _     _ _   _ _                 + + ′ ′ ′ ′ ′ ′ − ′ − = × + × ′ ′ ₋ _′ + ′ ′ ₋ ′ + ′ ′ (2.5) where Lσ =L L₁ ′₂+L L₁ ′m+L L′ ′₂ m.

For large networks all above-mentioned operations are done using a matrix form. The matrices are formed based on certain rules related to the network topology. These rules are out of the scope of this thesis, but interested reader can find them, for instance in [Chu 75]. The incidence matrix F is used for the description of the KCL (

Fi

=

0

) and the loop (or contour) matrix

H

for the description of the KVL (

H v

=

0

). Algebraic operations are applied on the submatrices inside these matrices, and then the state-space equations in an implicit form are obtained. They are solved with re-spect to the states derivatives using Gaussian elimination.

If difference equations are chosen for the network representation, the procedure is a somewhat different. The equations for each passive element of the equivalent circuit (resistor, capacitor, inductance) are discretized according to the solution method for differential equations. An example of such a discretization in case of a trapezoidal integration rule is shown in Table 2.1 (voltages are directed in a similar way as on Figure 2.2). The mathematical model of the network depends on the tion method. This is, of course, not really a nice property, since a change in the solu-tion method would require rebuilding of the whole system, and different relasolu-tion- relation-ships have to be derived for the description of the passive elements.

From Table 2.1 it is obvious that the equations for the whole network can be brought together into the following matrix form:

(

)

( ) ( ) ( ), ( )

G v t =i t −f i t−∆t v t−∆t (2.6) where f is a function based on values of the voltages and currents at the preceding time step and G is the conductance matrix characterizing the network elements.

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Table 2.1. Discretization of the element equations for a trapezoidal rule of integration v_R R i_R + _v R R i_R + 1 ( ) ( ) R R i t v t R = ⋅ v_L L i_L + v_L L i_L + ( ) ( ) , ( ) ( ) ( ) ( ) , 2 ( ) ( ) ( ) ( ). 2 2 L L L L L L L L L L di t v t L dt v t v t t i t i t t L t t t i t v t i t t v t t L L = ⋅ + −∆ − −∆ = ⋅ ∆ ∆ ∆ = ⋅ + −∆ + ⋅ −∆ v_C C i_C + v_C C i_C + ( ) ( ) , ( ) ( ) ( ) ( ) , 2 2 2 ( ) ( ) ( ) ( ). C C C C C C C C C C dv t i t C dt i t i t t v t v t t C t C C i t v t i t t v t t t t = ⋅ + −∆ − −∆ = ⋅ ∆ = ⋅ − −∆ − ⋅ −∆ ∆ ∆

Normally, some nodes have known voltages either as voltage sources are con-nected to them, or when the node is grounded. In this case (2.6) is partitioned into a set A of nodes with unknown voltages, and a set B of nodes with known ones. The unknown voltages v_A(t) are then found by solving:

(

)

( ) ( ) _B( ) ( ), ( )

AA A A AB

G ⋅v t =i t −G ⋅v t −f i t−∆t v t−∆t (2.7) This is done through the matrix GAA being triangularized with ordered elimination

and exploitation of the sparsity [Dom 86].

The overall dynamic system model is typically formed as illustrated in Figure 2.3. The nonlinear parts and the control systems are placed in feedback between the

Sources (inputs) Network (linear state-space)

u

y

Nonlinear models and control systems

Outputs

i

v

Sources (inputs) Network (linear state-space) Network (linear state-space)

u

y

Nonlinear models and control systems

Outputs

i

v

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voltage outputs and the current inputs of the linear model [Sim 04]. A typical exam-ple of a nonlinear model is the electrical machine. It contains nonlinearity in the mechanical equation, where the electrical torque is a nonlinear term.

An object-oriented approach, as it appears from its name, describes each system element separately. The computation is based on a set of blocks (systems) of differ-ential equations rather than a single system. The algorithm of an object-oriented ap-proach is shown in Figure 2.4.

3. Calculate the initial conditions

4. Run the numerical integration

2. Connect the subsystems in one global system 1. Create a model of each element (subsystem)

3. Calculate the initial conditions 3. Calculate the initial conditions

4. Run the numerical integration 4. Run the numerical integration

2. Connect the subsystems in one global system 2. Connect the subsystems in one global system 1. Create a model of each element (subsystem) 1. Create a model of each element (subsystem)

Figure 2.4. Algorithm of the object-oriented approach

The most important advantages of the object-oriented formulation are the fol-lowing [All 96]:

– it is better understandable;

– it clearly represents the relationships between the blocks of electrical, mecha-nical and control types;

– the addition of new blocks is easy and does not require explicitly to rebuild the whole system matrix.

