Wide area measurement-based approach for assessing the power ﬂow inﬂuence on inter-area oscillations

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Wide 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

Academic year 2013 - 2014

Thesis director: Prof. Jean-Claude Maun ULB, Brussels, Belgium

President of the committee: Prof. Michel Kinnaert ULB, Brussels, Belgium Members of the committee: Prof. Johan Gyselinck ULB, Brussels, Belgium

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Acknowledgments

First of all, I would like to thank Prof. Jean-Claude Maun 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 all the members of the Twenties project, especially the participants of the demonstration 5. It has been a real pleasure to work with all of you. It is diﬃcult 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 diﬃculty 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 have such a wonderful atmosphere at the oﬃce. Thank you for that and I wish you the best with your respective projects.123

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.

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Abstract

Power systems have been historically designed 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 integration 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.

All 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 all 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 have 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 prediction 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 damping estimates. This post-processing method consists in a noise reduction technique followed by a damping change detection algorithm.

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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 under 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 have always been present by nature in power systems and can not be avoided. During the last decades, 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 all modes of oscillation and for all the possible operating points. The reality is that power systems are continuously changing, former isolated systems are now interconnected and the increasing integration of renewables affects significantly the grid. All 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 European grid. Nowadays, signiﬁcant eﬀorts are performed to validate the dynamic model of the European power system; and an example is the current iTesla project 4_{. 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-4

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Introduction

ity limits. By doing this, the system can be operated closer to the limits without increas-ing the risk of instability. Two factors triggered the feasibility of usincreas-ing a measurement-based approach in Europe. Firstly, the need to have 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)5_. Coreso “merges” the snapshots of the participating TSOs in order to have a state esti-mation of a wide area of the grid. It is worth noting that the state estiesti-mation is available every 15 minutes at Coreso and does not allow to observe the power system dynamics. This latter can be partly observed using Phasor Measurement Units (PMU), which pro-vide synchronized measurements at a data rate between 10 and 50Hz. The maturity of PMUs facilitates 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 presents 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 specific 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 sufficient damping at all 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.

Figure 0.0.1: Flowchart of the proposed solution. The main contributions are emphasized in grey.

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Introduction

Twenties

Twenties is the acronym for “Transmission system operation with large penetration 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 is6_:

“demonstrating by early 2014 through real life, large scale demonstrations, 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 present level of reliability performance”

The Twenties project was divided into three main questions:

• What are the valuable contributions that intermittent generation and flexible load can bring 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 integration based on intelligent energy manage-ment of central Combined Heat and Power (CHP), oﬀ-shore wind, and local generation 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

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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

∗ 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 regional level (Central Western Europe) how much additional wind generation 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

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Introduction

Demonstration 5: NETFLEX

Figure 0.0.2: Power system planning and operation time frame. Image from [Michiels, 2013].

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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 region.

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 operators to determine corrective or preventive 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 integration of renewable energy while, on the other hand, the flows can be adapted by operators.

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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 inherent to a measurement-based approach and the solutions proposed to process the measurements.

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Contributions and structure of the

thesis

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 estimates. These two parts are carried out in parallel and are independent of each other. Finally, Part IV details the implementation and the results obtained using data from the Continental European grid.

Part I : Power system modeling

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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 lines for each mode of oscillation.

Part III: Oscillation monitoring

• Chapter 6 is a state of the art of existing oscillation monitoring techniques that have been tested in control rooms. Then, the current status of inter-area oscillations in Continental Europe is described using the PMUs available at Coreso.

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Introduction

Objective and contributions: To consider the errors resulting from the

estima-tion of the oscillaestima-tion damping ratios using measurements. The proposed method post-processes the “blackbox” damping estimates in order to detect sudden damping changes and select the periods during which the damping ratio stays constant.

Part IV: Implementation 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 II and Part III using real grid mea-surements from several months of data. Issues concerning the temporal evolution of the grid and the inﬂuence of the measurement set on the data mining algorithm are also addressed.

Objective and contributions: To illustrate that the tool combining the

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Publications

Conference communication

• O. Antoine, J-C. Maun and J. Warichet, "Building low-dimensional damping pre-dictors of the power system modes of oscillation", Transmission and Distribution Conference and Exposition, 2012.

