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Thèse de doctorat/ PhD Thesis Citation APA:
Antoine, O. (2013). Wide area measurement-based approach for assessing the power flow influence on inter-area oscillations (Unpublished doctoral
dissertation). Université libre de Bruxelles, Ecole polytechnique de Bruxelles – Electricien, Bruxelles.
Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/209368/4/c5142a56-d6e5-4c9b-a3a5-aec691b2c2a8.txt
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D 04062
ULB
.JEDE BRUXELLES
UNIVERSITÉ LIBRE DE BRUXELLESWide area measurement-based approach
for assessing the power flow influence
on inter-area oscillations
Olivier Antoine
A thesis submitted for the degree of PhD in Engineering Sciences
Academie year 2013 - 2014
Thesis director: Prof. Jean-Claude Mann ULB, Brussels, Belgiiun
President of the committee: Prof. Michel Kinnaert Members of the committee: Prof. Johan Gyselinck Prof. Luigi Vanfretti
Dr. Patrick McNabb Dr. Jacques Warichet
ULB, Brussels, Belgium ULB, Brussels, Belgium KTH, Stockholm, Sweden Statnett SF, Oslo, Norway Psymetrix Ltd, Edinburgh, UK Elia, Brussels, Belgium
Acknowledgments
First of ail, 1 would like to thank Prof. Jean-Claude Mann for having given me the opportunity to participate to the Twenties project, which has led to this thesis. I do not forget to mention Fabien which noticed me the vacancy as well as Michael, who had to support me during a few years in the LT120.
I would also like to thank ail the members of the Twenties project, especially the participants of the démonstration 5. It has been a real pleasure to work with ail of yoii. It is difhcult to thank everybody but I would like to emphasize the nice collaboration with Elia (Jacques, Wim and Christophe), the good contacts with Coreso (Aubry, Martin and Jeroen), the valuable advice of Psymetrix (Paddy, Douglas, Karine) and the profes- sionalism of RTE (Jean and Patrick). I thank also Minh and Pryianko for having shared the difhculty to work for the project and write a thesis at the same time.
I do not know what to say about my colleagues, which are actually more friends than colleagues. Actually I did not imagine that it was possible to hâve such a wonderful atmosphère at the office. Thank you for that and I wish you the best with your respective projects.^^^
The support of my friends from the Polytech Around The World Association (PATWA) is also deeply acknowledged. In particular, the François brothers for the coffee (or beer) breaks and the others for the parties as well as the interesting engineer discussions.
My family, my parents and my brother, for everything.
Last but not least, Carine for her small cute face and for having let me work really hard during this last year.
Abstract
Power Systems hâve been historically désignée! at a time when the production was centralized and the electricity had to be transmitted to the loads from the closest power plant. Nowadays, there is an increasing intégration of decentralized and intermittent pro duction. Moreover, the energy market coupling has enabled the transfer of electric power for economical purposes. Also, former isolated power Systems are now interconnected for reliability and financial reasons.
AU of these changes make difficult to predict the future behavior of the grid. Studies are done in order to plan for the future needs of the System. However, building new in frastructures takes time and it is expected that these needs will not be completely fulfilled in ail the parts of the grid. Therefore, transmission of active power could be limited by the existing infrastructure. For example, the presence of inter-area oscillations is often the limiting factor when a high active power is transmitted on a long transmission line between two groups of generators. Since higher levels of active power are exchanged on longer distances, problems of inter-area oscillations may arise in power Systems previously not affected by this phenomenon.
In this Work, a measurement-based approach, able to predict in the short-term the future behavior of oscillations, is presented. This approach is complementary to the long-term planning of the grid.
The mandatory first step towards a measurement-based approach is to hâve the ability to extract useful information among a huge quantity of data. To face this issue, some comparisons of data mining algorithms are performed. The proposed method combines two decision tree algorithms to obtain both prédiction accuracy and comprehensibility.
The second required step for building a measurement-based model is to take into account the limitations of the measurements. Two types of wide area measurements are used, synchronized measurements from PMUs and traditional unsynchronized data from the SCADA/EMS System. Oscillation monitoring using PMUs is especially of interest and an approach is presented to post-process dumping estimâtes. This post-processing method consists in a noise réduction technique followed by a dumping change détection algorithm.
Contents
Introduction
2
I
Small-signal stability and power System modeling
15
1 Background on small-signal stability 17
1.1 Classification of stability... 17
1.2 Rotor angle small-signal stability... 18
1.3 Electro-mechanical oscillations... 20
1.3.1 Local oscillations... 20
1.3.2 Mechanical équivalent of an inter-area oscillation ... 23
1.3.3 Parameters influencing the damping torque... 24
1.4 Summary and risks related to inter-area oscillations... 25
2 Power System modeling 27 2.1 Model of power System components... 27
2.1.1 Synchronous generator... 27
2.1.2 Excitation System and power System stabilizer... 29
2.1.3 Turbine and governor... 30
2.1.4 Load characteristics... 31
2.1.5 Transmission lines and network équations... 31
2.2 Modal analysis ... 32
2.3 Power flow influence on mode damping... 35
2.3.1 Generating random power flows... 35
2.3.2 4-machine test System ... 36
2.3.3 16-Machine test System... 39
Summary 42
II Data Mining
43
3.1 Review of data mining for damping prédiction and improvement... 47
3.2 Problem formulation... 49
4 Data mining approach 52 4.1 Algorithms... 52
4.1.1 Multiple linear régression... 53
4.1.2 Decision tree ... 55
4.1.3 Ensemble of trees... 58
4.1.3.1 Parameters... 60
4.1.4 Testing the model... 61
4.1.4.1 Hold-out ... 61
4.1.4.2 K-fold cross-validation... 61
4.1.4.3 Error rate and mean squared error... 61
4.1.4.4 Evaluating the influence of the learning set... 62
4.2 Feature sélection ... 62
4.2.1 Filter methods ... 63
4.2.2 Wrapper methods... 64
4.2.2.1 Stepwise linear régression... 64
4.2.2.2 Analysis of the tree structure... 64
4.3 Comparative example... 65
4.3.1 Géométrie interprétation... 65
4.3.2 Feature sélection illustrated on a spring-mass System... 66
4.3.2.1 Stepwise linear régression... 68
4.3.2.2 Backward élimination... 68
4.3.2.3 Testing the model ... 69
4.3.3 Influence of the learning set... 69
4.4 Proposed niethod... 70
4.5 Conclusion... 73
5 Simulation results 75 5.1 Database construction (implémentation)... 75