At the same time, the variable information flow between subsystems has to be defined. This has to be done in a such way that the outputs of one subsystem serve as inputs for the other. It might be not easy and, actually, it is the main disadvantage of this approach.

Sometimes there is no matching of input/output signals of the interconnected components. This situation is not uncommon in the simulation of complex systems,

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constructed using standard modules of components or templates. When there is no match between the signals of certain electrical components, the typical solution is to introduce a small capacitor or a large resistor at the junction of the connected com-ponents [Ong 98].

Such a problem and its solution are illustrated in Figure 2.5. A generator, a former and a grid are connected in series. Models of both the generator and the trans-former written in the object-oriented form (with explicitly defined inputs and out-puts) have voltages as the inputs and currents as the outputs. It can be easily under-stood, for example, by observation of the differential equations describing the trans-former (2.5). This selection of inputs and outputs is suitable since in this case (2.5) is formulated in a clear state-space form, convenient for the numerical integration. However, it is impossible to connect the subsystems in one global system since the voltage vgen (the desired input for both the transformer and the generator) is not

de-termined.

The object-oriented model

of the system

Transformer Model v_mv

Input (from the grid)

1

Output (to the grid)

Generator Model v_gen i_tr Fictitious Resistor Model i_gen v_gen i_mv

The one-line diagram

of the system

Grid v_mv i_mv i_gen v_gen

G

R_f i_f i_tr

The object-oriented model

of the system

Transformer Model v_mv

Input (from the grid)

1 1

Output (to the grid)

Generator Model v_gen i_tr Fictitious Resistor Model i_gen v_gen i_mv

The one-line diagram

of the system

Grid v_mv i_mv i_gen v_gen

G

R_f i_f i_tr

Figure 2.5. Use of a shunt fictitious resistor Rf to get the junction voltage vgen

Artificially adding a shunt fictitious resistor Rf , which is much larger than the

impedance of the actual circuit elements, would allow to develop v_gen without intro-ducing too much inaccuracy, and to obtain the overall system model. The required voltage input to the connected modules is given by the voltage equation of the fictitious resistor:

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(

)

gen f f gen tr f

v

=

i R

=

i

− ⋅

i

R

(2.8)

Alternatively, a small capacitor

C

f can be used instead:

(

)

1

gen gen tr f

v

i

i dt

C

=

_∫

−

(2.9)

Although, using a capacitor will introduce an extra state, its voltage does not am-plify the noise in the currents as with resistor. In some situations, it may result in a better approximation than with a

R

f. There is an advantage of being able to use

well-established templates, especially for large and complicated modules. The conven-ience of not having to reformulate the problem, sometimes may justify a small inac-curacy caused by the introduction of either a capacitor or a resistor across a junction to develop the common junction voltage required by the connected modules. While choosing an extremely small capacitor or an extremely large resistor does minimize inaccuracies caused by the use of such artificial elements; too extreme values may result in system equations that are overly stiff for the used solution method. In practice, the smallest capacitor or largest resistor that one can use for such an ap-proximation will be limited by numerical stability.

In this thesis the object-oriented approach was selected for modeling of DG and the elements connecting it to the grid (cables and transformers up to the point of common coupling), while the grid itself is modeled using a system-oriented ap-proach. The overall system model is created in a similar way as shown in Figure 2.3. All modeling was done with Matlab/Simulink.

2.3 Transformation of state-space matrices from

abc to dq0 reference frame

One of the most important questions in power system dynamic modeling is the selection of an appropriate reference frame (or coordinate system) for the representa-tion of electrical variables. The coordinate systems typically used for power system modeling are the

abc

and

dq0

reference frames.

The representation of the electrical network in a

dq0

reference frame gives the following advantages:

1. The capabilities of a variable-step integration method can be used fully. The speed of simulation at steady-state or in a near steady-state situation is increasing rapidly then.

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2. The system matrices become nonsingular. This also contributes to the simulation speed and the numerical stability issues.