• O. Antoine and J-C. Maun "Inter-area oscillations: identifying causes of poor damp-ing usdamp-ing phasor measurement units", IEEE PESGM, 2012.

• O. Antoine, P. Janssen, Q. Jossen and J-C. Maun, "A Laboratory Microgrid for Studying Grid Operations with PMUs", IEEE PESGM, 2013.

• O. Antoine, J. Warichet, C. Druet, G. Jacobs and J-C. Maun, “Inter-area oscillation monitoring experience 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 Maun, 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 presentations

• Twenties Dissemination events, Copenhagen, 2011.

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Introduction

• “Applications of PMUs and current results for Twenties”

– RTE, Paris, 2012 – Coreso, Brussels, 2013

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Part I

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Introduction

The first part of this thesis aims at providing sufficient information to understand the complex nature of oscillation phenomena, what are the effects on the grid and how oscillations are usually handled.

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Chapter 1 Background on small-signal stability

1.1 Classification of stability

A deﬁnition 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.

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Chapter 1: Background on small-signal stability

Voltage stability

“Voltage stability is the ability of a power system to maintain steady acceptable voltages at all 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 power.”[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.”[Kundur, 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.”[Kundur, 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

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Rotor angle stability can be divided into transient stability following a severe distur-bance and small-signal stability in the case of small disturdistur-bances (e.g. load switching). To have 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 deviation of each machine is comprised of two components, one in phase with the rotor speed deviation and the other in phase with the rotor angle:

∆Te = KS∆δ + KD∆ω (1.2.1)

where KS is the synchronizing coeﬃcient in phase with the rotor angle and KD is the

damping coeﬃcient in phase with the rotor speed deviation.

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 negative damping [DeMello and Concordia, 1969]. For example, voltage regulators at the excitation of the machines increase the synchro-nizing torque but have 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 (ζ) which characterizes the decrease of the oscillation amplitude [Kundur, 1994]. The amplitude decreases to 1/e or (37%) in _2πς1 period. Thus, a damping ratio of 3% means that the oscillation amplitude decreases to 1/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 lies between 1Hz and 2Hz.

• Inter-area modes: frequency between 0.1 and 1Hz. 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.

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• 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 inﬁnite bus system.

In a power system composed of n machines, the equation of motion for the i-th machine can be written as:

dωi

dt = ωi

2Hi

(Tmechi − Tei− TDi) (1.3.1)

where H is the constant of inertia of the turbine and generator rotor, ω is the rotor speed and Tmechi, Tei represent respectively the mechanical torque produced by the turbine and

the electrical torque. The damping torque is given by TDi which is a ﬁctitious torque

representing the damping contribution of the i-th machine. It has to be noted that if no simpliﬁcations are performed in modeling the electrical torque, the term TDi contains

only mechanical frictions.

The equation of motion shows that when the sum of the torques is equal to zero, the machine speed is constant. Hence, the rotor speed variations are dependent on the diﬀerence between the mechanical torque and the electrical torque. To solve (1.3.1), the equations of the three torques need to be determined.

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constant, the expression of the electrical torque is shown on a simple system composed of one machine connected to an inﬁnite 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 all resistances are neglected.

According to Figure 1.3.1, the system of equations can be written as:

It= E′_∠0_{− E} B∠− δ jXT (1.3.2) S′ = P + jQ′ = E′It∗ = E′_E_B_sinδ XT + jE′(E′− EBcosδ) XT (1.3.3) where EBis the voltage at the inﬁnite bus, E′ is the voltage behind the transient reactance

and Xt is the sum of the machine and transmission line reactance. The voltage angle δ

is the diﬀerence between the machine’s internal point and the inﬁnite bus.

In per unit representation 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 δ:

Te(δ) = Pe(δ) =

E′_E_B_sinδ

XT

(1.3.4) The electrical power is plotted in Figure 1.3.2. It can be seen that the electrome-chanical oscillations are present by nature in power systems and that they depend on the slope of the P − δ characteristic. For the same ∆P , a steeper slope will require a smaller ∆δ and therefore the frequency of the oscillation will be higher [Farmer, 2001].

Starting from (1.3.4), the derivative of the electrical torque can be expressed as:

∆Te(δ) = ∂Te ∂δ ∆δ = E′_E_B_cosδ 0 XT ∆δ = KS∆δ (1.3.5)

where Ks is the synchronizing coeﬃcient.