5.2 Simulation on the 16-machine network... 77
5.2.1 Blind approach... 81
5.2.2 Feature sélection... 84
5.2.3 Identifying a list of actions... 85
5.2.4 Using the model to increase the damping ratio... 89
5.2.5 Limitations of the algorithms... 90
Summary 93
III
Oscillation monitoring
95
6 Oscillation monitoring in power Systems 98
6.1 Oscillation monitoring worldwide... 98
6.2 Current situation in Continental Europe... 102
6.2.1 Description of the WAMS... 102
6.2.2 Oscillation analysis... 105
6.2.2.1 Choice of the input signal... 105
6.2.2.2 Analysis of the mode frequency... 105
6.2.2.3 Analysis of the mode shapes... 108
6.2.2.4 Summary tables of the modes... 114
6.2.2.5 Damping distribution ... 116
6.3 Conclusion... 117
7 Integrating System identification in the data mining approach 118 7.1 Review of modal identification techniques... 120
7.1.1 Non-parametric method... 121 7.1.1.1 Frequency estimation... 121 7.1.1.2 Damping estimation... 122 7.1.2 Parametric method... 123 7.1.2.1 Ringdown algorithm... 123 7.1.2.2 Mode-meter algorithms ... 125
7.2 Accuracy of “classical” mode-meters... 129
7.2.1 Using synthetic signais... 130
7.2.2 Using real data... 135
7.3 Proposed solution... 139
7.3.1 CUSUM... 139
7.3.2 Sudden changes... 142
7.3.2.1 Noise réduction... 143
7.3.2.2 Change of mean détection... 143
7.3.2.3 Time alignment... 144
7.3.2.4 Illustration on real measurements... 146
7.3.2.5 Discussion... 149
7.3.3 Slow-change analysis... 149
7.3.3.2 Change of mean détection... 151
7.3.3.3 Time alignment... 154
7.3.3.4 Discussion ... 155
7.4 Conclusion... 156
Summary 158
IV
Results on the European Continental grid
159
8 Gathering the data 163 8.1 Load flow variables... 1648.1.1 State estimation... 164
8.1.2 Day-ahead congestion forecast files... 167
8.1.3 PTDF... 167
8.2 WAMS... 169
8.3 Implémentation at Coreso [Antoine et al, 2013]... 170
8.3.1 Learning phase... 170
8.3.1.1 Inputs... 170
8.3.1.2 Outputs... 171
8.3.2 Forecasting phase... 172
8.4 Conclusion... 173
9 Results in Continental Europe 174 9.1 Testing procedures... 175
9.1.1 “Group by days” dataset... 175
9.1.2 “Sliding” and “Growing” dataset ... 175
9.1.3 Accuracy indicator: MSE vs corrélation coefficient... 176
9.2 Slow-change analysis results... 176
9.2.1 The database in numbers... 176
9.2.2 Blind approach... 177
9.2.2.1 Grouping vs hold-out testing ... 177
9.2.2.2 Growing vs sliding testing... 180
9.2.2.3 “DACF” vs “snapshot” prédiction... 184
9.2.3 Feature sélection... 186
9.2.4 Identifying a list of actions... 187
9.2.4.1 Computing a single decision tree on the reduced dataset 187 9.2.4.2 Computing the masking score... 189
Conclusion and discussion of the results 192
Summary, contributions and perspectives
196
Bibliography
203
Appendices
212
A Details about the implémentation 213
A.l Import of PDX2 data to Matlab... 213
A.2 Split the modes based on the mode frequency... 214
A.3 Select static data for the timestamps of interest... 215
A.4 Learning algorithm... 215
A. 5 Visualization ... 216
B Data transmission, WAMS and PhasorPoint 218 B. l Data transmission... 218 B.1.1 PMU... 219 B.2 Phasor computation ... 219 B.2.1 Concept of phasor... 219 B.3 Laboratory microgrid... 220 B.3.1 GPS dock... 222 B.3.2 PMU <-> PDG ... 222 B.3.3 Wireshark ... 224
B.3.4 PMU Connection Tester... 224
B.3.5 PDC <-> internet... 227
B.4 PhasorPoint... 227
Context
Power System stability is a complex field involving a wide range of phenomena. Among the different types of instabilities, small-signal stability is concerned with the stability of the power System iinder small disturbances. Insufficiently damped oscillations are one of the major concerns of small-signal stability [Kundur, 1994], In the case of a negatively damped oscillation, its amplitude will grow and induce line and/or generator tripping. This situation can lead to a split of the power System into separate islands and, in the worst case, to a blackout [Kosterev et al., 1999].
Oscillations hâve always been présent by nature in power Systems and can not be avoided. During the last décades, these oscillation problems were handled by properly tuning controllers and by installing sources of positive damping such as Power System Sta- bilizers (PSS). However, this solution cannot be optimized for ail modes of oscillation and for ail the possible operating points. The reality is that power Systems are continuously changing, former isolated Systems are now interconnected and the increasing intégration of renewables affects significantly the grid. Ail of these changes impact oscillations in the sense that modes are changing and new modes are emerging. Moreover, low-frequency oscillations and especially inter-area oscillations are affected by numerous power System components. Therefore, the understanding of this phenomenon is not trivial and it is extremely challenging to predict in which direction the modes may move.
The typical method to analyze inter-area oscillations is to use a dynamic model of the power System. In this model, each power System component has to be accurately characterized. This represents a huge task, especially for large power Systems such as the Continental Eiiropean grid. Nowadays, significant efforts are performed to validate the dynamic model of the European power System; and an example is the current iTesla Project Assuming that an accurate dynamic model is available, the System can be linearized around an operating point for the purpose of small-signal stability analysis. Modal analysis can then be applied on the linearized System so as to tune controllers and increase the margin to the stability limits.
Another method is to use a measurement-based approach to better monitor the stabil-
Introduction
ity limits. By doing this, the System can be operated doser to the lirnits without increas- ing the risk of instability. Two factors triggered the feasibility of using a measurement- based approach in Europe. Firstly, the need to hâve a wide area view of the grid. Nowa- days, Transmission System Operators (TSO) no longer operate their own national grid without considering other networks. In Central Western Europe (CWE), a coordination center called CORESO (Coordination of Electricity System Operators) has been created in 2009 and is owned by five TSOs (Elia, RTE, 50HerzT, Terna and National Grid)®. Coreso “merges” the snapshots of the participating TSOs in order to hâve a state esti mation of a wide area of the grid. It is worth noting that the state estimation is available every 15 minutes at Coreso and does not allow to observe the power System dynamics. This latter can be partly observed using Phaser Measurement Units (PMU), which pro vide synchronized measurements at a data rate between 10 and 50Hz. The maturity of PMUs facilitâtes wide area applications such as low-frequency oscillation monitoring and is the second factor that makes possible the use of a measurement-based approach. Nowadays, Coreso gets data at each node via the “static” state estimation and observes low-frequency oscillations using a few numbers of PMUs.
