Dynamic Analysis and Control of Multi-machine Power System ...€¦ · Dynamic Analysis and Control...

Dynamic Analysis and Control of Multi-machine Power

System with Microgrids: A Koopman Theory Approach

Ibrahima Diagne

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Electrical Engineering

Lamine Mili, Chair

Virgilio Centeno

Robert Broadwater

Jih Lai

Arnold Urken

September 27, 2016

Falls Church, Virginia

Microgrids, Voltage Control, Modal Analysis, Koopman Mode Analysis.

c©Copyright 2016, Ibrahima Diagne

Dynamic Analysis and Control of Multi-machine Power System

with Microgrids: A Koopman Theory Approach

Ibrahima Diagne

ABSTRACT

Electric power systems are undergoing significant changes with the deployment of large-scale wind

and solar plants connected to the transmission system and small-scale Distributed Energy Resources

(DERs) and microgrids connected to the distribution system, making the latter an active system. A

microgrid is a small-scale power system that interconnects renewable and non-renewable generating

units such as solar photo-voltaic panels and micro-turbines, storage devices such as batteries and fly

wheels, and loads. Typically, it is connected to the distribution feeders via power electronic converters

with fast control responses within the micro-seconds. These new developments have prompted growing

research activities in stability analysis and control of the transmission and the distribution systems.

Unfortunately, these systems are treated as separated entities, limiting the scope of the applicability

of the proposed methods to real systems. It is worth stressing that the transmission and distribution

systems are interconnected via HV/MV transformers and therefore, are interacting dynamically in a

complex way. In this research work, we overcome this problem by investigating the dynamics of the

transmission and distribution systems with parallel microgrids as an integrated system . Specifically,

we develop a generic model of a microgrid that consists of a DC voltage source connected to an inverter

with real and reactive power control and voltage control. We analyze the small-signal stability of

the two-area four-machine system with four parallel microgrids connected to the distribution feeders

though different impedances. We show that the conventional PQ control of the inverters is insufficient

to stabilize the voltage at the point-of-common coupling when the feeder impedances have highly

unequal values. To ensure the existence of a stable equilibrium point associated with a sufficient

stability margin of the system, we propose a new voltage control implemented as an additional feedback

control loop of the conventional inner and outer current control schemes of the inverter. Furthermore,

we carry out a modal analysis of the four-machine system with microgrids using Koopman mode

analysis. We reveal the existence of local modes of oscillation of a microgrid against the rest of

the system and between parallel microgrids at frequencies that range between 0.1 and 3 Hz. When

the control of the microgrid becomes unstable, the frequencies of the oscillation are about 20 Hz.

Recall that the Koopman mode analysis is a new technique developed in fluid dynamics and recently

introduced in power systems by Suzuki and Mezic. It allows us to carry out small signal and transient

stability analysis by processing only measurements, without resorting to any model and without

assuming any linearization.

Dynamic Analysis and Control of Multi-machine Power System

with Microgrids: A Koopman Theory Approach

Ibrahima Diagne

GENERAL AUDIENCE ABSTRACT

Electric power systems are undergoing significant changes with the deployment of large-scale wind

and solar plants connected to the transmission system and small-scale Distributed Energy Resources

(DERs) and microgrids connected to the distribution system, making the latter an active system. A

microgrid is a small-scale power system that interconnects renewable and non-renewable generating

units such as solar photo-voltaic panels and micro-turbines, storage devices such as batteries and fly

wheels, and loads. Typically, it is connected to the distribution feeders via power electronic converters

with fast control responses within the micro-seconds. These new developments have prompted growing

research activities in stability analysis and control of the transmission and the distribution systems.

Unfortunately, these systems are treated as separated entities, limiting the scope of the applicability

of the proposed methods to real systems. It is worth stressing that the transmission and distribution

systems are interconnected via HV/MV transformers and therefore, are interacting dynamically in a

complex way. In this research work, we overcome this problem by investigating the dynamics of the

transmission and distribution systems with parallel microgrids as an integrated system . Specifically,

we develop a generic model of a microgrid that consists of a DC voltage source connected to an

inverter with real and reactive power control and voltage control. We show that the conventional PQ

control of the inverters is insufficient to stabilize the voltage at the point-of-common coupling when

the feeder impedances have highly unequal values. Furthermore, we carry out a modal analysis of

the four-machine system with microgrids using Koopman mode analysis. Koopman mode analysis

is a new technique developed in fluid dynamics and recently introduced in power systems by Suzuki

and Mezic. It allows us to carry out small signal and transient stability analysis by processing only

measurements, without resorting to any model and without assuming any linearization.

Dedication

To the memory of my late parents.

To my wife, Awa , who has supported me in all my endeavors.

i

Acknowledgments

In the name of God, the most Gracious and the Most Merciful.

I wish to express my deepest gratitude and appreciation to my academic advisor Prof. Mili

who has assisted me both academically and financially. I appreciate his support, patience,

and his dedication to research and excitement in regards to teaching. Without his guidance

and persistent help this work would not have been possible and I am grateful to him for his

supportive and friendly attitude. I would like to express my sincere thanks to Prof. Centeno,

Prof. Broadwater, Prof. Lai, and Prof. Urken for accepting to serve as members of my PhD

committee.

I would like to thank my family members, specially my wife, Awa, and my children, Oumou

and Mohamed, for their support and encouragement.

Special thanks to my friend Marcos Netto for his help. Marcos has always been there for

me anytime I needed him. I own him a depth of gratitude for all his assistance.

ii

Contents

1 Introduction 1

1.1 Problem Statement and Research Objectives . . . . . . . . . . . . . . . . . . . . 1

1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Power System Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Microgrid Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.3 Potential Benefits of Microgrids to Bulk Power Systems . . . . . . . . . 7

1.3.4 Proposed Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.5 Dissertation Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Power System Dynamic Modeling Using Koopman Mode Analysis 10

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Taxonomy of Modal Analysis Techniques . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Model Analysis Based on first Order Approximation . . . . . . . . . . . . . . . . 11

2.4 Linear Koopman Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Computation of Koopman Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 22

iii

2.5.1 Koopman Operator from a Linear Algebra Perspective . . . . . . . . . . 25

2.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6.1 Small Signal Analysis With Microgrids . . . . . . . . . . . . . . . . . . . 26

2.6.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6.3 Assumptions, Description of Test-Bed, and Computation of State Matrix 27

2.6.4 Small Signal Analysis using KMA . . . . . . . . . . . . . . . . . . . . . . 31

2.6.5 Transient Stability Analysis using KMA . . . . . . . . . . . . . . . . . . 33

2.6.5.1 Synchronous Machine Model . . . . . . . . . . . . . . . . . . . 33

2.6.5.2 IEEE Type-1 Exciter Model . . . . . . . . . . . . . . . . . . . . 34

2.6.5.3 Turbine Governor Model . . . . . . . . . . . . . . . . . . . . . 34

2.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Modeling of Microgrid Elements 40

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Microgrid Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Classification of Power Converters in Microgrid Applications . . . . . . . 41

3.2.2 Parallel Operation of Microgrids: Issues and Challenges . . . . . . . . . . 42

3.2.2.1 Park Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.2.2 Synchronization in AC Microgrids With Phase-locked loop . . . 45

3.2.3 Photovoltaic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.4 Dynamical Model of the Voltage Source Converter (VSC) . . . . . . . . . 48

3.2.5 Average Model of the switches . . . . . . . . . . . . . . . . . . . . . . . . 49

iv

3.2.6 Microgrid Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Doubly Fed Induction Generator 60

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Model and Control of DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.1 Aerodynamical Model of the Wind Blades . . . . . . . . . . . . . . . . . 61

4.2.2 Dynamical of the Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.3 Model of the Induction Generator . . . . . . . . . . . . . . . . . . . . . . 64

4.2.4 Design of the Rotor Side Controller (RSC) . . . . . . . . . . . . . . . . . 67

4.2.4.1 Design of the Inner Current Control Loop . . . . . . . . . . . . 68

4.2.5 Design of the Grid Side Controllers (GSC) . . . . . . . . . . . . . . . . . 69

4.2.5.1 Design of the Inner Current Control Loop . . . . . . . . . . . . 69

4.2.5.2 Design of the DC link Control . . . . . . . . . . . . . . . . . . 71

4.2.5.3 Reactive Power Control . . . . . . . . . . . . . . . . . . . . . . 72

4.2.6 Design of the Pitch Controller . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Inertia Emulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 Kinetic Energy Stored in DFIG . . . . . . . . . . . . . . . . . . . . . . . 73


4.4.1 Frequency Control by DFIG . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

v

5 Koopman Mode Analysis of a Multi-Machine System with Parallel Micro-

grids Using a New Voltage Control 76

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 Generic Model of the Power Electronic Interface Generating Units . . . . . . . . 78