The equation of motion can be written as:

d dt∆ωr = 1 2H(Tm− Te− KD∆ωr) (1.3.6) d dtδ = ω0∆ωr (1.3.7)

where ∆ωris the speed deviation in per unit, δ is the rotor angle, ω0 is the base rotor

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Chapter 1: Background on small-signal stability Pe= E ′_E_B Xt sin(δ) Pmech1 Pmech0 2 1 0 A ct iv e p ow er angle δ

Figure 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 equation of motion. When the new equilibrium point (1) is reached, the rotor begins to decelerate until point (2) where the speed deviation changes sign. Then the rotor speed deviation goes in the opposite direction until point (0). The rotor will continue to oscillate around point (1) unless a damping contribution dissipates the kinetic energy.

This gives the equations:

d dt   ∆ωr ∆δ   =   −KD 2H −K S 2H ω0 0     ∆ωr ∆δ   +   1 2H 0  ∆Tm (1.3.8)

where A is the state matrix, whose eigenvalues can be computed as λ = −KD+√K_D2−8HKsω0

4H .

The behavior of the system depends if the term under the square root is positive or negative. Here, we consider the case where this term is negative and leads to an oscillatory behavior of the system. This can be represented by the following complex pair of eigenvalues (λ = σ ± jω): λ1 = λ∗2 = −KD + j q 8HKsω0− KD2 4H (1.3.9)

The imaginary part represents the frequency of the oscillation. Hence, the frequency for an undamped oscillation (KD = 0) is equal to:

ω =

s

ω0Ks

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This equation 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 δ0 and for a lower external reactance.

The damping ratio (ζ) is given by:

ζ = q −σ (σ2_{+ ω}2₎ = KD √ 2ω0HKS (1.3.11)

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 comes from the damping coefficient KD.

1.3.2 Mechanical equivalent of an inter-area oscillation

The low-frequency (between 0.1Hz and 1Hz) inter-area oscillations typically occur between two coherent groups of generators linked by a weak tie-line. They involve complex mechanisms and for the sake of clarity, the mechanical equivalent proposed in [Samuelsson, 1997] [Messina, 2009] is used.

The mechanical equivalent is a two body spring-mass system representing one inter-area mode of oscillation (see Figure 1.3.2). Each body is the equivalent of one coherent group of generators and the connection between them is similar to electrical transmission lines. The two bodies exchange kinetic energy through an interconnection represented by a spring.

Figure 1.3.3: Spring-mass model equivalent to an inter-area mode

The mechanical forces F1 and F2 are equivalent to the mechanical input of the ma-chines while the springs stiﬀness k represents transmission lines. The set of equations can be written as:

M1x¨1 = k(x2− x1) + F1

M2x¨2 = k(x1− x2) − F2

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By considering the state variables x (position) and v (velocity) comparable to the rotors angle and speed, a state space representation of the system (composed of a set of diﬀerential equations) can be written as:

˙x = Ax + Bu (1.3.13) d dt         M1v1 M2v2 x1 x2         =         0 0 −k k 0 0 k _−k 1 0 0 0 0 1 0 0                 v1 v2 x1 x2         +         1 0 0 −1 0 0 0 0           F1 F2   (1.3.14)

The state matrix A has four eigenvalues:

λ1 = λ∗2 = j

r

k_M1₁ +_M1₂

λ3 = λ4 = 0

(1.3.15)

The pair of conjugate eigenvalues represents the inter-area mode while the two last form a rigid body mode. It can be observed that the mode frequency is dependent on the spring stiffness and on the masses. An increase of the stiffness will increase the frequency of the oscillation. The spring stiffness is equivalent to the inverse of the electrical distance for power systems. The use of a mechanical equivalent 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 have an impact on oscillation damping. The influence of these parameters is explained more in detail in the next chapter.

Automatic voltage regulators & control systems

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Turbine-governor dynamics

Since the oscillations of interest have a low-frequency (<1Hz), turbine governor dy-namics has to be taken into account. The governor (or speed regulator) may have a significant damping influence. For example, the control design of hydro turbines is diffi-cult due to the presence of a right-half plan zero. The negative damping effect of poorly calibrated hydro turbines with a high governor gain has been observed in Continental Europe [CIGRE, 1996].

