Document [original]

Small Signal Analysis of

Converter-Dominated Power

Systems

vorgelegt von

M. Sc.

Huoming Yang

ORCID: 0000-0003-3518-5992

an der Fakultät IV - Elektrotechnik und Informatik

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

-Dr.-Ing.-

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr.-Ing. Uwe Schäfer

Gutachterin: Prof. Dr.-Ing. Sibylle Dieckerhoff

Gutachter: Prof. Dr. ir. Dr. h. c. Rik W. De Doncker

Gutachter: Prof. Keyou Wang, PhD

Tag der wissenschaftlichen Aussprache: 30. August 2022

Berlin 2022

Abstract

The legacy power systems are dominated by synchronous generators providing inertia support,

frequency and voltage regulation capabilities. The linear time-invariant (LTI) modal analysis

has been widely and successfully applied to their small-signal models for the stability and

resonance evaluation. Driven by concerns about global climate changes, the modern power

system is undergoing the transition in generation technology from fossil fuel-based generation

to renewable generation. The number of power electronic converters used for the integration

of renewable energy sources has gradually increased to a significant level. The multiple-

timescale control loops of converters cause cross interactions with dynamics of loads and

power networks. Moreover, frequency coupling effects caused by converter controllers, the

pulse-width modulation process and unbalanced operation conditions further increase the

system complexity, which makes the modern converter-dominated power grids become complex

nonlinear time-periodic (NLTP) systems. The classical LTI theory based modeling, analysis

and design approaches become frangible and less valid.

To precisely describe dynamics of converter-dominated power systems operated under

different conditions, a linear time-periodic (LTP) modeling framework is developed in the

complex-valued domain. The classical modal analysis for the LTI system is generalized to the

LTP system. Accordingly, definitions and expressions of damping ratio, participation factor and

eigenvalue sensitivity are modified. A time-domain physical interpretation of the LTP system is

proposed to quantitatively confirm the necessity of its application. Two accurate and efficient

LTP eigenvalue calculation methods are developed, paving the way for easier application

of LTP theory. The proposed stability analysis method has been tested with grid-following

converter to investigate the interaction between different phase-locked loops, current controllers

and power networks on the system stability, and with grid-forming converter to investigate the

interaction between its outer power loop and inner voltage-current loop.

To evaluate the impact of integration of converters on harmonic resonances of future

power systems, analytical closed form of the harmonic transfer matrix (HTM) is deduced

to describe the relation between Fourier series coefficients of inputs and outputs of the LTP

system. The classical resonance mode analysis (RMA) is generalized by replacing the constant

system impedance matrix with a time-periodic matrix obtained from the reformulation of the

HTM. The LTP modal analysis leads to the extension of the participation analysis and the

sensitivity analysis for resonance frequency identification and resonance severity evaluation of

LTP systems. The proposed method is tested with an exemplary multiple-converter system to

investigate the impact of the converter controller, the LCL filter and the passive network on

the resonance behavior, considering balanced and unbalanced operation conditions.

Generalized eigenvalue sensitivity and participation factor analysis provides guidelines for

the control parameter and structure optimization of the LTP system. A linear programming

problem is formulated by using eigenvalue and damping ratio sensitivity indices to shift

critical eigenvalues towards the left-half of the complex plane without degrading the damping

performance. Moreover, auxiliary damping loop is designed based on participation analysis

results to extend stability margins. The theoretical analysis is confirmed by cross-validation

between stability and resonance evaluation results obtained from numerical models, analytical

models and hardware measurements.

Zusammenfassung

Das konventionelle Stromnetz wird von Synchrongeneratoren dominiert, die Trägheitsunter-

stützung, Frequenz- und Spannungsregelung bieten. Die lineare zeitinvariante (engl.: linear

time-invariant, LTI) Modalanalyse wurde weithin und erfolgreich auf ihre Kleinsignalmodelle

für die Stabilitäts- und Resonanzbewertung angewandt. Vor dem Hintergrund der globalen

Klimaveränderung vollzieht sich in modernen Stromnetzen ein technologischer Wandel von

der Stromerzeugung aus fossilen Brennstoffen hin zur Stromerzeugung aus erneuerbaren

Energien. Die Zahl der leistungselektronischen Umrichter, die für die Integration erneuerbarer

Energiequellen eingesetzt werden, hat allmählich ein signifikantes Ausmaß erreicht. Die

mehrstufigen Regelkreise von Umrichtern verursachen Wechselwirkungen mit der Dynamik

von Lasten und Stromnetzen. Darüber hinaus erhöhen Frequenzkopplungseffekte, die durch

Umrichterregler, den Pulsbreitenmodulationsprozess und unsymmetrische Betriebsbedingungen

verursacht werden, die Systemkomplexität weiter, wodurch die modernen, von Umrichtern

dominierten, Stromnetze zu komplexen nichtlinearen zeitperiodischen (engl.: nonlinear time-

periodic, NLTP) Systemen werden. Die klassischen, auf der LTI-Theorie, basierenden

Modellierungs-, Analyse- und Entwurfsansätze werden angreifbar und verlieren an Gültigkeit.