3. Representation in a global rotating reference frame removes the

dq0

transforma-tion (with sin

ωt

and cos

ωt

terms) between the network and the electrical ma-chines. It makes further linearization of the overall system model possible, which is a necessary prerequisite for model reduction and small-signal stability analysis. The main disadvantages of a

dq0

representation are that it is not natural (i.e., it is in-troduced artificially to simplify analysis), and it is impossible to use it for a detailed simulation of unbalanced internal faults in power system components (for example, interturn short-circuits of generator windings). However, this research focuses on the behavior of the power system as a whole rather than on the behavior of a specific power system component. Therefore, the

dq0

representation is used in this thesis.

The SimPowerSystems toolbox in Matlab was used to obtain the state-space equations describing the grid. These equations are expressed in an

abc

reference frame and contain an additional grounding component (linearly dependent on the phase quantities). Such a representation slows down the simulation of a large net-work significantly. One of the reasons is the introduction of the linearly dependent grounding component, which makes the system matrices singular. The other reason is the representation in the

abc

reference frame, where the variables are changing continuously even during steady-state. This does not allow to use the potential of a variable-step integration method fully. Therefore, the obtained system was modified. First, the grounding components were removed from the system. Then the system was transformed from an

abc

to a

dq0

reference frame. The system matrices were splitted into submatrices of 3 by 3 size, and the transformation was applied to them. The transformation has to be done carefully since there are three types of state vari-ables participating in the system description obtained from SimPowerSystems. Phasor diagrams of these types are presented in Figure 2.6.

v_a v_b v_c v_a v_b v_c v_ab v_ca v_bc v_a v_b v_ca v_ab v_bc v_c Y D11 D1 v_a v_b v_c v_a v_b v_c v_ab v_ca v_bc v_a v_b v_ca v_ab v_bc v_c Y D11 D1

Figure 2.6. Phasor diagrams of different types of variables

The

D1

and

D11

notations refer to the clock convention. It assumes that the ref-erence

Y

voltage phasor is pointing at 12 o’clock on a clock display.

D1

and

D11

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o’clock (

∆

voltages leading

Y

voltages by +30°) [Sim 04].

D11

is usual

∆

con-figuration, for example capacitor banks can have such a scheme.

D1

might be used for connection of transformer windings. An

Y

connection has

v

a,

v

b,

v

c as state

vari-ables, while

D11

and

D1

–

v

ab,

v

bc,

v

ca. Variables for each connection type are

trans-formed in a slightly different way. Here only the transformation of the matrices from the

Y

-type variables to

dq0

will be derived. However, this can be done in a similar manner for all possible combinations of variable types. Suppose the network state-space equations are described by (inputs and some matrix elements are omitted for simplicity): 11 12 1 1 2 2 abc abc abc abc abc abc

i

A

i

d

i

dt

                             

=

×

⋯

(2.10)

where

i

_abc₁

=

_

i

_a₁

i

_b₁

i

_c₁_T

,

i

_abc₂

=

_

i

_a₂

i

_b₂

i

_c₂_T

,

11 12 13 14 15 16 11 21 22 23 12 24 25 26 31 32 33 34 35 36 , . abc abc a a a a a a A a a a A a a a a a a a a a                     = =

Phase variables iabc are then related to the dq0 variables idq0 by the following

equation:

sin

cos

1

2

2 sin

cos

1

3

2

2 sin

cos

1

3

a d q b c 0

t

i

t

i

t

ω

π

ω

π

ω

             _ _ _ _       _ _ _ _                    _ _ _ _ _ _                

=

−

×

+

(2.11)

where

ω

is the angular speed of the

dq0

reference frame. This transformation is commonly used in three-phase electric machine models, where it is known as the Park transformation. It is amplitude-invariant, implying that the length of the current and voltage vectors in both

abc

and

dq0

reference frame are the same, however in this case the conservation of power is lost. This amplitude-invariant transformation is mostly used in modeling of electrical machines [Paa 00].

Equation (2.11) can be rewritten in the other way:

0 sin 1 3 3 1 cos 2 2 2 2 1 1 3 3 1 2 2 2 2 q 0 d a q q 0 t b d d dq c q q 0 d d i i i i t i i i i i i t M M i i i i i i ω

ω

                                  _ _       = − + − − × = × − − − (2.12)

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Differentiating (2.12) results in: 0 0 dq t abc t dq

d M

di

M

dt

ω ω

=

×

+

×

(2.13)

Then substituting (2.12) and (2.13) in (2.10), and taking the first row from the obtained expression and solving the resulting equations in respect to the

dq0

vari-ables derivatives, the following equation can be formed:

1 ₁₁ ₁₂ 1 2 2 dq0 dq0 dq0 dq0 dq0 dq0 i _A _A i d i i dt   _ _     _ _     _ _     _ _     _ _       = × ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (2.14) 11 11 12 13 1 12 14 15 16 dq0 T phase phase dq0 T phase phase A I a T a T a E A I a T a T a

ω

= ⋅ + ⋅ + ⋅ + ⋅ = ⋅ + ⋅ + ⋅

where aij is an element from the matrices 11 abc

A

or 12 abc

A

, 1

1

3

0 ₀

_{1 0}

2

2 ,

1 0 0

3

1

0

0 0 0

2

0

1

phase

T

E

    _ _   _ _   _ _   _ _        

−

=

= −

−

It can be seen from (2.14) that if a block of the system matrix corresponds to the same variable, which has its derivative on the left side, then an additional

ω

-term ap-pears in the expression ( 11

dq0

A

_{is an example of such a block in (2.14)).}

The transformation equations are derived in a similar manner for all possible combinations of variable types. They are summarized in Table 2.2. The additional

ω

-term has to be omitted when transformation is done for different variables of the same type.

The following transformation matrices are used in Table 2.2:

2 _

1

3

1

0

0 _{1 0 0}

2

2 ,

,

0 1 0

3

1

3

0

0 0 0 0

2

0

line phase line

T

E

        _ _     _ _     _ _     _ _              

−

=

−

It is important that the proposed way of reference frame transformation makes use of matrices of the whole network in an

abc

reference frame. These matrices can be obtained from standard power system simulation software like SimPowerSystems.

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Table 2.2. Transformation equations for all possible combinations of variable types Left side variable type Right side variable type Transformation equation

Y

11 12 13 1 dq0 T phase phase

A

= ⋅

I a

+

T

⋅

a

+

T

⋅

a

+ ⋅

E

ω

Y

D11

3

(

_ 11 1 12 _ 13

)

dq0 T

phase line phase line

A = ⋅T ⋅a + ⋅E a −T ⋅a

Y D1 3

(

_ 11 _ 12 1 13

)

dq0 T

A = ⋅T ⋅a −T ⋅a − ⋅E a

D11 Y 1 ₃

(

1 21 _ 22 _ 23

)

dq0 T T

A = ⋅ E a⋅ +T ⋅a −T ⋅a D11 D11 21 2 22 23 1 dq0 T line line A =T ⋅a + ⋅E a +T ⋅a + ⋅E

ω

D11 D1 21 22 2 23 dq0 T line line A =−T ⋅a −T ⋅a − ⋅E a D1 Y 1 ₃

(

1 31 _ 32 _ 33

)

dq0 T

A = ⋅ E a⋅ −T ⋅a +T ⋅a D1 D11 31 2 32 33 dq0 T line line A =−T ⋅a − ⋅E a −T ⋅a D1 D1 31 32 2 33 1 dq0 T line line A =T ⋅a +T ⋅a + ⋅E a + ⋅E

ω

Thus, the procedure does not require a reformulation of the whole network model in the dq0 reference frame from scratch (i.e., the definition of elements models in a dq0

reference frame and the build up of overall network model). The process of state-space matrices transformation can be easily automated.

The standard procedure for a dq0 transformation consists of two steps:

1. Transform the models of each power system component from an abc refer-ence frame into models in dq0.

2. Connect models of elements in dq0 according the power system topology and formulate the model of the whole system in dq0.

Dynamics and stability of distribution networks with dispersed generation

Dynamics and stability of distribution networks with dispersed

generation

Dynamics and Stability of

Distribution Networks with Dispersed

Generation

Contents

Summary

Dynamics and Stability of Distribution Networks

with Dispersed Generation

Samenvatting

Chapter 1

Introduction

1.1 Background and problem definition

1.2 Objective and research questions

1.3 Methods

dq0

1.4 Outline

abc

abc

dq0

1.5 Intelligent power systems research project

Chapter 2

Dynamic Modeling of Power Systems

with Dispersed Generation

2.1 Introduction

2.2 Dynamic modeling and simulation of power

systems

(

)

(

)

Fi

=

0

H

H v

=

0

(

)

(

)

u

y

i

v

u

y

i

v

The object-oriented model

of the system

The one-line diagram

of the system

G

The object-oriented model

of the system

The one-line diagram

of the system

G

(

)

v

=

i R

=

i

− ⋅

i

R

C

(

)

1

v

i

i dt

C

=

_∫