This thesis présents a measurement-based approach able to extract the information contained in the wide area “static” state estimation that explains changes in inter-area oscillations. By analyzing each inter-area mode separately, it is expected to approxi- mate the relationship between active power flows in spécifie corridors and the oscillation damping. This relationship could be used to predict future oscillation behaviors and to provide comprehensive advice for operators in order to keep a sufiieient damping at ail time. This task is the challenge posed to the Université libre de Bruxelles in the scope of the European project Twenties. The solution proposed in this thesis is illustrated in Figure 0.0.1. f
PMU
----►
Mode-meterV
JN
y
"Static’’ snapshotN
r \ SCADA/EMSState
estimation
V
y
V
y
uddcd-valucs :- Dctcction of sudden chemges - Detectiou nf constant dauipiug perioclR
Post-processing of the damping estimâtes
added-values :
- Sélection of rritical corridors - Pnxliction
Measurement-based
modül
I
Figure 0.0.1: Flowehart of the proposed solution. The main contributions are emphasized in grey.
Introduction
Twenties
Twenties is the acronym for “Transmission System operation with large pénétration of Wind and other renewable Electricity sources in Networks by means of innovative Tools and Integrated Energy Solutions”. The Twenties project was launched in 2009 and in volves 26 partners in the whole Europe. The main objective is®:
demonstrating by early 2014 through real life, large scale démonstrations, the benefits and impacts of several critical technologies required to improve the pan-European trans mission network, thus giving Europe a capability of responding to the increasing share of renewable in its energy mix by 2020 and beyond while keeping its présent level of reliability performance''
The Twenties project was divided into three main questions:
• What are the valuable contributions that intermittent génération and
flexible load can brHng to System services?
— Demo 1
* Title : SYSTEM SERVICES PROVIDED BY WIND FARMS * Leader : IBERDROLA
* Objectives: Tests to provide new active and reactive power control services to the System, using improved Systems, devices and tools, but keeping the current hardware at wind farm level.
— Demo 2
* Title : LARGE SCALE VIRTUAL POWER PLANT INTEGRATION * Leader : Dong Energy
* Objective : Improve wind intégration based on intelligent energy manage ment of central Combined Beat and Power (CHP), off-shore wind, and local génération and load units in the distribution grid.
• What should the network operators implement to allow for off-shore
wind development?
— Demo 3
* Title : TECHNICAL SPECIFICATIONS TOWARDS OFFSHORE HVDC NETWORKS
Introduction
* Leader : RTE
* Objective : Assess main drivers for the development of off-shore HVDC networks.
- Demo 4
* Title : OFFSHORE WIND FARM MANAGEMENT UNDER STORMY CONDITIONS
=t= Leader : Energinet
* Objective : Demonstrate shut down of wind farms under stormy conditions without jeopardizing safety of the System.
• How to give more flexibility to the transmission grid?
— Demo 5
* Title : NETWORK ENHANCED FLEXIBILITY (NETFLEX)
* Leader : ELIA
* Objective : Demonstrate at régional level (Central Western Europe) how much additional wind génération can be handled thanks to dynamic line rating, coordination of controllable devices (PSTs & HVDCs) and usage of WAMS.
- Demo 6
* Title : IMPROVING THE FLEXIBILITY OF THE TRANSMISSION GRID
* Leader: REE
* Objective : Demonstrating that current transmission network can meet de- mands of renewable energy by extending System operational limits, main- taining safety criteria
Introduction
Démonstration 5: NETFLEX
Planning Real time
Y 20 - Y 2 Gnd Development Y 2 -* Y -1 System Adequacy Forecast, European Outage planning, Génération planning
V
W-x — D-1 Reserve Control, Outage planning review. Génération constraints Force starting/stopping,Cancal grid outage(s}, Act on RFC device(s} No time to reinforcc the infrastRictuns,
Install Ampacimons, PMUs
Génération & Grid plannfng, Mitigation measures New gnd infrastructure and new PFC devices (2-ll)y to be comiViissioned),
Create political consaousness of adequacy issues -» new power plants (2-lOy to be commissioned)
î
c n >27
Congestion Mgt, Load shedding, European Emergency procedures, Emergency plan, Restoration planFigure 0.0.2: Power System planning and operation time frame. Image from [Michiels, 2013].
Introduction
Figure 0.0.3: Location of the PMUs and PSTs considered in this research. The high production of wind in the north of Germany may provoke high flows from the North of Germany through the Benelux région.
Aims of this work
In the context of Twenties, the final objective is to integrate a new tool into the Coreso daily process in order to take into account the possible occurence of low-damped situations. This tool must provide the two following functionalities:
1. Forecasting in day-ahead the damping ratio of each critical mode to evaluate whether a planned operating point is poorly damped.
2. Helping opéra tors to détermine corrective or préventive actions so as to avoid low- damped situations
It is worth emphasizing that operators usually take actions that influence power flows in the network (e.g. by changes of topology, PST taps, generator rescheduling, etc.). Hence, on the one hand, the flows are influenced by the intégration of renewable energy while, on the other hand, the flows can be adapted by operators.
Introduction
using measurements, is provided. For the sake of clarity, the document is structured in four parts, each of them faces one of the following sub-objectives:
1. Evaluating whether low-damped oscillations may be caused by changes of the power flow conditions only.
2. Proposing a solution to recognize previous critical power flow conditions in order to avoid them in the future. To reach this second sub-objective, a measurement-based approach was chosen.
3. Determining the issues inhérent to a measurement-based approach and the solutions proposed to process the measurements.
Contributions and structure of the
thesis
^ Part II
Data mining
Part III
Oscillation monitoringPart IV
Results in Continental Europe / \/
Figure 0.0.4; Structure of the document.
This document is composed of four parts as illustrated in Figure 0.0.4. The first Part introduces the topic and some concepts that will be used in the other parts. Then, Part II describes our data mining approach to approximate the relationship between the power flows and oscillation damping. Before being able to implement this method using measurements from the real grid, Part III reviews oscillation monitoring techniques and proposes a solution to post-process mode-meter damping estimâtes. These two parts are carried out in parallel and are independent of each other. Finally, Part IV details the implémentation and the results obtained using data from the Continental European grid.
Part I : Power system modeling
Introduction
• Chapter 2 makes a short review of power System modeling and modal analysis which is used to illustrate some concepts such as mode shapes and participation factors. Then, simulations on two test Systems are performed to show the influence of the power flow conditions on the eigenvalues of the System.
Objective: The subject of low-frequency oscillations and basics of modal analysis are introduced because these concepts will be used in the rest of the thesis. Then, the objective is to evaluate, using a dynamic model, whether a change of power flows can lead to an insufficient damping situation.
Part II: Data mining
• Chapter 3 begins with a review of data mining techniques applied to inter-area oscillations and formalizes the problem of mapping a relationship between a single output (e.g. the mode damping) and a high-dimensional vector of inputs (e.g. active power flows).
• Chapter 4 describes data mining algorithms and compares them using illustrative examples. Then, a method that takes into account the advantages and limitations of different algorithms is proposed.
• Chapter 5 tests the proposed data mining approach via simulation using a 16- machine network.