Um die Dynamik von umrichterdominierten Stromnetzen, die unter verschiedenen Bedingun-

gen betrieben werden, genau zu beschreiben, wird ein linearer zeitperiodischer (engl.: linear time-

periodic, LTP) Modellierungsrahmen im komplexwertigen Bereich entwickelt. Die klassische

Modalanalyse für das LTI-System wird auf das LTP-System verallgemeinert. Dementsprechend

werden die Definitionen und Ausdrücke des Dämpfungsgrads, des Partizipationsfaktors und

der Eigenwertsensitivität modifiziert. Es wird eine physikalische Interpretation des LTP-

Systems im Zeitbereich vorgeschlagen, um die Notwendigkeit seiner Anwendung quantitativ

zu bestätigen. Es werden zwei genaue und effiziente LTP-Eigenwertberechnungsmethoden

entwickelt, die den Weg für eine einfachere Anwendung der LTP-Theorie ebnen. Die

vorgeschlagene Stabilitätsanalysemethode wurde mit netzfolgenden Umrichtern getestet, um die

Wechselwirkung zwischen verschiedenen Phasenregelkreisen, Stromreglern und Stromnetzen auf

die Systemstabilität zu untersuchen, und mit netzbildenden Umrichtern, um die Wechselwirkung

zwischen dem äußeren Leistungsregelkreis und dem inneren Spannungs-Strom-Regelkreis zu

untersuchen.

Um die Auswirkungen der Integration von Umrichtern auf die harmonischen Resonanzen

künftiger Stromnetzen zu bewerten, wird eine analytisch geschlossene Form der harmonischen

Übertragungsmatrix (engl.: harmonic transfer matrix, HTM) abgeleitet, um die Beziehung

zwischen den Fourierkoeffizienten der Ein- und Ausgänge des LTP-Systems zu beschreiben. Die

klassische Resonanzmodenanalyse (engl.: resonance mode analysis, RMA) wird verallgemeinert,

indem die konstante Systemimpedanzmatrix durch eine zeitperiodische Matrix ersetzt wird, die

sich aus der Umformulierung der HTM ergibt. Die LTP-Modalanalyse führt zu einer Erweiterung

der Partizipationsanalyse und der Sensitivitätsanalyse zur Identifizierung der Resonanzfrequenz

und zur Bewertung des Einflusses dieser Resonanzfrequenzen. Die vorgeschlagene Methode

wird mit einem beispielhaften Mehrfachumrichtersystem getestet, um die Auswirkungen des

Umrichterreglers, des LCL-Filters und des passiven Netzwerks auf das Resonanzverhalten zu

untersuchen, wobei symmetrische und unsymmetrische Betriebsbedingungen berücksichtigt

werden.

Die verallgemeinerte Eigenwertsensitivitäts- und Partizipationsfaktoranalyse liefert Leit-

linien für die Optimierung der Regelparameter und der Struktur des LTP-Systems. Ein

lineares Programmierproblem wird unter Verwendung von Eigenwert- und Dämpfungsgrad-

Sensitivitätsindizes formuliert, um kritische Eigenwerte in Richtung der linken Halbebene

zu verschieben, ohne die Dämpfungsleistung zu verschlechtern. Darüber hinaus wird auf der

Grundlage der Ergebnisse der Partizipationsanalyse eine Hilfsdämpfungsschleife entworfen, um

die Stabilitätsbereiche zu erweitern. Die theoretische Analyse wird durch eine Kreuzvalidierung

von Stabilitäts- und Resonanzbewertungsergebnissen aus numerischen Modellen, analytischen

Modellen und Hardwaremessungen bestätigt.

Acknowledgement

First, I would like to express my most sincere gratitude to my supervisor Prof. Sibylle

Dieckerhoff, for offering me the chance to pursue my PhD at FG Leistungselektronik and for

her trust and support in the sharpening and realization of my research ideas. Her enthusiasm

and attitude towards research have always inspired me to develop critical thinking and conduct

meticulous researches with high motivations. I am also grateful for her understanding and

encouragement, allowing me to explore possibilities for the next step in my career.