Load dynamics

The loads have their own dynamics and also contribute to the damping of each mode of oscillation [Milanovič and Hiskens, 1995]. Unfortunately, the type and dynamics of the loads are generally diﬃcult to evaluate on real power systems and a static representation 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 have a great influence on electromechanical modes. It is known that operating con-ditions become very diverse with the increasing integration 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 equivalent 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

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Risks

As illustrated in Section 1.3.1, oscillations are inherent to power systems due to the interconnection of synchronous generators. Since these oscillations cannot be eliminated, it is important to guarantee suﬃcient damping at all 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 all 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 have 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 agree 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 separation 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 adequate 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 comes 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.

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Chapter 2 Power system modeling

Power system small-signal stability assumes 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 depends 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 equations representing each component and the interactions between them. Inter-area oscillations are difficult 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

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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 inﬁnite 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 acceleration 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 multiplying a feedback gain (Ks) on the rotor angle while the damping torque is obtained by multiplying gain

(KD) on the rotor speed.

Third order model

Compared to the classical model, the third order includes the eﬀect of the ﬁeld winding along the d-axis. The state variable ∆Ψf dtakes into account the variation of the reactance

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of the rotor angle. The coefficient K4 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 dependent on the rotor angle ∆δ and on the ﬁeld ﬂux ∆Ψf d.

∆ ˙ωr =

1

2H(∆Tm− K1∆δ − K2∆Ψf d− KD∆ωr) (2.1.1) where K1 = ∆Telec

∆δ with constant Ψf d and K2 = ∆Telec

∆Ψf d with constant angle δ. 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 have 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

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is positive and the damping contribution can be negative [DeMello and Concordia, 1969]. To overcome this negative 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 deviation as shown in Figure 2.1.3. The PSS increases signiﬁ-cantly the damping torque. It should be noted that a PSS may have 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

Load ref.

Figure 2.1.4: Block diagram turbine/governor

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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 diﬀerences 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 gate opening. This introduces a right-half plan zero which is usually mitigated by using a transient droop compensator. Typical values of the turbine parameters are:

• For steam turbines: FHP = 0.3, TRH = 7s, TCH = 0.3s

• For hydro turbines: Tw = 1s

2.1.4 Load characteristics

Load characteristics also have a great inﬂuence on inter-area oscillations and should be modeled properly. Ideally, load dynamics should also be considered [Banejad, 2004] [Milanovič and Hiskens, 1995]. However, load dynamics is diﬃcult to determine accu-rately due to the distributed nature of loads. In this thesis, the following static exponen-tial load model is used:

Pl = Pl0( Vl Vl0 )α _(2.1.2) Ql = Ql0( Vl Vl0 )β (2.1.3)

2.1.5 Transmission lines and network equations

The influence of the external system on the damping coefficient of the generators has been emphasized in Section 1.3.1. Therefore, transmission lines have to be modeled accurately as well. The most common representation is the equivalent π-model. The power flow equations at node i are expressed as:

PG,i− PL,i= n X m=1 ViVm(Gimcos(θi− θm) + Bimsin(θi− θm)) (2.1.4) QG,i− QL,i= n X m=1 ViVm(Gimsin(θi − θm) − Bimcos(θi− θm)) (2.1.5)

where PG,iis the generated active power and PL,ithe active power load at node i; Vi and

Vm represent the voltage at the nodes and; θi and θm are the voltage angles with respect

to a common reference. For i diﬀerent from m, Gim and Bim are the conductance and

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2.2 Modal analysis

A power system can be represented by a set of diﬀerential algebraic equations (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. f and g are the diﬀerential and algebraic

equations and h is a vector of output equations.