Objective and contributions: To propose a new method able to model the rela tionship between the power flows and the damping of oscillations. This method allows to predict the damping based on power flow variables only and to identify the critical Unes for each mode of oscillation.
Part III: Oscillation monitoring
• Chapter 6 is a state of the art of existing oscillation monitoring techniques that hâve been tested in control rooms. Then, the current status of inter-area oscillations in Continental Europe is described using the PMUs available at Coreso.
Introduction
Objective and contributions: To consider the errors resulting from the estima tion of the oscillation damping ratios using measureinents. The proposed method post processes the “blackbox” damping estimâtes in order to detect sudden damping changes and select the periods during which the damping ratio stays constant.
Part IV: Implémentation and experimental testing using measurements from the Continental European grid
• Chapter 8 details the data available at Coreso and how they are gathered. The tool implemented at Coreso handles data coming from two channels, from PMU measurements processed by PhasorPoint OSM and from state estimation every 15 minutes. The last section describes more in detail how the tool has been imple mented.
• Chapter 9 adopts the method proposed in Part 11 and Part III using real grid mea surements from several months of data. Issues concerning the temporal évolution of the grid and the influence of the measurement set on the data mining algorithm are also addressed.
Publications
Conférence communication
• O. Antoine, J-C. Mann and J. Warichet, "Building low-dimensional dumping pre- dictors of the power System modes of oscillation", Transmission and Distribution Conférence and Exposition, 2012.
• O. Antoine and J-C. Mann "Inter-area oscillations: identifying causes of poor dump ing using phasor measurement nuits", IEEE PESGM, 2012.
• O. Antoine, P. Janssen, Q. Jossen and J-C. Mann, "A Laboratory Microgrid for Studying Grid Operations with PMUs", IEEE PESGM, 2013.
• O. Antoine, J. Warichet, C. Druet, G. Jacobs and J-C. Mann, “Inter-area oscillation monitoring expérience in the European Continental grid and study of the load flow influence on the oscillations damping”, submitted to CIGRE Belgium 2014.
Twenties publications
• P. Guha Thakurta, H-M. Nguyen, O. Antoine, J. Maeght, A. Dejong, J. D’Hoker, M. Godemann, D. Van Hertem, P. Schell, F. Skivee, B. Godard, S. Doutreloup, J. Warichet, J-J. Lambin, J-C Mann, R. Belmans and J-L. Lilien, “Final report on NETFLEX Demo”, Twenties, deliverable D7.3, 2013.
• O. Antoine et al, “Ground operational rules for Power flow control devices and DLR”, Twenties, deliverable D13.3, submitted.
Oral présentations
• Twenties Dissémination events, Copenhagen, 2011.
Introduction
• “Applications of PMUs and current results for Twenties”
— RTE, Paris, 2012
— Coreso, Brussels, 2013
Part I
Introduction
The first part of this thesis aims at providing sufRcient information to understand the complex nature of oscillation phenomena, what are the effects on the grid and how oscillations are usually handled.
Chapter 1
Background on small-signal stability
1.1
Classification of stability
A définition of power System stability can be found in [Kundur et al., 2004]:
“Power System stability is the ability of an electric power System, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most System variables bounded so that practically the entire System remains intact.”
Power System instabilities can be divided into different groups (see Figure 1.1). It is worth noting that these groups are not independent and there exists relationships between the different types of instabilities. The three types of instabilities shown in Figure 1.1.1 (i.e. voltage, frequency and rotor angle stability) are successively defined.
Chapter 1: Background on small-signal stability
Voltage stability
“Voltage stability is the ability of a power System to maintain steady acceptable voltages at ail buses in the System under normal operating conditions and after being subjected to a disturbance. The main factor causing instability is the inability of the power System to meet the demand of reactive poiyer. ”[Kundur, 1994]
The voltage instability is mainly a local problem and, may trigger a sequence of events that can lead to major blackouts.
Frequency stability
“Frequency instabilities can occur when the equilibrium between load and production is not satisfied. Therefore, the generator speed will increase or decrease to find an equilib rium point. If the generator speed is too far from the synchronism, it can lose it and provide instability.1994]
In some cases, the network can be divided into islands where each island has his own frequency (depending on the equilibrium between load and production).
Rotor angle stability
Rotor angle stability is the ability of interconnected synchronous machines of a power System to remain in synchronism.”[Kund\iT, 1994]
At steady state, the mechanical torque of each generator is equal to the electrical torque. When a disturbance occurs, the rotors accelerate or decelerate to a new equilib rium point. This type of instability is greatly influenced by the generator angle dynamics and power-angle relationships.
1.2
Rotor angle small-signal stability
Local
modes
\____________ /Inter-area
modes
Control
Torsional
modes
V___________
>
k
modes
y
Chapter 1: Background on small-signal stability
Rotor angle stability can be divided into transient stability following a severe distur bance and small-signal stability in the case of small disturbances (e.g. load switching). To hâve a good understanding of the rotor angle small-signal stability, the concept of synchronizing and damping torques is introduced.
At steady State, the mechanical torque of each generator is equal to the electrical torque and the rotor angle is constant. When a small disturbance occurs, the rotors angle and speed move to reach a new equilibrium point. The electrical torque déviation of each machine is comprised of two components, one in phase with the rotor speed déviation and the other in phase with the rotor angle:
ATe = Ks^ô + KdAüj (1.2.1)
where K s is the synchronizing coefficient in phase with the rotor angle and Kq is the damping coefficient in phase with the rotor speed déviation.
Small-signal instabilities coming from an insufficient synchronizing torque lead to a steady increase of rotor angles. This is well mitigated with the help of automatic control such as automatic voltage regulators (AVR). On the other hand, an inappropriate tuning of automatic controllers can provide a négative damping [DeMello and Concordia, 1969]. For example, voltage regulators at the excitation of the machines increase the synchro nizing torque but hâve the side-effect to decrease the natural damping torque of the machine [Schleif et al., 1968]. Therefore, the major concern of small-signal stability is to damp out electromechanical oscillations. The criteria used in small-signal stability stud- ies is the damping ratio (<^) which characterizes the decrease of the oscillation amplitude [Kundur, 1994]. The amplitude decreases to l/e or (37%) in ^ period. Thus, a damping ratio of 3% means that the oscillation amplitude decreases to l/e in around 5 cycles of this oscillation.
Rotor angle oscillations can be divided in different groups depending on their nature.
• Local modes: typically between one machine and the rest of the network or between two machines in the same local area. The oscillation frequency fies between IHz and 2Hz.
• Inter-area modes: frequency between 0.1 and IHz. The mode involves groups of machines oscillating against each other. It is especially the case when these groups are connected by a weak tie-line, each group being composed of electrically close generators. •
Chapter 1 : Background on small-signal stability
• Torsional modes: involve the turbine-generator shaft System. An interaction can exist with excitation controllers or regulators. The frequency is usually higher than 15Hz.