I would like to thank Prof. Keyou Wang with Shanghai Jiao Tong University and Prof. Rik W.

De Doncker with RWTH Aachen University for their interest in my work and for taking the

time to evaluate it.

In addition, I want to thank all my colleagues at FG Leistungselektronik. I feel very lucky

to be part of this great team. The time we spent together, Mensa lunches, coffee breaks,

department trips and technical conferences, etc. will all be treasured memories for me. Special

thanks go to our Netzgruppe for sharing your experience with hardware, so that I can avoid

the risk of learning from explosions.

Last but not least, I want to thank my parents for their understanding and mental support.

Though they know very little about my research, they are always willing to listen to my ups

and downs. Their unconditional love is my power source during these stressful academic years.

Huoming

Berlin, September 2022

Table of Contents

1 Introduction 1

1.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 ThesisStructure ................................... 2

1.3 ListofPublications.................................. 3

2 State of the Art 5

2.1 Modeling ....................................... 6

2.2 Small-Signal Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Harmonic Resonance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 ThesisContribution ................................. 16

3 Modeling of Converter-Dominated Power Systems 17

3.1 Complex-Valued Large-Signal Model . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Grid-Following Voltage Source Converters . . . . . . . . . . . . . . . . . 20

3.1.2 Grid-Forming Voltage Source Converters . . . . . . . . . . . . . . . . . . 27

3.1.3 Power Networks and Passive Loads . . . . . . . . . . . . . . . . . . . . . 31

3.1.4 Steady-State Power Flow Calculation . . . . . . . . . . . . . . . . . . . 32

3.2 Small-SignalModel.................................. 35

3.2.1 Small-Signal Model of Grid-Following Converters . . . . . . . . . . . . . 35

3.2.2 Small-Signal Model of Grid-Forming Converters . . . . . . . . . . . . . . 40

3.2.3 Small-Signal Model of Power Networks and Loads . . . . . . . . . . . . 43

3.2.4

Complete Small-Signal Model of the Converter-Dominated Power System

3.3 General Description of the Simulation Platform and Experimental Test Setup . 46

3.4 CaseStudy ...................................... 48

3.4.1 Single-Converter System . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.2 Multiple-Converter System . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Summary ....................................... 55

4 Small-Signal Stability Analysis 59

4.1 Modal Analysis for the Linear Time-Invariant System . . . . . . . . . . . . . . 59

4.2 Time-Domain Physical Interpretation of the LTP System . . . . . . . . . . . . 63

4.3 Generalized Modal Analysis for Linear Time-Periodic Systems . . . . . . . . . . 64

4.3.1 Eigenvalues and Eigenvectors of LTP Systems . . . . . . . . . . . . . . . 65

4.3.2 Stability and Damping Ratio . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.3 Participation Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . . 70

TABLE OF CONTENTS

4.3.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4 LTP Eigenvalue Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5 CaseStudy ...................................... 84

4.5.1 Grid-Following Converter . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5.2 Grid-Forming Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.6 Summary ....................................... 96

5 Harmonic Resonance Analysis 99

5.1 Forced Response of the LTP System . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2 Impedance Modeling of Converter-Dominated Power Systems . . . . . . . . . . 103