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:

∆x = x − x0 _(2.2.2)

∆xa= xa− x0a

∆u = u − u0 ∆y = y − y0

By incorporating the input and output vectors in the algebraic variables, the new vector ∆˜xa is formed. The following linear set of diﬀerential algebraic equations is found:

E   ∆ ˙x ∆ ˙˜xa  = Af ull   ∆x ∆˜xa   (2.2.3)

The state matrix Af ull is called the full state matrix while a reduced state matrix

A can be obtained by expressing the diﬀerential equations in function of the algebraic

equations. The matrix E from 2.2.3 is represented as follows:

E =   I 0 0 0   (2.2.4)

The power system modeling software Eurostag [Stubbe et al., 1989] is used in this thesis. Eurostag computes matrices Af ull and E. This allows to use the full state or

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diﬀerential equation can be found [Kundur, 1994]:

∆ ˙x = A∆x + B∆u (2.2.5)

∆y = C∆x + D∆u

where ∆x is the state vector, ∆u the input vector and ∆y 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 ω. The damping ratio ζ and the eigenvalue λ are expressed as follows:

λ = σ + jω ζ = _√ −σ

σ2_{+ ω}2

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 evolution of the eigenvalues is influenced by all state variables [Abed et al., 1999]. One possibility is to decouple the problem by introducing modal coordinates.

Transformation in modal coordinates

The adequate choice of reference frame may simplify some complex problems. In this case, it is possible to decouple the problem by diagonalizing the state matrix A. This can be done by introducing the concept of eigenvectors as follows:

φ−1Aφ = Λ (2.2.6)

ψAψ−1 = Λ (2.2.7)

Where φ is a matrix containing the right eigenvectors and ψ contains the left eigen-vectors. The eigenvectors are scaled so as to satisfy the following condition:

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Chapter 2: Power system modeling ∆x = φz z = ψ∆x (2.2.9) φ ˙z = Aφz + Bu y = Cφz + Du (2.2.10) ˙z = Λz + ψBu y = Cφz + Du (2.2.11)

Equation 2.2.11 shows that the modal controllability matrix ψB expresses the influ-ence of the inputs on the modes. Similarly, the modal observability matrix Cφ defines the influence of the modes on the outputs.

Free 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:

∆ ˙x = A∆x

φ ˙z = Aφz (2.2.12)

By inverting the right eigenvectors and simplifying, we obtain n ﬁrst-order uncoupled equations:

˙

zi = λizi (2.2.13)

The solution of this equation can be found and the modal variables, and z can be expressed in function of the physical state variables x:

z(t) = n X i=1 ψi∆xi(0)eλit (2.2.14) ∆x(t) = n X i=1 φizi(0)eλit (2.2.15) Mode shape

The concept of “mode shape” is now introduced in order to help the understanding of the system. Equation (2.2.15) allows to express that the right eigenvector φi relates the

activity of the state variables xi to the mode zk, while the left eigenvector ψi indicates

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in a particular mode. However, this amplitude depends on the units and scaling of the state variables which makes the comparison diﬃcult between diﬀerent variable types.

Participation factor

To overcome the diﬃculty 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:

pik := ψkiφik (2.2.16)

This equation 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 all 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 Arnoldi method [Kundur, 1994]). The eigenvalues can be selected by looking at a maximum frequency and damping ratio (e.g. 1Hz, 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

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coeﬃcient. The random variation of the loads at node i can be expressed as follows [Archer et al., 2008]: PL(i) = T LL ∗ P (i) LO{1 + 2∆P [0.5 − ǫ (i) P L(k)]} (2.3.1)

Q(i)_L _{= T LL ∗ Q}(i)_LO_{{1 + 2∆Q[0.5 − ǫ}(i)_QL_(k)]} (2.3.2) where TLL is the total load level, P_LO(i) and Q(i)_LO are the base case at load bus i, ∆P and ∆Q are the maximum variation and ǫi

P L , ǫiQL 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 +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:

PG(i) = T LL ∗ P

(i)

G {1 + 2∆P [0.5 − ǫ

(i)

P G(k)]} (2.3.3)

Here also, the maximum variation ∆P is equal to 0.1. Power ﬂows which could not be solved were rejected, there is also a mandatory condition on the slack bus to guarantee that all generators vary in the same range. For each solved power ﬂow, 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

Figure 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 data1 are inspired from [Klein et al., 1991]. In the base case, there is a power ﬂow of around 400 MW from area A to area B. This small test system is used to illustrate an inter-area

1

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oscillation and how to carry out modal analysis. Then, the inﬂuence of the power ﬂow conditions is shown by randomly changing the generation and load patterns.

First, a dynamic simulation is performed in order to illustrate the inter-area oscil-lation. A line opening occurs at t = 1 s and the line is closed 0.1 s after. The speed deviation 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.