In this thesis, only inter-area modes are studied. The characteristic of these oscillations is that they involve large areas of the grid. Therefore, it is required to model accu- rately numerous power System components to reproduce the inter-area behavior using simulations.
1.3
Electro-mechanical oscillations
1.3.1
Local oscillations
Figure 1.3.1: Single machine infinité bus System.
In a power System cornposed of n machines, the équation of motion for the z-th machine can be written as:
where H is the constant of inertia of the turbine and generator rotor, OJ is the rotor speed and Tmechi, le* Topresent respectively the mechanical torque produced by the turbine and the electrical torque. The dumping torque is given by T^. which is a fictitious torque representing the dumping contribution of the z-th machine. It has to be noted that if no simplifications are performed in modeling the electrical torque, the term T/j. contains only mechanical frictions.
The équation of motion shows that when the sum of the torques is equal to zéro, the machine speed is constant. Hence, the rotor speed variations are dépendent on the différence between the mechanical torque and the electrical torque. To solve (1.3.1), the équations of the three torques need to be determined.
Chapter 1 : Background on small-signal stability
constant, the expression of the electrical torque is shown on a simple System composed of one machine connected to an infinité bus (see Figure 1.3.1). This simplified System is used to represent the oscillation of one machine against the rest of the network. To facilitate understanding, the generator is modeled using the classical model (i.e. constant voltage behind a transient reactance) and ail résistances are neglected.
According to Figure 1.3.1, the System of équations can be written as:
It = E'Z.0 - Eb^ - S Wt
(1.3.2)
= P + 0 =
=
(1.3.3)X.'j' y\x
where Eb is the voltage at the infinité bus, E' is the voltage behind the transient reactance
and Xt is the sum of the machine and transmission line reactance. The voltage angle ô is the différence between the machine’s internai point and the infinité bus.
In per unit représentation and considering small variations around the synchronous speed, the electrical torque is equal to the electrical power. Therefore, the electrical torque can be expressed as a function of the angle ô:
Z(i)
=P,(i) =
(1.3.4)The electrical power is plotted in Figure 1.3.2. It can be seen that the electrome- chanical oscillations are présent by nature in power Systems and that they dépend on the slope of the P — ô characteristic. For the same AP, a steeper slope will require a smaller A(5 and therefore the frequency of the oscillation will be higher [Farmer, 2001].
Starting from (1.3.4), the dérivative of the electrical torque can be expressed as:
ATe(^) -
^Aô
-= KsAÔ
(1.3.5)uo Xt
where Kg is the synchronizing coefficient. The équation of motion can be written as:
j^Aiür = - 'le
-
KoAtür) (1-3.6)j^ô = LJoAuJr
(1.3.7)Chapter 1 : Background on small-signal stability
Figiire 1.3.2: Initially, there is an equilibrium between the mechanical and the electrical power (point 0). If there is a variation in the mechanical torque, the rotor speed will accelerate according to the équation of motion. When the new equilibrium point (1) is reached, the rotor begins to decelerate until point (2) where the speed déviation changes sign. Then the rotor speed déviation goes in the opposite direction until point (0). The rotor will continue to oscillate around point (1) unless a damping contribution dissipâtes the kinetic energy.
This gives the équations:
d Aüür dt AS 2H iÜQ -Ks 2H 0 AcJr A(5
+
1 2H 0 AT^ (1.3.8)where A is the State matrix, whose eigenvalues can be computed as A = shk
^
uioThe behavior of the System dépends if the term under the square root is positive or négative. Here, we consider the case where this term is négative and leads to an oscillatory behavior of the System. This can be represented by the following complex pair of eigenvalues {X = a ± ju>):
Xi = x; =
-KD + jsj^s U)o Kl
AH (1.3.9)
The imaginary part represents the frequency of the oscillation. Hence, the frequency for an undamped oscillation (Kp = 0) is equal to:
Chapter 1: Background on small-signal stability
This équation shows the influence of the synchronizing torque and the machine inertia on the oscillation frequency. From (1.3.5) and (1.3.10), it can be seen that the synchro nizing torque and therefore the oscillation frequency is higher for a lower value of for a lower external reactance.
The damping ratio (^) is given by:
c= , = , "= (1.3.11)
^(<72 + w2)
V2üüoHKs
In this expression, we can see that there is an influence of the synchronizing coefficient and of the generator inertia. Nevertheless, the main damping contribution cornes from the damping coefficient Kp.
1.3.2
Mechanical équivalent of an inter-area oscillation
The low-frequency (between O.lHz and IHz) inter-area oscillations typically occur between two cohérent groups of generators linked by a weak tie-line. They involve complex mechanisms and for the sake of clarity, the mechanical équivalent proposed in [Samuelsson, 1997] [Messina, 2009] is used.
The mechanical équivalent is a two body spring-mass System representing one inter area mode of oscillation (see Figure 1.3.2). Each body is the équivalent of one cohérent group of generators and the connection between them is similar to electrical transmission fines. The two bodies exchange kinetic energy through an interconnection represented by a spring.
Xo
Figure 1.3.3: Spring-mass model équivalent to an inter-area mode
The mechanical forces Fi and F2 are équivalent to the mechanical input of the ma
chines while the springs stiffness k represents transmission fines. The set of équations can be written as:
MiXi = k{%2 - a:i) -I- F\ M2X2 = k{xi - X2) - F2
Chapter 1 : Background on small-signal stability
By considering the state variables x (position) and v (velocity) comparable to the rotors angle and speed, a state space représentation of the System (composed of a set of differential équations) can be written as:
X = Ax + Bu
MiVi 0 0 -k k Vi 1 0
d M2V2 0 0 k -k V2 0 -1 Fl
dt X'i 1 0 0 0 Xi + 0 0 F2
. ^2 0 1 0 0 52 0 0
The State matrix A has four eigenvalues:
A3 — A4 — 0
(1.3.13)
(1.3.14)
(1.3.15)
The pair of conjugale eigenvalues represents the inter-area mode while the two last form a rigid body mode. It can be observed that the mode frequency is dépendent on the spring stiffness and on the masses. An increase of the stiffness will increase the frequency of the oscillation. The spring stiffness is équivalent to the inverse of the electrical distance for power Systems. The use of a mechanical équivalent is helpful to understand how the oscillation frequency can change.
1.3.3
Parameters influencing the damping torque
It should be emphasized that it is difficult to evaluate the impact of the different factors on oscillation damping [Klein et al., 1991] [Grigsby, 2007]. Hence, the parameters influencing inter-area oscillation damping are listed and their qualitative influence is described. From (1.3.1), it is clear that each parameter influencing the mechanical or the electrical torque may hâve an impact on oscillation damping. The influence of these parameters is explained more in detail in the next chapter.