5.2.1 Impedance Modeling of the Voltage Source Converters . . . . . . . . . . 103

5.2.2 Impedance Modeling of Power Networks and Loads . . . . . . . . . . . . 106

5.3 LTP-Theory Based Harmonic Resonance Analysis . . . . . . . . . . . . . . . . . 107

5.3.1 Review of the Conventional Resonance Mode Analysis . . . . . . . . . . 107

5.3.2 LTP-Theory Based Generalized RMA method . . . . . . . . . . . . . . . 110

5.4 CaseStudy ......................................116

5.5 Summary .......................................124

6 Sensitivity-Based Stability Improvement 125

6.1 Sensitivity-Based Parameter Optimization . . . . . . . . . . . . . . . . . . . . . 125

6.2 Sensitivity-Based Damping Loop Design . . . . . . . . . . . . . . . . . . . . . . 131

6.3 Summary .......................................133

7 Conclusion and Outlook 135

7.1 Conclusions......................................135

7.2 FutureWork .....................................136

References 137

List of Figures 147

List of Tables 153

Appendix A Wirtinger Calculus 155

Appendix B Small-Signal Model of Induction machine 157

Glossary of Acronyms

DDSRF-PLL decoupled double synchronous reference frame 14

DP dynamic phasor 9

DSOGI-PLL dual second order generalized integrator PLL 20

DSRF-PLL dual synchronous reference frame PLL 20

EHD extended harmonic domain 9

FLL frequency-locked loop 5

FSA frequency scanning analysis 15

GA generalized averaging 8

GFL grid-following 5

GFM grid-forming 5

HSS harmonic state-space 10

HTM harmonic transfer matrix 10

LPF low-pass filter 21

LTI linear time-invariant 2

LTP linear time-periodic 6

NLTP nonlinear time-periodic 8

OAT one-at-a-time 116

PHIL power-hardware-in-the-loop 47

PI proportional-integral 21

PLL phase-locked loop 5

PoC point of connection 5

PR proportional-resonant 21

PSCD positive-sequence component detector 22

PWM pulse-width modulation 6

RMA resonance mode analysis 15

SCR short-circuit ratio 20

SG synchronous generator 5

SRF-PLL synchronous reference frame PLL 7

VSC voltage source converter 19

VSG virtual synchronous generator 27

Introduction

1.1 Motivation and Objectives

The conventional power system is dominated by fossil fuel and nuclear power generation. The

growth in electricity consumption brings increased greenhouse gas emissions and the risk

of nuclear power plant accidents [1], which force people to reconsider the energy structure.

To achieve efficient, safe and environmentally friendly energy generation and use, renewable

generation technologies (e.g., photovoltaic and wind energy systems) are considered promising

solutions. In Figure 1.1, it can be observed that electricity production with renewable sources in

Germany is gradually replacing coal-based and nuclear-based generation. The overall ambition

is to achieve a 100% renewable German energy system by 2050 [2].

1985 20211990 1995 2000 2005 2010 2015

0 TWh

100 TWh

200 TWh

300 TWh

400 TWh

500 TWh

600 TWh

Other

renewables

Solar

Wind

Hydropower

Nuclear

Oil

Gas

Coal

1985 20211990 1995 2000 2005 2010 2015

20%

40%

60%

80%

100%

Figure 1.1: Electricity production in Germany by sources [3]

With the increasing share of renewable based power generation, the large synchronous

generators providing rotational inertia, frequency and voltage stability are gradually replaced

with power electronics converters with small time constants but flexible and fast control

properties. The control flexibility facilitates converters to emulate dynamics of the synchronous

generators. However, physical differences, such as limited energy storage capacity and

overcurrent withstand capability, make it still difficult for converters to fully shoulder the

frequency and voltage regulation responsibilities in all operation conditions. The multi-

timescale control loops of converters can interact with the dynamics of loads, synchronous

1. Introduction

generators and the electric network, which brings unexpected harmonics, resonances or even

instability over a wide frequency range. Those unique characteristics of converters raise new

challenges to the stability of future converter-dominated/only power systems (see Figure 1.2).

Moreover, the time-periodic steady-state operation trajectory caused by unbalanced conditions

and switching frequency harmonics makes the classical linear time-invariant (LTI) theory based

modeling, analysis and design approaches become frangible and less valid.

Power System Stability

Resonance Stability Converter-driven Stability Rotor Angle Stability Voltage Stability Frequency Stability

Electrical Torsional Fast

Interaction

Slow

Interaction

Small-

disturbance

Transient Small-

disturbance

Large-

disturbance

Short term Long term Short term Long term

New categories of stability related to power electronics converters

Figure 1.2: Extended classification of power system stability [4]

To pave the way towards converter-dominated or even converter-only power systems, this

thesis aims to develop efficient and accurate modeling and stability analysis methods to reveal

instability mechanisms and develop countermeasures. The main objectives are summarized as

follows:

•

Develop a modular analytical and numerical modeling framework for converter-dominated

power systems considering different types of converters and operation conditions.

•

Develop small-signal stability and resonance analysis methods to evaluate the impact of

high penetration levels of converters on dynamics of future power systems.

•

Develop control parameters and structures optimization methods to improve system

stability margins and damping performance.

1.2 Thesis Structure

This thesis is composed of seven chapters:

•Chapter 1 is the Introduction.

•

Chapter 2 reviews the state-of-the-art modeling, stability and resonance analysis

methods for converter-dominated power systems, after which the main contributions of

this thesis are clarified.

•

In Chapter 3, the nonlinear average models of different converters, loads and power

networks are developed in the complex value domain. Then, steady-state power flow

analysis and small-signal linearization techniques are proposed to build a modular and

flexible numerical and analytical modeling framework. A general description of the

software platform and hardware test setup is given.