5.2.1 VSC Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.1.1 Inner Current Control . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.1.2 Outer Current Control . . . . . . . . . . . . . . . . . . . . . . . 82

5.3 Voltage Control at the PCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.1 Grid Impedance Impact on Voltage Regulation . . . . . . . . . . . . . . . 83


5.4.1 Small Signal Stability Analysis Procedures . . . . . . . . . . . . . . . . . 85

5.4.1.1 Impact of the microgrid on the small signal stability . . . . . . 87

5.4.1.2 Impact of the microgrid control strategy on the small signal

stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4.1.3 Impact of the microgrid mode of operation on the small signal

stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Conclusions 90

6.1 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Bibliography 91

vi

List of Figures

2.6.1 Single line diagram of the test-bed for the small-signal analysis . . . . . . . 28

2.6.2 Schematic two-area system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6.3 Comparison between KMA and small signal based on linearization. . . . . . 32

2.6.4 Single-line diagram of the 10-machine New-England Power System . . . . . 35

2.6.5 Time-domain of the generator rotor speeds. . . . . . . . . . . . . . . . . . . 36

2.6.6 Illustration of the 20 largest Koopman modes. . . . . . . . . . . . . . . . . 37

2.6.7 Illustration of the 20 largest Koopman modes (contd.). . . . . . . . . . . . 38

2.6.8 Illustration of the 20 largest Koopman modes. . . . . . . . . . . . . . . . . 39

2.6.9 Koopman eigenvalues in the Z-domain. . . . . . . . . . . . . . . . . . . . . 39

3.2.1 Schematic representation of an inverter operating in grid-feeding mode [1] . 42

3.2.2 Schematic representation of an inverter operating in grid-forming mode [1] 42

3.2.3 Structure of PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.4 Schematic of the Photovoltaic Cell . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.5 Half-bridge VSC system connected to AC source. . . . . . . . . . . . . . . . 49

3.2.6 Pulse Width Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.7 Average Model Single Phase VSC system connected to AC source . . . . . 53

vii

3.3.1 Three Phase VSC connected to the grid . . . . . . . . . . . . . . . . . . . . 54

3.3.2 Two-area test system with microgrids. . . . . . . . . . . . . . . . . . . . . . 54

3.3.3 Power generated after a load step change at 0.25 s . . . . . . . . . . . . . . 55

3.3.4 Modulation indexes in the abc reference frame . . . . . . . . . . . . . . . . 55

3.3.5 Modulation indexes in the (dq) reference frame . . . . . . . . . . . . . . . 56

3.3.6 Voltage Profile at the Point of Coupling . . . . . . . . . . . . . . . . . . . . 56

3.3.7 Active power exporting under FFC control . . . . . . . . . . . . . . . . . . 56

3.3.8 Reactive power exporting under FFC control . . . . . . . . . . . . . . . . . 57

3.3.9 Active power importing under FFC control . . . . . . . . . . . . . . . . . . 57

3.3.10 Reactive power importing under FFC control . . . . . . . . . . . . . . . . . 57

3.3.11 Active power neither importing nor exporting; Load demand matches mi-

crogrid generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.12 Reactive power neither importing nor exporting; Load demand matches

microgrid generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 Power-coefficient versus tip-speed-ratio using the exponential model . . . . 64

4.2.2 Schematic diagram of the variable-speed wind -power system . . . . . . . . 67

4.2.3 Vector Control Method of the RSC . . . . . . . . . . . . . . . . . . . . . . 68

4.2.4 Vector Control Method of the GSC . . . . . . . . . . . . . . . . . . . . . . 70

4.2.5 Wind turbine Pitch Angle Controller . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 Active power and pitch angle controllers of DFIG for frequency control. [2] 73

4.4.1 Frequency deviation at bus 39 with and without supplementary control . . 75

viii

5.2.1 Schematic of the VSC Active/-Reactive Power Control . . . . . . . . . . . 79

5.2.2 Voltage Source Converter (VSC) controls: (1) PQ control. (2) Feeder Flow

Control (FFC). (3) Voltage control. . . . . . . . . . . . . . . . . . . . . . . 79

5.4.1 Modified two-area test system including microgrids. . . . . . . . . . . . . . 85

5.4.2 Eigenvalues loci in the complex plane. Microgrids exporting power to the

main grid, and using the feeder flow control (FFC). . . . . . . . . . . . . . 86

5.4.3 Area-2 local mode for Xvar.=7.54 (Ω) and Rvar.=3.77 (Ω), microgrids ex-

porting power to the main grid. (a) FFC. (b) UPC. (c) PV control. . . . . 86

5.4.4 System eigenvalues for FFC. (a) no power transferring between microgrid

and the main grid. (b) microgrid importing power from the main grid. . . . 88

ix

List of Tables

2.6.1 Impact of the microgrid on rotor oscillatory modes.. . . . . . . . . . . . . . 30

2.6.2 Eigenvalues of the system with and without a microgrid. . . . . . . . . . . 31

2.6.3 Koopman eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Data VSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4.1 VSC setpoint values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.2 VSC setpoint values (continued). . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.3 VSC electrical and control parameters. . . . . . . . . . . . . . . . . . . . . 85

5.4.4 Microgrid data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

x

Chapter 1

Introduction

1.1 Problem Statement and Research Objectives

In the USA, electric power systems are being operated closer to their stability limits. Indeed,

over the last decades, the load demand has been steadily increasing while new transmission

capacity has not been expanded at a sufficient pace to meet the safety margin requirements

needed for the secure operation of the power grids. Efforts to enhance the voltage and rotor an-

gle stability of the power transmission system have been relying on the installation of advanced

control systems such as fast automatic voltage regulators (AVRs) of synchronous generators for

the improvement of transient stability, power systems stabilizers (PSSs) for damping inter-areas

mode of oscillations, static vars compensator (SVCs) for reactive power support of transmis-

sion lines, HVDC links for wheeling bulk power over long distances, and FACTS devices for

providing system flexibility, to name a few.

With the deregulation of the power industry, the growing environmental concerns about

water and air pollution induced by thermoelectric power plants (classical and nuclear), and the

advancement in power electronics, we are witnessing a growing penetration of Distributed Gen-

eration (DG) in power systems, mainly at the distribution systems level. When organized in

microgrids, small-scale DGs and their associated storage devices have the capability to deliver

power to local load centers within acceptable voltage and frequency limits while improving the

voltage and rotor angle stability of the power grid. A microgrid has two modes of operation: (i)

1

2

a grid-connected mode where microgrids may absorb or supply power to the grid and provide

ancillary services, and (ii) an islanded mode of operation where a single or a group of micro-

grids operate in an isolated manner during planned and emergency conditions. When properly

controlled, microgrids have the potential to interrupt the propagation of cascading failures that

may otherwise lead to large-scale blackout, such as the one that occurred in August 2003 in the

Northeastern part of the USA and Ontario in Canada. Statistics of power outages have shown

that their frequency is steadily increasing over the years. For instance, from 2000 till 2004, the

number of blackouts amounted to 149, while from 2004 till 2008 it had more than doubled,

reaching 349. Many other blackouts have been recorded around the world in the last decades

[2]. Among them, we can list the Tokyo blackout in 1987 and the 2003 blackout in Sweden,

Denmark, and Italy. All these blackouts are signs of increasing power system vulnerability to

cascading failures.

The connection in a near future of a large number of small-scale generation along the feeders

of a distribution power system raises several interesting research problems, which include (i)

the analysis of the local modes of oscillation among the microgrids connected to the same

feeder or to different feeders related to the same bus bar; (ii) the analysis of the control modes

of oscillation between the controllers of the microgrids and the AVRs of the turbo-generator

systems, the SVC devices, and the FACT devices connected to the transmission systems, which

all have close time constants; (iii) the mitigation of the reduced total system inertia due to

storage devices and photovoltaic panels that do not have rotating masses; (iv) the development

of controllers of microgrids’ converters with response times of the same order as those of the

turbo-generator systems.

There exists a large literature on the development of various control schemes for microgrids

when the latter are either operating in isolated mode or connected to a stiff transmission system

modeled as an infinite bus-bar. By contrast, there exist a handful of papers that study the

reliability or small-signal stability of a multi-machine system with microgrids. Tools available

for the design of controllers rely on linearized models, which neglect the nonlinear effect on

system dynamics. Therefore, there is a need to develop new tools that allow us to analyze

both small signal and transient stability analysis of a multi-machine system with microgrids.

In our research, we will resort to Koopman Mode Analysis (KMA) to carry out such analyses.

3

Because this method does not require any linearization of the system model, it accounts for

both linear and nonlinear oscillations that may take place in the system.