Automatic voltage regulators &; control Systems
Chapter 1 : Background on small-signal stability
Turbine-governor dynamics
Since the oscillations of interest hâve a low-frequency (<lHz), turbine governor dy namics has to be taken into account. The governor (or speed regulator) may hâve a significant damping influence. For example, the control design of hydro turbines is diffl- cult due to the presence of a right-half plan zéro. The négative damping effect of poorly calibrated hydro turbines with a high governor gain has been observed in Continental Europe [CIGRE, 1996].
Load dynamics
The loads hâve their own dynamics and also contribute to the damping of each mode of oscillation [Milanovic and Hiskens, 1995]. Unfortunately, the type and dynamics of the loads are generally difflcult to evaluate on real power Systems and a static représentation of the loads is used in this thesis.
Power flow conditions
The power flow conditions determined by the generator outputs, loads, and grid topol- ogy hâve a great influence on electromechanical modes. It is known that operating con ditions become very diverse with the increasing intégration of renewable energy sources. Therefore, the operating conditions influence will be analyzed in the following chapters.
Transmission system
The transmission System can be seen as the path between generators that exchange kinetic energy through electrical power. The strength of this path is of great importance and especially the équivalent reactance between groups of machines. Some authors refer the lines along which an oscillation is transmitted as the “dominant oscillation path” [Chompoobutrgool and Vanfretti, 2012].
1.4
Summary and risks related to inter-area oscilla
tions
Summary
Chapter 1 : Background on small-signal stability
Risks
As illustrated in Section 1.3.1, oscillations are inhérent to power Systems due to the interconnection of synchronous generators. Since these oscillations cannot be eliminated, it is important to guarantee sufficient damping at ail time in order to avoid line or generator tripping, islanding and, in the worst case, a blackout. Historically, oscillation problems were related to local modes and the use of fast-excitation with high gain. In addition, the lack of damper windings, especially in hydro turbines, has caused some issues [CIGRE, 1996]. However, the damping ratio is usually neglected in daily power System operation. The damping ratio is assessed in simulation and it is assumed that PSSs will guarantee sufficient damping at ail time. The first risk is that the predicted damping ratio is based on an inaccurate model. The most famous example occurred in 1996 in California [Kosterev et al., 1999]. At that time, the operators did not hâve real- time oscillation monitoring Systems and the dynamic model was not able to reproduce the oscillations. The same sequence of events has been simulated on the WSCC dynamic database and the simulations did not agréé with the recordings.
Nowadays, the problem of electromechanical oscillations is generally associated with the growth of power System. For example, weak transmission lines and heavy power transfers between formerly separated networks may lead to poorly damped oscillations. In this case, the transfer has to be reduced so as to avoid trips of tie-lines, and the séparation of the System into several islands. In this context, it is important to mitigate inter-area oscillations in order to avoid islanding but also to be able to increase the power flow transferred between areas. This can be done by using power system stabilizers which can provide adéquate damping. However, the tuning of a PSS has to be foreseen well in advance. Moreover, damping control design using an inacurrate model will not be able to damp oscillations. The second risk cornes from the fact that the power system is evolving fast and that tuning of a PSS requires changes and redesign that may take too long to put into the field.
Chapter 2
Power System modeling
Power System small-signal stability aasumes that the System can be linearized around an operating point for the purpose of analysis. The use of modal analysis on a linearized power System model is the most common method to study oscillation problems. However, this method dépends on the accuracy of a dynamic model. This chapter describes how modal analysis can be used to understand the influence of network components and to illustrate the impact of power flow conditions on oscillation damping.
First, models of power System components are described. Then, the modal analysis method is presented. Finally, the influence of power flow conditions on the eigenvalues is illustrated using simulation.
2.1
Model of power System components
Power Systems involve several components (e.g. generators, loads, regulators, trans mission lines, etc.) and can be modeled by a set of differential and algebraic équations representing each component and the interactions between them. Inter-area oscillations are diflîcult to analyze because their behavior is influenced by automatic Controls, gen erators dynamics, the strength of the network, loads, etc. [Pal and Chaudhuri, 2005]. Hence, it is important to correctly model each power System component.
2.1.1
Synchronous generator
Chapter 2: Power System modeling
can be chosen. In the case of small-signal stability and the analysis of oscillations, the electromagnetic transients at the stator can be ignored. It is assumed that the time con stant related to stator transients are much faster than swing dynamics. Therefore, the stator variables are considered as algebraic variables [Pal and Chaudhuri, 2005]. How- ever, rotor transients are not ignored and a sixth order model containing the description of the windings in the rotor is often used. In this section, three models are briefly ex- plained using the block diagrams introduced in [Heffron and Phillips, 1952], the first is the classical model composed of a constant voltage behind a transient reactance. Then, a third order model is shown and finally, a sixth order model taking into account the effect of damper windings is described.
Classical model
Figure 2.1.1: Block diagram of the classical model.
The classical model has been described on the single machine infinité bus System in Section 1.3.1. For the sake of clarity, the block diagram of the classical model is shown in Figure 2.1.1. It can be observed that the sum of the torques gives the accélération which can be integrated to compute the rotor speed. The rotor speed can be integrated as well to give the rotor angle. The synchronizing torque is obtained by niultiplying a feedback gain (Kg) on the rotor angle while the damping torque is obtained by multiplying gain {Kd) on the rotor speed.
Third order model
Chapter 2: Power System modeling
of the rotor angle. The coefficient represents the demagnetizing effect of the armature reaction and is usually positive. For oscillations having a frequency around 1 Hz, the field circuit dynamics has usually the effect to reduce the synchronizing torque and increase the damping torque. [Kundur, 1994]
Figure 2.1.2: Block diagram of the third order model
In this model, the electrical torque is dépendent on the rotor angle Ad and on the field flux
Ach, = :-^(Ar^ - /CiAd - K2/\^sd - KoAur) (2.1.1) Lfi
where K\ = with constant 4'/d and with constant angle 5.
Sixth order model
The sixth order model takes into account the effect of the damper windings which cause a decrease of the reactance during subtransient conditions. These windings hâve thus a positive damping contribution. However, the contribution of the damper windings is less significant for high values of the external reactance, which is the case for inter-area oscillations. The three additional State variables are the fluxes in the damper windings (usually one along the d-axis and two along the q-axis).
2.1.2
Excitation System and power System stabilizer
Chapter 2: Power System modeling
is positive and the damping contribution can be négative [DeMello and Concordia, 1969]. To overcoine this négative damping contribution, a power System stabilizer (PSS) can be added to the AVR input. The PSS gives a supplementary signal to the voltage regulator in phase with the speed déviation as shown in Figure 2.1.3. The PSS increases signifi- cantly the damping torque. It should be noted that a PSS may hâve multiple inputs and thus use more than one feedback signal (e.g. active power and frequency).
Figure 2.1.3: Block diagram of a third order machine with AVR and PSS.