1.3 List of Publications

•

Chapter 4 generalizes the classical modal analysis for linear-time periodic small-signal

models of the converter-dominated power systems. Stability analysis, damping ratio

analysis, participation factor analysis and eigenvalue sensitivity analysis are performed

to evaluate the dynamic performance of different types of power converters.

•

Chapter 5 focuses on the impact of high penetration levels of converters on resonance

characteristics of modern power systems.

•

In Chapter 6, the aforementioned analytical methods are used as design-oriented tools

to guide the optimization of control parameters and structures to improve stability

margins and damping performance of different converter systems.

•Chapter 7 concludes this thesis.

1.3 List of Publications

The journal and conference publications originating from this thesis are:

•

H. Yang, H. Just, M. Eggers and S. Dieckerhoff, "Linear Time-Periodic Theory-Based

Modeling and Stability Analysis of Voltage-Source Converters," in IEEE Journal of

Emerging and Selected Topics in Power Electronics, vol. 9, no. 3, pp. 3517-3529, June

2021, DOI: 10.1109/JESTPE.2020.3003379.

•

H. Yang, M. Eggers, H. Just, P. Teske and S. Dieckerhoff, "Linear Time-Periodic

Theory-Based Harmonic Resonance Analysis of Converter-Dominated Power System," in

IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 9, no. 6, pp.

7422-7435, Dec. 2021, DOI: 10.1109/JESTPE.2020.3041945.

•

H. Yang and S. Dieckerhoff, "Truncation Order Selection Method for LTP-Theory-

Based Stability Analysis of Converter Dominated Power Systems," in IEEE Transactions

on Power Electronics, vol. 36, no. 11, pp. 12168-12172, Nov. 2021, DOI:

10.1109/TPEL.2021.3076877.

•

H. Yang, M. Eggers, P. Teske and S. Dieckerhoff, "Comparative Stability Analysis and

Improvement of Grid-Following Converters using Novel Interpretation of Linear Time-

Periodic Theory," IEEE Journal of Emerging and Selected Topics in Power Electronics,

early access, 2022, DOI: 10.1109/JESTPE.2022.3194411.

•

H. Yang, M. Eggers, P. Teske and S. Dieckerhoff, "Modeling and Stability Analysis

of Converter-Dominated Grids with Dynamic Loads," 2021 6th IEEE Workshop on the

Electronic Grid (eGRID), 2021, pp. 01-07, DOI: 10.1109/eGRID52793.2021.9662151.

•

H. Yang, H. Just, M. Eggers and S. Dieckerhoff, "Modeling and Stability Analysis of

Grid-Following Voltage-Source Converters Utilizing Individual Channel Design Method,"

2020 IEEE 21st Workshop on Control and Modeling for Power Electronics (COMPEL),

2020, pp. 1-6, DOI: 10.1109/COMPEL49091.2020.9265666.

1. Introduction

•

H. Yang, H. Just, M. Eggers and S. Dieckerhoff, "Wirtinger Calculus Based Modeling

and Analysis of VSG-Dominated Grids," 2020 IEEE 21st Workshop on Control and

Modeling for Power Electronics (COMPEL), 2020, pp. 1-6, DOI: 10.1109/COM-

PEL49091.2020.9265753.

•

H. Yang and S. Dieckerhoff, "Modeling and Stability Analysis of Advanced PLLs Based

on LTP Theory," 2019 21st European Conference on Power Electronics and Applications

(EPE ’19 ECCE Europe), 2019, pp. P.1-P.10, DOI: 10.23919/EPE.2019.8915382.

•

H. Yang, H. Just and S. Dieckerhoff, "Identification of Critical Parameters Affecting

the Small-Signal Stability of Converter-based Microgrids," 2019 20th Workshop on

Control and Modeling for Power Electronics (COMPEL), 2019, pp. 1-6, DOI:

10.1109/COMPEL.2019.8769621.

State of the Art

Power systems are currently undergoing the transition in generation technology from fossil

fuel-based generation to renewable generation. The number of power electronic converters

required to integrate renewable energy sources has gradually increased to a significant level. In

three-phase power systems, the control of these converters can be generally classified into two

categories, namely the grid-following (GFL) control [5] and the grid-forming (GFM) control [6,

7, 8, 9, 10, 11].