The objectives of this dissertation are the following:

I. Developing a generic small signal model of a microgrid consisting of solar PV panels and

storage devices, which do not have rotating masses, and carrying out a small signal stability

analysis using KMA;

I. Developing a model of a microgrid with Doubly Fed Induction Generator (DFIG), which has

a considerable kinetic energy stored in the rotating mass of the wind blades, and carrying out

stability analysis of a power system with DFIG using the spinning reserve emulation proposed

by [2].

II. Investigating the dynamical interaction of parallel microgrids with the transmission grid

modelled as a multi-machine system using KMA.

III. Developing a controller that will coordinate the parallel operation of a collection of micro-

grids connected to the same bus-bar via feeders with different impedances.

1.2 Contributions

The major contributions of this dissertation include the following:

New application of Koopman Mode Analysis (KMA) to integrated power transmission

and distribution systems with parallel microgrids. Previous applications of KMA involve

only power transmission systems without power electronics converter-based DG units.

Development of a new voltage control scheme for a parallel operation of a collection of

microgrids connected in parallel to the same bus-bar via feeders with different impedances.

4

1.3 Background

The main difference between microgrids and backup small-scale generators is that the former

has controllable outputs that may improve the stability of a power system in many different

ways, while the latter do not have. Indeed, a microgrid has the capability to connect and

disconnect seamlessly from the main grid. It can meet, either partial or totally, the local load

demand in case of emergency and provide a support to the main grid via ancillary services [3]

such as reactive power support, improve power transfer capabilities of transmission lines, and

damp system oscillations by modulating its power. These special characteristics exemplify why

microgrids are so appealing to both energy producers and consumers. The benefits of microgrids

to the main grid include, but are not limited to: improved reliability [4–6], power quality [7],

resiliency [8], and stability via the control of the very fast response power electronics converters.

As is the case for the control of synchronous generators, the control of microgrids can be

hierarchical [1] with three levels of control. The first level, or primary control, is the fastest

control used to damp the frequency oscillations. The controller is typically made of a gain

block. Although the controller is fast to respond to frequency deviations, it cannot eliminate

the steady error on the frequency. The secondary control is used to eliminate the steady

state frequency error of the primary control and bring the frequency back to its nominal value.

Finally a tertiary control is used for economic dispatch; here, system operators set the operating

points of the power controllers of the microgrids and determine the power flow between the

microgrids and the utility. In other instances, where the DGs are intermittent such as wind

and photovoltaic generating units, the control design is more complex and requires the use of

two control methods, namely master-slave control and peer-to-peer control [9].

The control of voltage and frequency within the microgrid is very important for its stability.

While synchronous generators can take advantage of the large kinetic energy stored in their

rotating masses to improve their inertial response, DGs within a microgrid may be inertia-less

and are typically decoupled from the main grid via power electronics interfaces. Hence, as the

level of penetration of microgrids is increased, the overall inertia of the power system decreases,

with the possibility of inducing system instabilities. An important question is therefore what is

the maximum level of penetration of microgrids that is tolerable for maintaining the stability

5

of a power system.

1.3.1 Power System Security

Concerns of power systems security reached a tipping point with the US Northeast blackout of

1965. A committee led by Dy Liacco expressed the need to mitigate the risk of catastrophic

failures of power systems by constantly monitoring its operating state and taking preventive

actions if the system stability margins are deemed insufficient. The work on power system

security started with Dy Liacco in 1967 [10], who identified three operating states: normal,

emergency, and restorative. Later on, an improvement was made by Fink and Carlen with the

addition of two more states, namely alert and in-extremis states. The rational of this addition

is to identify and initiate corrective and/or heroic actions in the case where the system looses

its stability.

Fink and Carlen define the different system states as follows:

Normal State: In this state, all the equality and inequality constraints of the system are

satisfied and the system has enough security margins to withstand the loss of a critical

element, such as a major transmission line, transformer, or generator, without losing

stability. This is the so-called N-1 security;

Alert State: In this state, the equality and inequality constraints are satisfied while the

power system has narrow security margins. A single contingency could make the system

lose its stability. A preventive action such as increasing the redundancy of the critical

elements may move the system to a normal state;

Emergency State: It is a state where some inequality constraints are not satisfied. Cor-

rective action should be initiated in a timely manner to bring back the system to an alert

or normal state. These actions include load shedding, transmission line tripping, among

others;

In Extremis State: It is characterized by both equality and inequality constraints being

violated following load shedding and outages of generators to maintain the integrity of

the transmission system;

6

Restorative State: It is a state where the power system is fragmented while some of its

parts or its totality are down. Restorative actions need to be taken to bring the state to

the normal or alert state.

The security of a power system results from its robustness, which is defined as its ability

to remain in a normal state when subjected to some predefined contingencies. In other words,

given a class of disturbances, a power system is said to be robust if it has the ability to maintain

its function when subjected to disturbances of that class. But recent blackouts show that it is

insufficient to have a robust power system as there are major unexpected disturbances, which

are not foreseen by the contingency analysis carried out at the control center, that can trigger

cascading failures leading to large-scale blackouts. If the system loses its stability, one wants

it to gracefully degrade and to quickly recover its equilibrium. This brings a new concept in

engineering, which is that of system resiliency [11].

1.3.2 Microgrid Concept

According to the Consortium for Electric Reliability Technology and Solutions (CERTS) [12],

a microgrid as quoted is defined as “an aggregation of loads and microsources operating as a

single system providing both power and heat. The majority of the microsources must be power

electronic based to provide the required flexibility to insure operation as a single aggregated

system. This control flexibility allows the CERT microgrid to present itself to the bulk power

system as a single controlled unit that meets local needs for reliability and security.” A microgrid

has the following characteristics that define its different modes of operation, and with the help

of smart breaker it can operate in

direct connection to the main grid

island mode (autonomous)

switching mode between the grid connected and the island mode of operation

The major differences of microgrids and conventional back-up generators are multiple and

diverse. The DG units used in microgrids are much more diverse than those utilized as back-up

7

generation in data centers, hospitals, and administration buildings. A microgrid encompasses

low-scale generating units such as diesel reciprocating engines, micro-turbines, fuel cells, to

name a few, along with intermittent units such as photovoltaic panels and wind generating

units. Most of these units have either no inertia or low inertia, and they require power electronic

interface to be securely synchronized to the grid. Also with tangible progress made in the fields

of power electronics, communications and computers, the grid is getting ‘smarter’ and more

flexible. The equipment used range from flexible AC transmission systems (FACTS) devices,

high-voltage DC (HVDC) links with advanced control monitoring using synchronized phasor

measurement units (PMUs) to enhance system angle and voltage stability. Recently, microgrids

connected to the distribution system via converters have gained a great deal of attention by

power utilities and private businesses due to their ability to provide increased resilience of

a power grid to natural disasters such as hurricanes, tornadoes, and ice-storms. With the

progress made in power electronics and storage devices such as ion-lithium batteries, fly-wheel

and compressed air units, the intermittency and variability of renewable generating units can

be mitigated.

The incorporation of these low-scale generating units within a microgrid constitutes a de-

parture from centralized electrical energy generation. This system reconfiguration has been

prompted by the following facts: (i) environmental concerns due to the emission of greenhouse

gas and pollutants by fossil-fuel-fired power plants controlled by utilities, municipalities, and

generation companies; (ii) a steady increase in fuel cost resulting from the depletion of natural

energy resources, worldwide; (iii) rapid technological advancement of Distributed Generations

(DG) units; (iv) installation of power electronic devices at the transmission and distribution

level; (v) increased vulnerability of power systems due to the fact that the addition of trans-

mission capacity is not keeping pace with the increase of the load, resulting in a shrinking of

system security margins.

1.3.3 Potential Benefits of Microgrids to Bulk Power Systems

There are several advantages of using microgrids in a power system. Firstly, they can provide

peak shaving during normal operation conditions, resulting in increased stability margins and

reliability of the power system. Secondly, they can provide ancillary services to a power system,

8

including voltage and reactive power control, energy balancing, and load following, to name a

few. Thirdly, they can achieve ride-through capability when a fault occurs at their terminals,

a function that is provided by their converter interfaces. For instance, this function is an

IEEE requirement (known as IEEE 1547) imposed on wind farm for grid connection. In this

dissertation, we will examine under which conditions microgrids can facilitate the achievement

of all these services and functions.

1.3.4 Proposed Approach

The simulation results described in this dissertation are mostly based on KMA. While PSCAD

software is used to implement the case studies, a KMA code has been written in MATLAB

environment for eigen-analysis.

1.3.5 Dissertation Layout

The next five chapters of the dissertation are organized as follows:

Chapter 2 deals with KMA as applied to power system dynamics. The analysis is not model

based, but processes metered values on a collection of system variables. The chapter begins

by reviewing the fundamental theory of modal analysis based on linearization,then KMA is

introduced. Simulation results obtained from a multimachine systems namely the two-area

power system are presented. The objective of this chapter is to validate KMA as a tool to

perform both modal analysis and transient stability analysis.