2.1.3
Turbine and governor
Figure 2.1.4: Block diagram turbine/governor
Chapter 2: Power System modeling
Figure 2.1.4 shows the turbine-governors System for a steam turbine with a single reheater and for an hydro turbine. One of the main différences between the hydro and steam governor is that the hydro governor needs to take into account the fact that the velocity of the water does not vary directly with the gâte opening. This introduces a right-half plan zéro which is usually mitigated by using a transient droop compensator. Typical values of the turbine parameters are:
• For steam turbines: F^p = 0.3, Tpp = 7s, Tch = 0.3s
• For hydro turbines: = Is
2.1.4
Load characteristics
Load characteristics also hâve a great influence on inter-area oscillations and should be modeled properly. Ideally, load dynamics should also be considered [Banejad, 2004] [Milanovic and Hiskens, 1995]. However, load dynamics is difîicult to détermine accu- rately due to the distributed nature of loads. In this thesis, the following static exponen- tial load model is used:
Pi = Pioi^r (2.1.2)
Qi = (2.1.3)
2.1.5
Transmission lines and network équations
The influence of the external System on the damping coefficient of the generators has been emphasized in Section 1.3.1. Therefore, transmission lines hâve to be modeled accurately as well. The most common représentation is the équivalent 7r-model. The power flow équations at node i are expressed as:
Pc,i - PL,i = E ViVm{GimCOS{ei - 9m) + - 9m)) (2.1.4) m=l
Qc,i - Qb,i = E ^iym{GimSin{9i - 9m) - BimCos{9i - 9m)) (2.1.5) m=l
Chapter 2; Power System modeling
2.2
Modal analysis
A power System can be represented by a set of differential algebraic équations (DAE) [Samuelsson, 1997].
X = f{x,Xa,u)
0 = g{x,Xa,u) (2.2.1)
y = h{x, Xa, u)
where
x
is the vector containing the differential variables (e.g. rotors speed and angle),Xa
is the vector containing the algebraic variables (e.g. bus voltages and voltage angles), U is the input vector and y is the output vector. / and g are the differential and algebraic équations andh
is a vector of output équations.For small-signal stability analysis, it is assumed that the disturbance is sufficiently small and that the System can be linearized at an operating point:
Ax-
=
x- x-° (2.2.2)^a Au = U —
Ay = y - y°
By incorporating the input and output vectors in the algebraic variables, the new vector Ax'a is formed. The following linear set of differential algebraic équations is found:
E
AxAXa — ^full Ax AXa
(2.2.3)
The State matrix
Afuii
is called the full state matrix while a reduced State matrixA
can be obtained by expressing the differential équations in fonction of the algebraic équations. The matrixE
from 2.2.3 is represented as follows:E =
I 00 0 (2.2.4)
The power system modeling software Eurostag [Stubbe et al., 1989] is used in this thesis. Eurostag computes matrices
Af^u
andE.
This allows to use the full State orChapter 2: Power System modeling
differential équation can be found [Kundur, 1994];
Ai; = AAx + B Au (2.2.5)
Ay = CAx + DAu
where Ax is the State vector, Au the input vector and Ay the output vector, A is the State matrix, B is the control matrix, C is the output matrix and D is the feed-forward matrix.
The linearized State matrix A can be used to compute the eigenvalues of the System. The imaginary part of the eigenvalue gives the frequency
u).
The damping ratio(
and tiie eigenvalue A are expressed as follows:\ = a + jüJ
\/a^
-\-uP-Each pair of complex conjugate eigenvalues represents one mode of oscillation. The analysis of eigenvalues is of great importance to assess the stability of the System. How- ever, the influence of the State variables is difficult to evaluate because the évolution of the eigenvalues is influenced by ail State variables [Abed et al., 1999]. One possibility is to découplé the problem by introducing modal coordinates.
Transformation in modal coordinates
The adéquate choice of reference frame may simplify some complex problems. In this case, it is possible to découplé the problem by diagonalizing the state matrix A. This can be done by introducing the concept of eigenvectors as follows:
(f)~^A(t) =
A (2.2.6)^pAip-^ = A (2.2.7)
Where 0 is a matrix containing the right eigenvectors and ijj contains the left eigen vectors. The eigenvectors are scaled so as to satisfy the following condition:
^p = (j)-^ (2.2.8)
Chapter 2: Power system modeling Ax = 4>z Z = ■0Ax (2.2.9) (pz = A(f)Z -1- Bu y = C(f>z -f Du (2.2.10) Z = Az + ipBu y = C(pz + Du (2.2.11)
Equation 2.2.11 shows that the modal controllability matrix ipB expresses the influ ence of the inputs on the modes. Similarly, the modal observability matrix C4> defines the influence of the modes on the outputs.
Pree response of the system
Modal analysis is a useful tool but it is worth relating this analysis to the temporal response of the system. The system free response can be expressed as:
Ax = 2lAx
(j)Z = A4>z (2.2.12)
By inverting the right eigenvectors and simplifying, we obtain n first-order uncoupled équations:
(2.2.13)
The solution of this équation can be found and the modal variables, and z can be expressed in function of the physical State variables x:
z{t) = (2.2.14)
t = l
A.x{t) = j^<j>iZi[Q)e^'^ (2.2.15) i=l
Mode shape
Chapter 2: Power System modeling
in a particular mode. However, this amplitude dépends on the units and scaling of the
State variables which makes the comparison difhcult between different variable types.
Participation factor
To overcome the difffculty of comparing the contribution of State variables in one mode, the concept of participation factor has been introduced [Verghese et al., 1982]. The participation factor is defined as:
Vik ■= 'fpkàik (2.2.16)
This équation shows that the participation factor Pki describes the relative contribu tion of the State variable k in the mode i. The product of the left and right eigenvectors makes the participation factor dimensionless. For each mode, the sum of the pki is equal to one. Similarly, the sum of the participation factor for one State variable is equal to one.
Computing the eigenvalues of interest
For large power Systems, it is time-consuming to compute ail eigenvalues. Since we are especially interesting in low-frequency oscillations having a low-damping, it is possible to compute eigenvalues in the area of interest in the complex plane (e.g. by using the Arnold! method [Kundur, 1994]). The eigenvalues can be selected by looking at a maximum frequency and damping ratio (e.g. IHz, 10%). Once the eigenvalues are selected, their corresponding eigenvectors can be computed.
2.3
Power flow influence on mode damping
The review of power System modeling and modal analysis has shown that the eigenval ues can be affected by numerous parameters. This thesis focuses mainly on the influence of power flow conditions on mode damping. Therefore, this influence is now evaluated through simulations on two test Systems. The first is the well-known “Klein-Rogers- Kundur” 4-machine System [Klein et al., 1991] and the second is a 16-machine System rep- resenting the interconnection between New York and New England [Pal and Chaudhuri, 2005].