The GFL converter aims to inject the reference current determined by the maximum

available power of the regenerative source and the estimation of the voltage at the point

of connection (PoC). This simple concept has been widely used in the integration of wind

parks and photovoltaic power plants. However, GFL converters are not able to undertake

the frequency and voltage regulation responsibility of the conventional synchronous generator

(SG). Instability issues caused by voltage synchronization units, namely phase-locked loop

(PLL) [12] and frequency-locked loop (FLL) [13] units, hinder the further integration of GFL

converters under weak grid conditions. To pave the way towards stable and resilient operation

of future 100% renewable grids, the GFM control emulating the primary control and swing

dynamics of synchronous generators is considered a feasible option. Existing GFM control

schemes can be classified into four categories. First, the droop control [6] derived from the

primary control of synchronous generators is a simple and robust operation approach for

parallel connected converters in standalone grids. To limit the change rate and nadir of

frequencies after large disturbances, the second group (VISMA [7], synchronverter [8] and

virtual synchronous generator [9]), mimicking the inertia response through the emulation of

electromechanical and electromagnetic dynamics of synchronous generators at different levels

of detail, have been proposed. Inspired by the similarities between the DC-link capacitor of the

three-phase DC/AC converter system and swing dynamics of the synchronous generator, the

matching control is proposed in [10], which requires only measurements of the DC-link capacitor

voltage. The fourth category is the virtual oscillator control [11] which makes converters

reproduce dynamics of a weakly nonlinear oscillator. The virtual oscillator control guarantees

almost global asymptotic stability, nevertheless, its active and reactive power control capability

2. State of the Art

remains to be solved. Although GFM control algorithms enable converters to share similar

dynamics with rotational generators, two major physical differences, namely

GFM converters require extra energy storage devices to absorb and inject power during

transients,

Constrained by the thermal capacity of power electronic switches, converters are not

comparable to synchronous generators to withstand the significant overcurrent during

faults,

make the replacement of synchronous generators with GFM converters still a challenge.

VDC

LgRg

L R

Converter Three-phase Grid

Δiin

fin

Δv

fsw

Δv

0fgfsw

-fg

-fsw

fin 2fg-fin fsw-fin fsw+fin

-fsw-fin -fsw+fin

Δiin

0fgfsw

-fg

-fsw

fin

Spectrum:

Figure 2.1: Qualitative illustration of the frequency coupling effect, current disturbance ∆

iin

the frequency of fin can excite voltage response ∆vat multiple frequencies.

Both GFL and GFM converters are equipped with fast and coupled controls that can

dynamically interact with loads and electric networks. Such interaction may result in

unexpected harmonics, resonances or even instability over a wide frequency range. Moreover,

the asymmetric controller (e.g., the PLL brings only the

-component of input voltages to

zero) and the pulse-width modulation (PWM) cause a frequency coupling effect [14, 15], shown

in Figure 2.1. In summary, the high penetration level of converters poses new challenges to

the stability and resonance analysis of future power systems [4].

2.1 Modeling

The small-signal analysis has been widely used in the stability and resonance analysis of

the conventional power systems [16, 17]. To gain insightful understanding of the impact of

the increasing penetration of power electronic converters on the dynamic behavior of the

modern power grids, continuous efforts have been made to develop adequate and efficient

small-signal models for converter systems. Those models can be grouped into two categories,

time-domain state-space models and frequency-domain impedance models. On the other hand,

depending on whether the linear approximation is performed around a constant operation

point or a time-periodic trajectory, small-signal models are divided into LTI models and linear

time-periodic (LTP) models.

2.1 Modeling

Linear Time-Invariant Model

Based on experience in modeling multiple-synchronous-generator power systems, a linear

state-space model in the rotational

reference frame is proposed in [18] for microgrids with

droop-controlled GFM converters. This modeling method divides the microgrid into converters,

networks and loads. The state-space model of each converter is first developed in its individual

reference frame. Then, one of the converters is selected to provide the common

frame.

The complete state-space model of the microgrid is obtained by translating that of other

converters to the common

frame and combining state-space models of networks and loads

on the common

frame. This modularity concept is called component connection method.

It is adopted and extended in [19, 20, 21, 22, 23, 24] by including other types of converters

and loads, such as GFL converters, electric springs and induction motor loads. Based on

state-space models, participation and sensitivity analysis can be performed to gain insightful

evaluation of the system dynamic performance [25, 26]. Nevertheless, to build such full-order

state-space models, all details of the converter topology, control structure and parameters need

to be known, which are generally confidential information of the converter vendors. To tackle

this limitation, the frequency-domain impedance model draws increasing attention in power

electronics community. The basic idea is to derive the terminal equivalent impedance of the

converters, namely the transfer function between the converter output voltage and current.