Chapter 3 provides the details modeling of the different components of the microgrids, which

include the modeling of solar photovoltaic generators, power electronic converters. Issues and

challenges for parallel microgrids in island mode are addressed when feeder impedances have

different values. Specifically, the inaccuracy of reactive power sharing are discussed.

Chapter 4 describes the detailed model of a DFIG wind turbine and highlights some of the

auxiliary services of DFIG such as frequency support to the grid. It includes some simulation

results carried out on the New-England 39-Bus system to which is attached a wind farm. The

9

excited modes of this system via a KMA are analyzed.

Chapter 5 presents a new control scheme for four parallel microgrids equipped with PQ

control that are connected to a multi-machine system via distribution feeders . The transmis-

sion system is subjected to a small disturbance induced by a 1% load change followed by an

incremental increase of the impedance value of one feeder. Then, transient stability analysis is

carried out on the same testbed. Stability analysis is performed for the following two scenarios:

(i) the microgrids are equipped with a new voltage control; (ii) microgrids are equipped with

PQ control. These two scenarios allow us to demonstrate the superiority of our proposed new

voltage control scheme compared to the conventional PQ control.

Chapter 6 presents the conclusions of the dissertation and suggests research topics for future

work.

Chapter 2

Power System Dynamic Modeling

Using Koopman Mode Analysis

2.1 Introduction

The power system network is a vast interconnection network with a load demand that is steadily

increasing while new transmission lines are not being built at an acceptable rate. Furthermore,

the penetration of new technologies and control equipment are pushing the power system to

operate near of its stability limits. Hence, a risk of increase power system oscillations might

occur for such highly stressed power system network. These oscillations are detrimental to the

security of power system operation and need to be eliminated. The identification of the excited

modes requires linearizing the system model which is highly non-linear, as it is the case in

power system. Furthermore, the design of controllers to mitigate power oscillations resort to

linearization .For example, the power system stabilizer (PSS) can be adequately tuned using

classical linear theory tools to dampen power system inter-area oscillations. In this work,

we propose the identification of inter-area oscillations by the use of Koopman mode Analysis

(KMA) for small signal and transient stability analysis. This chapter will begin by reviewing

the current modal analysis techniques available with emphasis on the method of linearization.

Then, we will introduce the Koopman operator for the identification of linear and non-linear

modes. The chapter will conclude with simulation results.

10

11

2.2 Taxonomy of Modal Analysis Techniques

A stable power system is paramount importance for the security of its operation. For example

some inter-area oscillations can lead to system splits [13] or to major power outages [14]. Before

proceeding any further, it is very important to clarify some basics definition of power system

stability. These definitions are well addressed in the IEEE/CIGRE joint task force led by

Kundur and al. [15]. Power system stability classification depends on three main factors [15]:

(i) the action that creates the instability as recorded by one of the system variables ; this can be

the rotor angle, the voltage, and the frequency. (ii) The size of the disturbance, which influences

the method of calculation used to predict instability: a short circuit has more impact on the

degree of instability of a power system network than a closing or opening of a circuit breaker;

calculation methods such as Lyapunov’s method are required to capture non-linear modes when

the system is highly non-linear, while linearization techniques of the system dynamical model

around a equilibrium point is enough to study stability of the power network when subjected

to a small disturbance.(iii) The time span, equipment and mechanism all affect the stability of

the system.

2.3 Model Analysis Based on first Order Approximation

A definition of power system stability given by the IEEE task-force chaired by Kundur is

as follows:“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” When the

power system is subject to a small disturbance and remains steady (i.e the oscillations created by

the disturbance are eliminated), then the power system is stable. Sometimes these oscillations

grow in amplitudes to the point of inducing instability in the system. There are many factors

that affect the dynamic behavior of the system under small signal stability condition, including

device control schemes and the electric distance between the different components of the power

network. When analyzing the dynamic behavior for small perturbations, the dynamical model

can be linearized around an operating point. The following theory is developed based on [16,17].

12

Assume the nonlinear dynamical model of power systems described by n first order nonlinear

differential equations expressed as

ẋ(t) = f(x,u, t), (2.1)

y(t) = g(x,u, t), (2.2)

with initial conditions, (x0,y0), i.e 0 = g(x0,y0) where

- state vector x ∈ Rn ;

- input vector u ∈ Rm;

- vector-valued nonlinear system function f(·) ∈ Rn ;

- output vector y ∈ Rm;

-g(·) is a vector value function and

- t is the time variable.

The non-linearity of (2.1) and (2.2) stem from the fact that f and/or g have terms (i)with

non-linear functions of states and/or inputs (i.e. sin,cos),(ii)f and/or g have terms with states

and/or inputs appearing as power of something other than 1 or 0 (iii) f and/or g have terms

with cross products of states and/or inputs.

When the derivatives of the state variables are not explicit functions of time, the system is

said to be autonomous, and (2.1) simplifies to

ẋ = f(x,u). (2.3)

In practice, we are often interested in output variables that are measurable because states

are not directly accessible through measurements; hence an observer for the state is usually

designed through measurement of system output.

Equations (2.1) and (2.2) may be defined for any continuous, nonlinear, time-invariant

13

dynamic system. Now, non-linear dynamic system can be linearized around an equilibrium

point (x0,u0) defined as

ẋ = f(x0,u0) = 0. (2.4)

Assuming that the perturbations to which this system is submitted are sufficient small i.e

(x̃ = x− x0); (ũ = u− u0), its dynamic response can be modeled in terms of Taylor’s series

expansion about the operating point, with terms of second and higher order neglected. Hence,

applying Taylor expansion at the origin on (2.1) gives

ẋ = f(x0,u0) + Jf ,x(x0,u0)x̃+ Jf ,u(x0,u0)ũ+ (h.o.t.). (2.5)

By neglecting the higher order terms, the Taylor series expansion simplifies to

∆ẋ = A(x0,u0) ∆x+B(x0,u0) ∆u,

∆y = C(x0,u0) ∆x+D(x0,u0) ∆u,

(2.6)

where

- state matrix A ∈ Rn×n;

- input matrix B ∈ Rn×p;

- output matrix C ∈ Rm×n;

- feed-forward matrix C ∈ Rm×p;

- ∆ is a prefix to denote small deviation.

Matrices B, C and D hold important information about the system. Some of their impor-

tant properties, specially for control theory, were put forth by Kalman in the 1960s.

To study the dynamic behavior of the linear system, it is important to understand the system

state matrix A, which is also known as the Jacobian matrix, whose elements aij are given by

the partial derivatives ∂fi(x0,u0)/∂xj. The state matrix embed the stability of the system. In

order words, for small perturbations, the stability of the system is given by the eigenvalues of

14

A, which is referred to as small signal stability analysis. To compute its eigenvalues for large

systems, QR-factorization and algorithms based on Krylov subspace have been recommended

due to their numerical stability and ability to deal with very large matrices in an efficient way.

Formally, one has

Aφi = λiφi, i = 1, . . . , n, (2.7)

where φi is the right eigenvector associated with the i-th eigenvalue λi of the matrix A. In a

similar way, the row vector ψTi which satisfies

ψTi A = λiψTi , i = 1, . . . , n, (2.8)

is the left eigenvector associated with the i-th eigenvalue λi. Next, some important properties

are highlighted:

left and right eigenvectors associated with different eigenvalues are orthogonal. Formally,

we have ψTi φj = 0 for all i 6= j;

left and right eigenvectors associated with the same eigenvalue imply ψTi φi = ci, where

ci is a non-zero constant, which is equal to 1 if the eigenvectors are normalized.

Finally, let us initiate the derivation to explicitly identify the modes of oscillation of the system.

For this purpose, define the following modal matrices, all square of dimension n:

Φ = [φ1 φ2 . . . φn], (2.9)

Ψ = [ψ1 ψ2 . . . ψn]T , (2.10)

Λ = diag{λ1, λ2, . . . , λn}. (2.11)

The following relations are important. Note that, from this point forward, we are assuming

15

that the eigenvectors are normalized.

AΦ = ΦΛ (2.12)

ΨΦ = I (2.13)

Ψ = Φ−1 (2.14)

Φ−1AΦ = Λ (2.15)

Now, assuming that no input is applied to the system, the first row of (2.6) may be re-written

as

∆ẋ = A∆x. (2.16)

The relationship (2.16) shows that the rate of change of each state variable is a linear

combination of all state variables. Indeed, for power systems, A is sparse, implying that

the rate of change of a state variable is a linear combination of a subset of the system state

variables. Now, the question that arises is how to solve (2.16). To this end, let us define the

linear transformation

∆x = Φz. (2.17)

Substituting (2.17) into (2.16), one has

Φż = AΦz

ż = Φ−1AΦz

ż = Λz. (2.18)

Note that the matrix Λ is diagonal. Therefore, the states are now completely decoupled in

terms of the new state vector z, which is related to the original state vector ∆x through (2.17).