2.3.1
Generating random power flows
Chapter 2; Power System modeling
coefficient. The random variation of the loads at node i can be expressed as follows [Archer et al., 2008];
- T LL * P^l{l + 2AP[0.5 - (2.3.1)
Qf = TLL * Q%{! + 2AQ10.5 - eg{k)]} (2.3.2)
where TLL is the total load level, P^q and are the base case at load bus i, AP and AQ are the maximum variation and e^pi , Cqp are chosen randomly between 0 and 1 with a uniform distribution. The maximum variation (for real and reactive power) is chosen equal to 0.1. The loads vary thus between -10 and -1-10% for each total load level. There are three base cases of TTL. The low-load case where TLL = 0.8, the medium-load case where TLL = 1 and the high-load case where TLL =1.2. The active power produced by generators is randomly modified as well according to;
P<5^ = TLL * P^\l + 2AP[0.5 - e%{k)]} (2.3.3)
Here also, the maximum variation AP is equal to 0.1. Power fiows which could not be solved were rejected, there is also a mandatory condition on the slack bus to guarantee that ail generators vary in the same range. For each solved power fiow, the System is linearized and the eigenvalues and eigenvectors are computed. The eigenvectors are used to split the modes and the damping ratios are given by the eigenvalues.
2.3.2
4-machine test System
NGAl GENAI NGA2 NLA NLB NGB2
l
J r 0—' GENA2 GENB2 NLA NLB NGBl GENBlFigure 2.3.1; 4-machine “Klein-Rogers-Kundur” test System represented in Eurostag.
The test System is composed of 2 generator areas linked by a weak tie-line. In each area, the two generators are electrically close from each other. The test System data^ are inspired from [Klein et al., 1991]. In the base case, there is a power fiow of around 400 MW from area A to area B. This small test System is used to illustrate an inter-area
Chapter 2; Power System modeling
oscillation and how to carry ont modal analysis. Then, the influence of the power flow conditions is shown by randomly changing the génération and load patterns.
First, a dynamic simulation is performed in order to illustrate the inter-area oscil lation. A line opening occurs a,t t = Is and the line is closed 0.1 5 after. The speed déviation of the machines are plotted in Figure 2.3.2. Observe that the machines in area A oscillate in antiphase against the machines in area B.
50.02 r 50.015 ■ 50.01 • 50.005 ■ 50 49.995 ■ -OMEGA — GENB2 -OMEGA 49.99 • 49.985 0 23456789 10 time (s)
Figure 2.3.2: Speed of the machines. A line opening has the effect to excite the inter-area mode. The local mode between GENAI and GENA2 can also be observed just after the event.
Secondly, the System is linearized and the mode shapes and participation factors are computed. Electromechanical oscillations can be identified by looking at the participation factors of ail the State variables, the speed and the angle of rotors should hâve the greatest participation factors. There are three electromechanical modes, the inter-area mode between area A and area B and two local modes (one between GENAI and GENA2 and the second between GENBl and GENB2). To make a distinction between these modes, the participation factors (or the mode shape amplitudes) of the speed of each machine can be used. Figure 2.3.3 shows the mode shapes and participation factors for the inter-area mode (the State variables shown are the speed of each machine).
gener-Chapter 2: Power System modeiing
ated power flow, the System is linearized and the three eigenvalues corresponding to the electromechanical modes are plotted. If the power fiow can not be solved, the System is not linearized and the eigenvalues are not computed. It can be observed in Figure 2.3.4 that there is a factor greater than 2 between the minimum and the maximum inter-area mode dumping.
Al A2 B1 B2 Gen name
Mode shape
90 0.002
Figure 2.3.3: The mode shape angles confirm that the machines in area A oscillâtes against those in area B. The participation factors of the machines speed show that the machines at the extremities participate the most in the oscillation.
Chapter 2: Power System modeling
2.3.3 16-Machine test System
Figure 2.3.5: 16-machine System (from [Pal and Chaudhuri, 2005])
The System (see Figure 2.3.5) is composed of 5 areas and the original data can be found in [Pal and Chaudhuri, 2005] (the Eurostag files can be found online at the authors webpage^). The dynamic équivalent of the New England Test System (NETS) area is composed of 9 generators and the New York Power System (NYPS) area is represented by 4 machines. Each of the three other areas are modeled by only one équivalent machine that has a much greater inertia than the others. The loads are considered as constant current. Smaller generators (1 through 12) are modeled with IEEE-AC4A type exciters while the large aggregate generators 13 to 16 hâve slow dc-exciters (type lEEE-DClA) [IEEE, 2006].
Modal analysis
The System is linearized and modal analysis is performed. The eigenvalues and partici pation factors show that there are three inter-area oscillations of interest in this System. The summary of the most participating machines and areas involved in each mode is shown in Table 2.3.1. The mode shapes of the 0.67 Hz mode are illustrated here because the two other modes will be analyzed more in detail in Section 5.2.
Chapter 2: Power System modeling
Mode i UJi G Generators vs Generators 1 0.35 Hz 6.8% 14,15,16 (areas 3,4,5) VS 1-13 (areas 1,2)
2 0.54 Hz 8.9% 1,2,3,4,5,6,7,8 (area 1) VS 11-13 (area 2)
3 0.67 Hz 5.5% 14 (area 3) VS 16 (area 5)
Table 2.3.1: Inter-area oscillations for the base case. The machines having the highest participation factor for each mode are in bold.
Gen number: State variable=machine speed
Figure 2.3.6: 0.67 Hz inter-area oscillations. Mode shapes and participation factors for the 16 machines speeds. It can be observed that machines 14 and 16 participate the most and that there is an angle différence of tt (or 180 degrees).
Random génération of operating points
Chapter 2: Power System modeling
Figure 2.3.7: Eigenvalues in the complex plane for 900 power flow scénarios. It can be observée! that the load level has a significant influence. However, this influence is different for each mode. It can also be seen that the damping ratios vary more for the high-load scénario
Observations
From modal analysis, three inter-area modes hâve been observed. These modes hâve a damping ratio lower than 10%, a frequency lower than 0.7 Hz and involve electrically distant areas of the grid. For the base case, illustrated in Table 2.3.1, the 0.67 Hz mode has the lowest damping ratio while the 0.54 Hz mode has the highest damping. Interestingly, Figure 2.3.7 shows that the damping of the latter can become low while the damping of the 0.67 Hz mode does not vary much. Hence, this emphasizes that the impact of power flow variations is different for each mode.
Discussion and summary
The first part of this thesis has explained the phenomenon of low-frequency oscillations and listed possible risks related to inter-area oscillations.
The second chapter introduces mathematical tools needed to study power System small-signal stability. The basics of modal analysis hâve been reviewed and the concepts of mode shapes and participation factors hâve been illustrated using simulations. Mode shapes allow to détermine the oscillating areas while participation factors give the con tribution of each State to a mode. The use of modal analysis to design controllers has not been shown but it is obvions that modal analysis is helpful to tune a power System stabilizer (PSS).