The

-frame impedance models of the grid-following converter are developed in [27, 28, 29,

30] to investigate the negative impact of the synchronous reference frame PLL (SRF-PLL) on

system stability. The impact of advanced PLLs is addressed in [31, 32, 33]. The impedance

model of GFM converters has been reported in [34]. The impedance modeling method provides

the possibility for black-box modeling with measurement results. However, it might give

inaccurate stability assessment results when the poorly damped or unstable oscillation modes

do not participate in the system outputs. Generally, the selection of the modeling method

depends on application scenarios and the purpose of applications. It is inappropriate to draw

conclusions about which modeling method is superior.

The models reviewed above are all in the rotational

reference frame. The

-frame

modeling enables to describe the system steady state with constant values, which guarantees

that the linearized small-signal model is time invariant. However, the physical meaning of

voltages, currents and impedances in the fictitious

frame can be confusing in practice

for engineers. Moreover, during impedance measurements, phase angles of the grid and the

converter under tests are unknown. The phase angle of the converter injecting voltage/current

disturbances is the only choice for the Park transformation. Additional compensation algorithms

are required to eliminate the deviation caused by the interaction between the converter

injecting perturbations and the system under tests [35]. To gain clear physical interpretation

and to simplify the impedance measurement procedure, the sequence-domain impedance

models are proposed in [36, 37, 38]. In the sequence-domain modeling framework, two-

dimensional real-valued voltage and current vectors [

xα

;

xβ

]are assembled into complex-

valued variables

xαβ

xα

jxβ

. Both positive-sequence and negative-sequence voltage (or

current) perturbations are injected into the system under tests. Corresponding steady-state

current (or voltage) responses are deduced by using the harmonic linearization principle for

the impedance calculation. This modeling framework becomes questionable when negative-

2. State of the Art

frequency components appear in the positive-sequence variables. To further eliminate such

confusion, the authors of [39] propose a complex-valued impedance modeling framework in

the stationary

αβ

reference frame by introducing so-called base vectors

xαβ

xα

jxβ

and

x∗

αβ

xα−jxβ

. Yet the mathematical and physical meaning of these base vectors are

unclear. Additionally, real-valued description is still needed for the linearization, complete

complex-domain modeling has not been achieved.

Inspired by the fact that similar stability analysis results were obtained from the

-frame

impedance model and the sequence-domain impedance model, some work has been motivated

to mathematically reveal their equivalence [40, 41, 42, 43, 44]. It is concluded that small-signal

models in different reference frames, giving the same stability assessment result, can be bridged

by linear transformations.

Linear Time-Periodic Model

The LTI models are only valid for three-phase balanced sinusoidal systems with classical

converter controllers, such as the standard SRF-PLL, grid-following and grid-forming control

without frequency-adaptive harmonics and negative sequence component extraction structures.

The real-world three-phase systems, experiencing voltage imbalance and harmonic distortion

[45], or containing converters with advanced grid synchronization units [46], imbalance and

harmonic estimation/compensation controllers [47], are actually complex nonlinear time-

periodic (NLTP) systems. Their dynamics cannot be fully captured by the LTI theory.

NLTP

model

GA method

LTP model

DP/EHD model

HSS/HTM model

Floquet

Theory

Nonlinear Linear

Time Domain

Freq. Domain

Laplace Trans.

Harm. Bal.

Linearization

Fourier Ser.

Linearization

Harm. Bal.

Fourier Ser.

Figure 2.2: Small-signal models of the nonlinear time-periodic system.

To overcome the limitation of the LTI models, linear time-periodic theory gains increasing

application in recent years. Various LTP-theory-based linearized models have been proposed

for the small-signal analysis of converter systems described with ordinary differential equations

x˙(t) = f(x(t), u(t)) (2.1)

where

(

)and

(

)denote state and input vectors. Those models share the same theoretical

basis, that is the first-order Taylor series approximation, Fourier series expansion and the

harmonic balance principle. As shown in Figure 2.2, they are derived through two paths.