The time-domain solution of system of first-order differential equations defined by (2.18) is

given by

zi(t) = zi(0)eλit i = 1, . . . , n. (2.19)

16

Expanding (2.17) and using the relation expressed by (2.19) yields

∆x(t) = Φz(t)

= [φ1 φ2 . . . φn]

z1(t)

z2(t)

...

zn(t)

=

n∑i=1

φizi(t)

=n∑i=1

φizi(0)eλit. (2.20)

Again from (2.17), we know that

z(t) = Φ−1∆x(t) = Ψ ∆x(t). (2.21)

Therefore, for each element of the vector z we have

zi(t) = ψTi ∆x(t), (2.22)

and for t = 0,

zi(0) = ψTi ∆x(0). (2.23)

Substituting (2.23) into (2.20) we get

∆x(t) =n∑i=1

φiψTi ∆x(0)e

λit.

Finally, the time-domain response of the i-th state variable is given by

4xi(t) = φi1ψTi14x(0)eλ1t + φi2ψTi24x(0)eλ2t + . . .+ φinψTin4x(0)eλnt. (2.24)

17

Thus, the free motion time-domain response of the system is given by (2.24), which express

4xi(t) in terms of eigenvalues, right and left eigenvectors, and system initial condition.

2.4 Linear Koopman Operator

Poincare’s geometric picture is the most widely used method for the graphical representation of

dynamical states. The graphical representation approach is primarily due to the impossibility

to solve closed form three body problem . The geometric approach is based on phase space

which is the glimpse of the study of chaos. While this visual representation have dominated

the study of dynamical systems for a century, it has shown strict limitations in handling high-

dimensional dynamical systems, which are more and more prevalent in engineering systems

design.

An alternative framework for the study of dynamical system based on the dynamics of observables

picture, is proposed in [18]. The main concept is the Koopman operator: an infinite-dimensional

linear operator that has the capability of capturing the non-linear modes of a dynamical system.

This is what KMA attractive: the use of well know linear tools to study non-linear dynamics.

The study of the dynamical system using KMA is based in visualizing the temporal evolution

of function on the state space, rather than looking directly at the space space trajectory. The

time evolution of the function of states can be studied by decomposing the function into a

basis of eigenfunctions of the Koopman operator. Classical tools of linear algebra are used

for the computation of linear and nonlinear modes. The scheme is identical to normal modes

from linear vibration theory. KMA starts by identifying/choosing a set of linear independent

observables, or equivalently a vector value-observable; in power system, these observables could

be the rotor angle of a generator, the speed of a generator, or any state variables that are of

interest. These are choices based on the objective of the study. The Koopman operator U is

then applied to the subspace spanned by the observables. All this theory need to be clarified

by use of mathematical concepts. In the space of analytic functions, the composition operator

Cφ with symbol φ is a linear operator defined as

Cφ(f) = f ◦ φ, (2.25)

18

where f ◦ φ denotes function composition. Similarly, we can use the same concept on the

following theory used by [19–21] Consider the dynamics described by a discrete-time nonlinear

equation evolving on a smooth manifold M expressed as

xk+1 = f(xk), (2.26)

where f : M →M is a nonlinear map. The Koopman operator is a linear, infinite-dimensional

operator acting on scalar-valued functions (observables) g : M → R in the following manner:

Ug(x) = g ◦ f(x) = g(f(x)). (2.27)

The Koopman eigenvalues λj ∈ C and Koopman eigenfunctions ϕj : M → C are defined as

Uϕj(x) = λjϕj(x), j = 1, 2, . . . . (2.28)

Now, consider a vector-valued observable g : M → Rp. In [18] the author shows that if the

dynamical system (2.26) possesses a smooth invariant measure, or the initial condition x0 of

(2.26) is on any attractor, then g(xk) = (g1(xk), · · · , gp(xk))T is exactly represented as

g(xk) =∞∑j=1

λkjϕj(x0)υj +

∫ 2π0

eikθdE(θ)g1(x0)

...∫ 2π0

eikθdE(θ)gp(x0)

, (2.29)

where E(θ) is a continuous, complex spectral measure; and the vectors υj are the Koopman

modes of the system.

The first term on the right-hand side of (2.29) represents the contribution of Koopman

eigenvalues to the time evolution of g(xk). In other words, this terms stands for the discrete

spectra of U . It describes the average and quasi-periodic parts of g(xk).

In turn, the second term on the right-hand side of (2.29) stands for the continuous spectra

of U , and describes the aperiodic part of g(xk).

From the interpretation of the right-hand side terms in (2.29) comes the first important

19

approximation, which takes into account the physical behaviour of the power systems: according

to practical experience, they have no continuous spectrum in frequency domain; unlike, they

show a finite number of discrete spectra. Thus, the second term in (2.29) is dropped, which

yields to

g(xk) =∞∑j=1

λkjϕj(x0)υj. (2.30)

Equation (2.30) describes the time evolution of observable g(xk) starting from g(x0). Now,

the question that arises is the following: How to solve for the Koopman modes?

In [18,22], the authors show that the terms ϕj(x0)υj are defined and computed with a pro-

jection operation associated with U applied to the observable g. They also show a relationship

between generalized Fourier analysis and eigenfunctions of the Koopman operator.

Define a family of operators PV : for g : M → R

PVg(x0) = limn→∞

1

n

n−1∑k=0

e−i2πkVg(xk) (2.31)

where V ∈ [−1/2, 1/2). When the initial condition x0 is on an attractor of (2.31), a nonzero PV

is the orthogonal projection operator onto the eigenspace of U associated with the Koopman

eigenvalue λ = ei2πV .

Finally, the projections of the p components of g on the j-th eigenspace are given by

PVjg1(x0)

...

PVjgp(x0)

= ϕj(x0)υj (2.32)

where Vj = =[lnλj]/2π; and =[z] stands for the imaginary part of a complex number z.

Remark 2.4.0.1 (Measure theory and the time-averaging operator [22]). Define a discrete-time

dynamical system as follows:

xk+1 = T (xk), yk = f(xk),

where k ∈ Z; xk ∈M , T : M →M is measurable; and f is a smooth real function on a compact

20

Riemannian manifold M endowed with the Borel sigma algebra. On a compact manifold, every

continuous dynamical system has an invariant measure µ. The function f ∗ is called the time

average of a function f under T if

f ∗(x) = limn→∞

1

n

n−1∑k=0

f(T kx)

almost everywhere with respect to the measure µ on M . The time average f ∗ is a function

of the initial state x. The operator PT : L1 → L1 such that PT (f) = f ∗ is referred as the

time-averaging operator.

Remark 2.4.0.2 (Fourier transform). In the discrete-time domain, without too much math-

ematical rigor, a sequence of n complex numbers {x0, x1, · · · , xn−1} is transformed into an

n-periodic sequence of complex numbers through

Xj =n−1∑k=0

xk · e−i2πjk/n,

where each Xj is a complex number that encodes both amplitude and phase of a sinusoidal

component of function xk. It is called the discrete-time Fourier transform, which is tipically

denoted by the symbol F , as in X = F{x}.

Now, keeping Remarks (2.4.0.1) and (2.4.0.2) in mind, let us look at (2.31) again with a

closer focus. Clearly, if Vj = =[lnλj]/2π, PVg(x0) represents the time-averaging operation

applied to the Fourier transforms of observations {g(x0), g(x1), · · · , g(xn−1)}. Thanks to this

important conclusion, the terms ϕj(x0)υj can be computed through general Fourier analysis.

However, it should be pointed out that, from Remark (2.4.0.1), the function f is assumed to

be smooth; in other words, the dynamics of the system described by (2.26) is assumed to be

on an attractor.

In [19], the authors show that even if the dynamics of the system represented by (2.26) is off

attractors, the Koopman modes oscillate with a single frequency. If each of the p components

of g lies within the span of eigenfunctions ϕj, then we may expand the vector-valued g in terms

of these eigenfunctions as

g(x) =∞∑j=1

ϕj(x)υj, (2.33)

21

where υj are regarded as the vector coefficients in the expansion. The time evolution {g(xk)}

starting at g(x0) is identically given by (2.30), that is,

g(xk) =∞∑j=1

ϕj(xk)υj =∞∑j=1

Ukϕj(x0)υj =∞∑j=1

λkjϕj(x0)υj. (2.34)

To finish this section, we present an important derivation taken from [19], which states that

the Koopman modes provide a nonlinear extension of linear oscillatory modes.