0 1 2 3 4 5 6 7 8 9 10 49.985 49.99 49.995 50 50.005 50.01 50.015 50.02 time (s) Machine speed (Hz) GENA1 −OMEGA GENA2 −OMEGA GENB1 −OMEGA GENB2 −OMEGA

Figure 2.3.2: Speed of the machines. A line opening has the eﬀect to excite the inter-area mode. The local mode between GENA1 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 all the state variables, the speed and the angle of rotors should have 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 GENA1 and GENA2 and the second between GENB1 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).

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gener-Chapter 2: Power system modeling

ated power ﬂow, the system is linearized and the three eigenvalues corresponding to the electromechanical modes are plotted. If the power ﬂow 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 damping. Mode shape 0 180 330 150 300 120 270 90 240 60 210 30 0.002 0.001 M o d e sh a p e p h a se (r a d ) Gen name P a rt ic ip a ti o n fa ct o r Gen name A1 A2 B1 B2 A1 A2 B1 B2 −2 −1 0 1 2 0 0.05 0.1 0.15 0.2

Figure 2.3.3: The mode shape angles conﬁrm that the machines in area A oscillates 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.

Im a g in a ry p a rt (r a d / s) Real part (1/s)

local mode A

inter-area

local mode B

−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0 1 2 3 4 5 6 7

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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 webpage2_{). The dynamic equivalent 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 equivalent 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 have slow dc-exciters (type IEEE-DC1A) [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.

2

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Mode i ωi ζi 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.

M o d e sh a p e p h a se (r a d )

Gen number: state variable=machine speed

P a rt ic ip a ti o n fa ct o r

Gen number: state variable=machine speed

Mode shape 0 180 330 150 300 120 270 90 240 60 210 30 0.005 0.0025 0 5 10 15 0 5 10 15 −4 −2 0 2 4 0 0.1 0.2 0.3 0.4

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 diﬀerence of π (or 180 degrees).

Random generation of operating points

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Chapter 2: Power system modeling 5% Low-load Mid-load High-load 10% m o d e fr eq u en cy (H z) real part (1/s) −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 2.3.7: Eigenvalues in the complex plane for 900 power flow scenarios. It can be observed 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 scenario

Observations

From modal analysis, three inter-area modes have been observed. These modes have 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.

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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 have been reviewed and the concepts of mode shapes and participation factors have been illustrated using simulations. Mode shapes allow to determine 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 obvious that modal analysis is helpful to tune a power system stabilizer (PSS).

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Part II

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Introduction

Electromechanical oscillation problems are usually studied using a dynamic model of the power system. A dynamic model is useful to design controllers in order to move the eigenvalues further in the left part of the complex plane. Controllers such as power system stabilizers (PSS) are typically tuned to damp out low-frequency oscillations. However, these controllers are optimized for local modes and for a range of operating points. Since “new” operating points are experienced due to the increases in load consumption and the integration of renewables, the influence of controllers on inter-area mode damping could be insufficient. Another possible use of dynamic models is to assess the damping of operating points in daily operation. However, dynamic models of large networks are difficult to be used in real-time because of high computational requirements. Moreover, a dynamic model could be inaccurate and lead to wrong conclusions.

A complementary approach to the use of a dynamic model is to perform a measurement-based approach using data mining algorithms. This allows to learn a “model” between a vector of measured inputs and one measured output (or target). After learning, this measurement-based model can be used for prediction of the target. For inter-area os-cillation purposes, the target must represent a stability indicator (e.g. mode damping ratio). Since the final aim is to provide a tool for control rooms, the inputs are limited to variables monitored by operators. In the context of an increasing integration of re-newable energy sources which leads to higher active power transfers, the impact of line active power flows on mode damping is of particular interest. In addition, the influence of active power generated as well as load consumption has to be analyzed. These values can be measured or computed from the results of state estimation and are therefore already available in control rooms.

By monitoring power ﬂow variables on the one hand and the mode damping ratio on the other hand, one can approximate the relationship between them. This approximated relationship (or model) can be used for three applications:

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3. Evaluating the impact of power ﬂow changes in order to adapt mode damping It is worth emphasizing that we use the term “adapt” and not “control” since the objective is to increase the damping ratio if needed, but not to control it using an automated feedback loop.

Wide area measurement-based approach for assessing the power ﬂow inﬂuence on inter-area oscillations