Remark 2.4.0.3 (Koopman modes for linear systems). Suppose M is an n-dimensional linear

space, and suppose the map f is linear, given by

f(x) = Ax. (2.35)

The eigenvalues of A are also eigenvalues of U , and the eigenvectors of A are related to eigen-

functions of U as well.

Proof. Let λj be the j-th eigenvalue of A, and υj its associated eigenvector, such as

Aυj = λjυj, j = 1, · · · , n. (2.36)

and let ωj be the corresponding eigenvector of the adjoint A∗, such that A∗ωj = λ̄jωj.

Consider that the eigenvectors are normalized so that

〈υj,ωk〉 = δjk (2.37)

where 〈·, ·〉 denotes an inner product on M , and δjk is the Kronecker delta.

Now, define scalar-valued functions

ϕj(x) = 〈x,ωj〉 j = 1, · · · , n. (2.38)

Recall that the Koopman operator acts on scalar-valued functions such as in (2.27). Thus, we

have

Uϕj(x) = ϕj(Ax), (2.39)

22

and then, from the previous equations we can write

Uϕj(x) = ϕj(Ax) = 〈Ax,ωj〉 = 〈x,A∗ωj〉 = λj〈x,ωj〉 = λjϕj(x), (2.40)

which shows that the eigenvalues ofA are also eigenvalues of U . However, note that the opposite

is not true: one cannot state that the eigenvalues of U are also eigenvalues of A because the

operator U is infinite and has a countably infinite number of eigenvalues. For example, λkj is

also an eigenvalue, with associated eigenfunction ϕ(x)k for any integer k.

Finally, assuming that A has a full set of eigenvectors, for any x ∈M , one has

x =n∑j=1

〈x,ωj〉 υj =n∑j=1

ϕj(x)υj, (2.41)

which shows that, for linear systems, the Koopman modes, υj, coincide with the eigenvectors

of A. �

2.5 Computation of Koopman Modes

As aforementioned, the general Fourier analysis allows to compute the Koopman modes when

the dynamics of the system is on an attractor. However, computation of Koopman eigenvalues

and Koopman modes is a more challenging problem when the system is off-attractor. To

overcome that, the authors in [19] showed that the so-called empirical Ritz values λ̃j and

empirical Ritz vectors υ̃j approximate the Koopman eigenvalues λj and factors ϕj(x0)υj in

(2.29) in terms of a finite truncation. The empirical Ritz values and vectors are computed

in [19] using a modified version of the Arnoldi algorithm.

Consider a set of N + 1 vectors {g(x0), g(x1), · · · , g(xN)}, containing data coming from

either measurements or simulations, where a given vector g(xk) holds the values of the observ-

ables at time tk. Now, a residual vector r is defined as

r = g(xN)−N−1∑j=0

cjg(xj), (2.42)

23

where the constants cj are chosen such that

r ⊥ span{g(x0), g(x1), · · · , g(xN−1)}. (2.43)

Therefore, pre-multiplying (2.42) by KT , and due to the condition made in (2.43), one has

KTr = 0. (2.44)

Now, define

g(xN) = z′, (2.45)

K = [g(x0), g(x1), · · · , g(xN−1)], (2.46)

c = [c0, c1, · · · , cN−1]T . (2.47)

Then, (2.42) may be rewritten as

r = z′ −Kc, (2.48)

and substituting (2.48) into (2.44) we have

KTz′ − (KTK)cT = b−Gc = 0. (2.49)

Because matrix G is not full rank, it is impossible to determine a unique minimizer c ∈ RN

of the norm ||b −Gc||. In [20], the authors proposed to minimize the norm using the Moore-

Penrose pseudo-inverse matrix G† of G. Thus,

c = G†b. (2.50)

24

Now, consider the Companion matrix

C =

0 0 · · · 0 c0

1 0 0 c1

0 1 0 c2...

.... . .

...

0 0 . . . 1 cN−1

. (2.51)

The empirical Ritz values λ̃j are solutions to det(C − λ̃I) = 0, where I is the identity

matrix.

Finally, define the Vandermonde matrix

T =

1 λ̃1 λ̃

21 · · · λ̃N−11

1 λ̃2 λ̃22 · · · λ̃N−12

......

.... . .

...

1 λ̃N λ̃2N · · · λ̃N−1N

. (2.52)

The empirical Ritz vectors υ̃j are defined as the columns of

V = KT−1. (2.53)

The empirical Ritz values λ̃j are good approximations of the Koopman eigenvalues λj.

Because of that, from now on, we will call the empirical Ritz values as Koopman eigenvalues.

The empirical Ritz vectors υ̃j, in turn, are approximations of the Koopman modes υj scaled by

constant values ϕj(x0). Nevertheless, they are computed for a finite sum as in (2.54) instead

of an infinite sum as in (2.30).

g(xk) =

N∑j=1

λ̃kj υ̃j, k = 1, · · · , N − 1,

g(xN) = r +N∑j=1

λ̃Nj υ̃j,

(2.54)

where r is a residual vector that accounts for the approximation errors.

25

2.5.1 Koopman Operator from a Linear Algebra Perspective

Suppose the system response is given as a sum of spatial structures υj evolving in time according

to a function αj(t),

s(t) =N∑j=1

αj(t)υj. (2.55)

The problem is to find pairs {αj(t),υj} that satisfy (2.55). Now, assume that 1) system

response is only composed of periodically oscillating structures; 2) system oscillations are ex-

actly decomposed in a finite number N of distinct modes oscillating at single frequencies; and

3) time step ∆t between two snapshots is constant. Then, we can write:

αj(t) = ajei(ωjt+θj) (2.56)

Rewriting (2.56) considering the time when snapshot is taken,

αj(tn) = ajeiθjeiωjtn [ϕj := aje

iθj ]

= ϕjeiωjn∆t

= ϕj(eiωj∆t)n [ψj := ωj∆t]

= ϕj(eiψj)n [λj := e

iψj ]

= ϕjλnj . (2.57)

Now, substituting (2.57) into (2.55), we have:

s(tn+1) = s(n+1) =N∑j=1

λ(n+1)j ϕjυj. (2.58)

In matrix form,

s(n+1) = Φσ(n+1) = ΦΛσn = ΦΛΦ−1Φσn = ΦΛΦ

−1sn

= Asn, (2.59)

26

where σ(n+1) = {λ(n+1)1 ϕ1, λ(n+1)2 ϕ2, · · · , λ

(n+1)N ϕN}T ; Φ = {υ1, υ2, · · · , υN}; Λ =

diag {λ1, λ2, · · · , λN}; and A = ΦΛΦ−1 is the Koopman operator, a push-forward oper-

ator that propagates observables over one time-step. Note that Φ is invertible since we have

assumed that the N structures υj are independent.

Remark: From (2.59), we notice that λj are the eigenvalues of A, which contain the fre-

quencies of the structures, fj, within their argument. Formally,

fj =ωj2π

=ψj

2π∆t. (2.60)

By measuring the argument inside ]−π ; π], we have

|ψj| ≤ |π| ,∣∣∣∣ ψj2πfj∣∣∣∣ ≤ ∣∣∣∣ π2πfj

∣∣∣∣ ,|∆t| ≤

∣∣∣∣ 12fj∣∣∣∣ . (2.61)

Drawing a parallel between (2.61) and the Nyquist criteria, the sampling frequency of snapshots,

(∆t)−1, has to be at least twice the largest frequency representable in the decomposition.

2.6 Simulation Results

2.6.1 Small Signal Analysis With Microgrids

This subsection addresses the benefit of microgrids to enhance the inter-area steady-state sta-

bility margins. This is being achieved by means of frequency and voltage droop controls of

the microgrid. The latter is a small-scale energy system that can operate in island mode or

in a grid-connected mode via converters while providing electricity and heat energy to a local

load. The steady-state stability of a two-area 3 machine 9-bus system is investigated with and

without microgrid. The microgrid is connected to a load bus and is provided with droop con-

trols acting on the inverter. Simulation results show that the gain of the frequency controller

of the inverter changes the location of the system eigenvalues. With sufficient high gain, the

27

eigenvalues move to the left-hand side of the complex plane, stabilizing the system.

2.6.2 Motivation

Power system oscillations have attracted a great deal of attention for many years. Different

modes of oscillations have been defined by the IEEE Task Force [15] . One of them is inter-

area mode, which occurs when coherent groups of generators swing against each other at a

frequency that ranges between 0.2 and 1 Hz. Its damping attribute is governed by the tie-line

strength and the type of the load. These oscillations are detrimental to the system operation.

Indeed, they can limit the power transfer capacity from one area to the other one and they

can create instability that can lead to power outages. Consequently, they have to be damped

out [23].Typically, inter-area modes are damped by means of Power System Stabilizers (PSSs).

In this work, we show that good damping can be achieved by means of microgrids instead of

PSSs, which are interfaced via inverters with high-gain frequency controllers and strategically

located close to some loads in the main grid. The study consists in analyzing the small-signal

stability of a two-area 3 machine 9-bus system, which exhibits undamped inter-area oscillations.

It is demonstrated that the system can be stabilized by means of a microgrid placed at a load

bus located in the area with a high participation factor.

2.6.3 Assumptions, Description of Test-Bed, and Computation of

State Matrix

The multi-machine being investigated is the IEEE WECC 9-bus 3-machine system, which is

a benchmark typically used for power system dynamic analysis. It is depicted in 2.6.1. All

the parameters were taken from [24], with some modifications to create a two-area system

connected with long tie-lines. Specifically, the reactances of Line 5-7 and Line 6-9 have been

increased to 0.861 pu and 0.870 pu, respectively. The loads are modeled as constant active

and reactive power. Each of the generators and their control are represented by a five-order

model. The static exciters are modeled by the IEEE-type 1 model with the assumption that

saturation is neglected, and no rate feedback. The dynamic of the boiler and steam turbines

are all neglected. The input torques of the generators are assumed to be constant. Regarding

28

the microgrid, only the dynamics of the inverter are to be modeled because the latter acts as

an interface between the power system and all the devices within the microgrid. The inverter is

represented by a voltage source with a controllable voltage and frequency [25]. When modeling

the test-bed with and without the microgrid for small signal stability analysis, we neglect the

time constant of the network elements and of the microgrids because they are much smaller

than those of the synchronous machines and their controllers

DC

Mic

rog

rid

2 7 8 9 3

65

10

4

1

11

Gen2 Gen3

Gen1

LC filter

Load C

Load A Load B

Figure 2.6.1: Single line diagram of the test-bed for the small-signal analysis

Model of the Multi-machines system and their controllers

The method used to derive the state matrix of the system is described in [26]. The synchronous

generators and their controllers are governed by a set of ordinary different equations (ODE)

29

given by

dδidt

= ωi − ωs, (2.62)

dωidt

=TMiMi−

(E ′qi −X ′diIdi)IqiMi

−(E ′di +X

′qiIqi)Idi

Mi− Di(ωi − ωs)

Mi, (2.63)

dE ′qidt

= −E ′qiT ′doi− (Xdi −X

′di)Idi

T ′doi+EfdiT ′doi

, (2.64)

dE ′didt

= −E′di

T ′qoi+

(Xqi −X ′qi)IqiT ′qoi

, (2.65)

dEfdidt

= −EfdiTAi

+KATAi

(Vrefi − Vi), (2.66)

for i = 1, . . . ,m.

The stator algebraic equations in polar form are

E ′di − Vi sin (δi − θi)−RsiIdi +X ′qiIqi = 0, (2.67)

E ′qi − Vi cos (δi − θi)−RsiIqi −X ′diIdi = 0, (2.68)

for i = 1, . . . ,m.

The network equations with loads modeled as constant power are as follows.

Generator bus-bars:

IdiVi sin (δi − θi) + IqiVi cos (δi − θi) + PLi −n∑k=1

ViVkYik cos (θi − θk − αik) = 0, (2.69)

IdiVi cos (δi − θi)− IqiVi sin (δi − θi) +QLi −n∑k=1

ViVkYik sin (θi − θk − αik) = 0, (2.70)

for i = 1, . . . ,m

Load bus-bars:

PLi −n∑k=1

ViVkYik cos (θi − θk − αik) = 0, (2.71)

QLi −n∑k=1

ViVkYik sin (θi − θk − αik) = 0 (2.72)

for i = m+ 1, . . . , n.

30

Model of the control of the inverter is given below

ω = ωrated +Kp(P∗ − P11), (2.73)

V c = Erated +Kv(Q∗ −Q11). (2.74)

Here P11 and Q11 are the total active and reactive power injected at bus 11 by the microgrid.

δ =

∫(ω − ωs)dt (2.75)

Table 2.6.1: Impact of the microgrid on rotor oscillatory modes..

Eigenvalues Freq. Damping Dominant statesNo. Real Imag (Hz) ratio1, 2 −1.1120 ±12.00 1.9800 0.0922 ∆ω2, ∆δ2, ∆ω3, ∆δ33, 4 −0.1667 ±6.96 1.1000 0.0239 ∆ω1, ∆δ1, ∆ω2, ∆δ2, ∆ω3, ∆δ35, 6 −0.1755 ±4.70 0.7480 0.0373 ∆ω1, ∆δ1, ∆ω2, ∆δ2, ∆ω3, ∆δ37, 8 −1.5823 ±3.63 0.5780 0.0399 ∆E ′q2, ∆Efd2, ∆E ′q3, ∆Efd39, 10 −3.9086 ±1.96 0.3121 0.8939 ∆Efd2, ∆E ′q311, 12 −2.6334 ±1.48 0.2356 0.8717 ∆E ′q1, ∆Efd1

13 −2.6876 0.00 − − ∆E ′d3, ∆E ′d2, ∆Efd3, ∆E ′q314 −1.1684 0.00 − − ∆E ′q2, ∆E ′d2, ∆Efd315 −37.6867 0.00 − − ∆eq of microgrid16 −37.6990 0.00 − − ∆ω of microgrid17 0.0000 0.00 − − ∆ed of microgrid18 −3.2200 0.00 − − ∆E ′d1

From Table 2.6.2, we observe that the original system without microgrid is unstable, whereas

when the system is provided with a microgrid at bus 11, the eigenvalues located on the right

part of the complex plane are displaced to the left side. The system stabilization stems from

the additional damping provided by the microgrid to the inter-area mode of oscillations via

the frequency-droop controller of the inverter. This controller exhibits a fast response to both

the frequency and voltage errors at the inverter terminal bus. A high gain of 2000 MW.s/rad

is necessary to stabilize the system. The dominant modes of the system are the generators

speed as shown in Table 2.6.1. The inter-area mode of oscillation between Generator 1 in one

area and Generator 2 and 3 in the other area has a frequency of 0.748Hz. Hence, we conclude

that this control scheme offers an alternative method for damping local and inter-area plant

modes of oscillations in power system using microgrids. Good damping of inter-area mode of

oscillation has been achieved through the modulation of the active and reactive power of the

31

DG unit within the microgrid via the inverter interface.

Table 2.6.2: Eigenvalues of the system with and without a microgrid.Eigenvalues without Microgrid Eigenvalues with Microgrid

−0.7832± 12.0902i −1.1119± 12.4796i0.0370± 4.2555i −0.1667± 6.9649i−0.4456± 1.3698i −0.1755± 4.7023i−1.5657± 4.2527i −1.5823± 3.6356i−4.7566± 2.7838i −3.9086± 1.9581i−0.4952± 0.5835i −2.6334± 1.4884i

−0.8479 −2.6876−0.0100 −1.1684−3.2210 −37.5887

−37.6971−0.0027

2.6.4 Small Signal Analysis using KMA

The objective is to perform modal analysis on the two-area power system shown in Fig. 2.6.2

using KMA. A small disturbance is generated at bus 9 by creating a step load change from

(1767 MW, -250 Mvar) to (1782 MW, -246 Mvar) at time t = 3 s. The rotor angles and speeds

of the generators are recorded for KMA using the theory described at the beginning of this

chapter.

Table 2.6.3: Koopman eigenvalues.Eigenvalue Frequency (Hz) Damping (%)

λ1,2 0.5313 3.66λ3,4 0.1814 24.49λ5,6 0.3422 13.53λ7,8 0.7226 9.31λ9,10 1.3016 6.40λ11,12 1.4782 6.10λ13,14 1.6411 6.12λ15,16 1.1251 9.22λ17,18 1.7996 6.05λ19,20 0.8883 12.19

G1 G3

1 35 116 7 1098

2 4

Area 1 Area 2

1767 MW-250 Mvar

967 MW-100 Mvar

G2 G4

15 MW6 Mvar

s1

Figure 2.6.2: Schematic two-area system.

KMA provides the results in discrete domain, and a transformation from z domain to s

32

domain is made for analysis purposes. The continuous domain provides information such as

damping ratio that be compared with studies made in [27] . Hence, the twenty Koopman

eigenvalues and their corresponding damping ratio listed in Table 2.6.3 represent the dominant

modes computed from the Koopman algorithm that are transformed in the s domain. Clearly,

the inter-area and local electromechanical modes are captured by KMA: the inter-area modes

λ1,2 with frequency f = 0.5313 Hz and damping ratio ζ = 3.66 % shown in Table 2.6.3 are

very close in values to those obtained from the linearization technique on the same two-area

system described in IEEE PES Task Force report [27]. The same comparison can be made for

the local modes λ15,16 with frequency f = 1.1251 Hz and damping ratio ζ = 9.22 % with those

obtained in the same report. The small discrepancy between the results is due to the use of

power system stabilizers in [27].

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2

{ }

-20

-10

10

20

{ }

Koopman mode analysisSmall signal analysis

=2.5%=5.0%

=5.0% =2.5%

Inter-areamodes of oscillation

Figure 2.6.3: Comparison between KMA and small signal based on linearization.

To evaluate the accuracy of KMA to perform modal analysis, modes obtained from KMA

are superimposed on those obtained from linearization as depicted in Fig. 2.6.3. The inter-

area modes are accurately detected by both methods. For higher frequencies, the linearization

method fails to detect the nonlinear modes as expected.

33

2.6.5 Transient Stability Analysis using KMA

In this subsection, KMA is used to study the transient stability analysis of a multimachine

system. The test-bed is the 10-machine New-England Power System depicted in Fig. 2.6.4 ,

a benchmark for transient stability studies. All the generators are modeled using the two-axis

model and are all equipped with an exciter and a turbine governor.

2.6.5.1 Synchronous Machine Model

The following assumptions are made in deriving the model of generators.

The fast dynamics associated with the stator, and network transients, and damper wind-

ings are neglected.

The machine rotor angles, δi, are assumed to be constant for a constant rated speeds, ωs,

of the synchronous machines.

Hence, the differential equations governing the two-axis model of the synchronous machines are

given by

dδidt

= ωi − ωs, (2.76)

dωidt

=TMiMi−

(E ′qi −X ′diIdi)IqiMi

−(E ′di +X

′qiIqi)Idi

Mi− Di(ωi − ωs)

Mi, (2.77)

dE ′qidt

= −E ′qiT ′doi− (Xdi −X

′di)Idi

T ′doi+EfdiT ′doi

, (2.78)

dE ′didt

= −E′di

T ′qoi+

(Xqi −X ′qi)IqiT ′qoi

, (2.79)

for i = 1, . . . ,m,

where m is the number of synchronous generators, which is ten in this case.

Equations, (2.76), (2.77), (2.78), and (2.79), represent the dynamic associated by the rotor

angles, δi, speeds, ωi , field winding flux linkages, E′qi , and damper windings, E

′di, respectively.

The constant, Mi = 2Hi, have typical values from 5 to 20 s for thermal units and 4.0 to 8.0 s for

hydraulic units. The time constants, T ′qoi, T′doi, for thermal units, have typical values between

34

0.5 to 2.0 s and 3.0 to 10.0 s, respectively; whereas typical values of T ′doi are between 1.5 to 9.0

s for hydraulic units [16].

2.6.5.2 IEEE Type-1 Exciter Model

The equations describing the model of the IEEE Type-1 exciter are given by

TEidEfdidt

= −(KEi + SEi(Efdi))Efdi + VRi, (2.80)

TFidRFidt

= −RFi +KFiTFi

Efdi, (2.81)

TAidVRidt

= −VRi +KAiRFi −KAiKFiTFi

Efdi +KAi(Vrefi − Vi), (2.82)

where the time constants, TEi, TFi, TAi, have typical value ranging from 0.50 to 0.95 s, 0.35 to

1.00 s, and 0.02 to 0.20 s, respectively [28]. The state variables Efdi, VRi, and RFi represent the

output voltage produced by the exciter, the exciter input, and the feedback rate of the voltage

regulator, respectively. The nonlinear function SEi(Efdi) = AieBiEfdi models the exciter iron

saturation.

2.6.5.3 Turbine Governor Model

The equations describing the model of steam turbine governor are given by

TCHidTMidt

= −TMi + PSV i, (2.83)

TSV idPSV idt

= −PSV i + PCi −1

RDi(ωiωs− 1), (2.84)

where PSV i is the steam valve position, TMi reprresents the mechanical input torque of the

machine.

The time-domain simulations is carried out in the PSCAD/EMTDC environment. Once

the system reaches the steady state at 5 s , a three-phase short circuit is applied at bus 27

for a duration of 83 ms and subsequently cleared at time t = 5.083 s without changing the

topology of the network. Figure 2.6.5 depicts the variation of the generator speeds with respect

35

to time. The oscillatory motion is decaying over time which is sign of stability. The first swing

amplitude is larger for machines closer to the fault (bus 27) than for the ones located far away

to the fault . For instance, machine 8 at bus 37 and machine 10 at bus 30 are electrically

closer to the fault location than machine machine 3 and machine 2 located at buses 32 and

31, respectively. Hence, the amplitudes of the first swing for machines 8 and 10 are larger

that for machines 3 and 2. The relationship between the amplitude of rotor speed and fault

strength/duration could be explained as follows: when there is a fault, a voltage dip occurs

at the vicinity of the fault, which affects both active-/reactive powers. Some machines are

accelerating or decelerating according to physical laws of a rotating body, described by the

swing equation. In their tendencies to reach a stable equilibrium point, the faster machines

will pick up part of the loads of slower machines to reduce their relative rotor angle differences.

However, beyond a certain critical limit of an insufficient power transfer to decrease the rotor

angle differences, instability will result. As far the reactive power is concerned, a three-phase

fault can create a voltage dip which causes a large reactive power circulation from high to low

voltage levels.

3

2

1

4 5

5

7

10

8

9

6

10

8

25 26 28 29

9

1

27

3738

13

14

15

17

18

19

16

1

31

20

3

33

3239

30 7

23

4

12

6

2221

36

35

342

Figure 2.6.4: Single-line diagram of the 10-machine New-England Power System

In order to study the dynamic behavior of the generators using KMA, a set of measurements

of the the speed of the generators depicted in Fig. 2.6.5 are recorded. The sampling rate of 120

samples/s is used, which is the maximum sampling rate of Phasor Measurement Unit (PMU)

36

6 7 8 9 10 11 12 13 14

time (s)

0.994

0.996

0.998

1

1.002

1.004

1.006Data effectively utilized for Koopman Mode Analysis (KMA)

G36G37G38G39G35G34G33G32G31G30

Figure 2.6.5: Time-domain of the generator rotor speeds.

devices. Once the samples are collected, then the KMA is applied to the scalar value functions as

described previously in this dissertation. With this sampling rate, KMA will detect thousand

modes. These modes are usually ordered according to their magnitudes which mirror their

energy levels. Figures 2.6.6 and 2.6.7 represent the 20 largest Koopman modes. These figures

give an intuitive understanding of the type of mode being displayed. For instance, looking these

compass plots, one can deduct if the mode is local or inter area mode. The main advantage in

this graphical display is to also to identify the coherent group of machines. In power systems,

coherency is a very important concept to built dynamic equivalents when modeling external

systems [29]. Looking at Fig. 2.6.7, for f=1.8 Hz, machine 9 and machine 8 are oscillating

against each other. These are two machines that belong to different areas. The frequency of

oscillation is a local mode.

Figure 2.6.8 represents the frequency spectrum of the different modes.The DFT is only

applicable if the dynamical system is stable, on an attractor. Figures 2.6.6 and 2.6.7 carry

more information than Fig.2.6.8, because they provide in addition to the frequency the growth

rate.

The stability of the system can be measured by looking at Fig. 2.6.9. All the eigenvalues

are on the unit circle, hence the system is stable.

37

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0

Delta38Delta37Delta36Delta35Delta34Delta33Delta32Delta31Delta30

KM #1 (f = 1.2448 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #2 (f = 1.4753 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #3 (f = 1.3594 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #4 (f = 1.0270 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #5 (f = 0.9060 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #6 (f = 0.5669 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #7 (f = 0.6885 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #8 (f = 0.7961 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #9 (f = 1.1211 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #7 (f = 1.5883 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #8 (f = 0.4525 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #9 (f = 1.6934 Hz.)

Figure 2.6.6: Illustration of the 20 largest Koopman modes.

2.7 Summary and Conclusions

The conclusions of this chapter are threefold.

The DG within the microgrid equipped with frequency/voltage droop control offers an

38

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #1 (f = 0.3363 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #2 (f = 1.8001 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #3 (f = 1.9059 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #4 (f = 2.0111 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #5 (f = 2.1165 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #6 (f = 2.2222 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #7 (f = 2.3279 Hz.)

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0


KM #8 (f = 2.4340 Hz.)

Figure 2.6.7: Illustration of the 20 largest Koopman modes (contd.).

alternative method for damping local and inter-area plant modes of oscillations in power

system . The analysis tool is based on linearization of the system model. Good damping

of inter-area mode of oscillation has been achieved through the modulation of the active

and reactive power of the DG unit within the microgrid via the inverter interface.

KMA for small signal analysis is introduced and validated using the two-area power

system. The local and inter-areas of the electromechanical modes are detected, which

make KMA a good analysis tool for modal analysis.

KMA for transient stability analysis is used for the detection of the non-linea

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