First, according to the generalized averaging (GA) method introduced in [48], the waveforms

(

)and

(

)can be approximated on the time interval [

t−T0, t

]to arbitrary precision with

2.1 Modeling

the Fourier series

x(t−T0+τ) = ∑︁kxk(t)ejkω0(t−T0+τ)

u(t−T0+τ) = ∑︁kuk(t)ejkω0(t−T0+τ), τ ∈[0, T0]

where

= 2

π/ω0

is the width of the moving time window, over which the time-dependent

Fourier coefficients xk(t)and uk(t)are calculated. Defining Fourier coefficient vectors

X(t)=[· · · , x−1(t), x0(t), x1(t),· · · ]T

and

U(t)=[· · · , u−1(t), u0(t), u1(t),· · · ]T,

the time-domain finite-dimensional system Eq. (2.1) can be equivalently reformulated as an

infinite-dimensional system in the frequency domain

˙(t) = g(X(t), U(t)) .(2.2)

When the steady-state operation trajectories

{xss(t), uss(t)}

are time periodic with fundamental

period of

{Xss(t), Uss(t)}

become constant vectors. Then, a theoretically infinite-

dimensional LTI small-signal model can be obtained

∆X

˙(t) = A∆X(t) + B∆U(t)

A=∂g

∂X Xss, Uss

B=∂g

∂U Xss, Uss

which is commonly referred to as dynamic phasor (DP) model [49, 50, 51, 52, 53] or extended

harmonic domain (EHD) model [54, 55, 56, 57].

The second route is inspired by the pioneer work of Wereley [58], whose primary thrust

is to develop an operator for the LTP system, which maps the Fourier coefficients of inputs to

those of outputs, comparable to the transfer function for LTI systems. To apply his method, the

NLTP model Eq. (2.1) is first linearized around the time-periodic trajectories

{xss(t), uss(t)}

∆x˙ (t) = A(t) ∆x(t) + B(t) ∆u(t)

A(t) = ∂f

∂xxss(t), uss(t)

B(t) = ∂f

∂uxss(t), uss(t)

.(2.3)

If the so-called exponentially modulated periodic inputs

∆u(t) = est ∑︂∞

k=−∞ Ukejkω0t

are fed to the LTP model Eq. (2.3), the steady-state response are also exponentially modulated

periodic signals

∆x(t) = est ∑︂∞

k=−∞ Xkejkω0t

2. State of the Art

satisfying the harmonic state-space (HSS) equation [59, 60, 61, 62, 63, 64, 65, 66] obtained

from the harmonic balance principle (namely Fourier coefficients of both sides of Eq. (2.3)

must be the same)

sX= (A−N)X+BU (2.4)

where sis a complex number and

X= [· · · , X−1, X0, X1,· · · ]T

U= [· · · , U−1, U0, U1,· · · ]T

N=diag[· · · , N−k,· · · , N0,· · · , Nk,· · · ]T

Constant matrices

and

are Toeplitz matrices of Fourier coefficients of

(

)and

(

is a diagonal square matrix with diagonal elements equal to

jkω0

, and it shares the same

dimension with A(t).

Let ∆

x(t)

be also the output vector, the infinite-dimensional harmonic transfer matrix

(HTM) [67, 68, 69, 70, 71, 72], describing the relation between harmonics in input and output

variables, is given by

GHT M (s)=(sI − (A−N))−1B(2.5)

where Iis the identity matrix.

It is seen that the common idea behind the DP/EHD and HSS/HTM models is to find an

LTI approximation of the NLTP system for small-signal stability analysis. To this end, the

DP/EHD model replaces time-domain variables with their Fourier series coefficients, resulting

in a nonlinear time-invariant model. Then, classical LTI techniques apply. Instead, the

HSS/HTM model initially performs first-order Taylor series approximation, resulting in an

LTP model. Then, based on Fourier series expansion and harmonic balance principles, a

frequency-domain LTI model that maps harmonics of input and output variables is derived.

Theoretically, the HSS/HTM models can be regarded as the Laplace transformation of the

DP/EHD models. Numerically, the difference between these two categories of models is that the

truncation order of DP/EHD models, namely the number of selected harmonics, is determined

before the linearization. Therefore, the linearization process must be repeated when more

harmonics need to be considered to guarantee the accuracy. Contrarily, the HSS/HTM method

is more convenient to adjust the truncation order. It should be mentioned that, different from

the HSS/HTM method, the GA method is also valid for the description of general time-varying

systems, since the Fourier coefficients can be calculated and updated by online integration.

The LTP small-signal models reviewed above are essentially developed in the frequency

domain. Deriving analytical expressions for their free and forced responses is not straightforward

because of the need to perform Fourier series expansion of input variables and Fourier series

synthesis of state/output variables. Though eigenvalue and participation analysis have been

applied to the truncated DP model [51] and HSS model [73] for small-signal stability analysis,

some inconveniences of these frequency-domain models can still be identified:

•

selection of the truncation order is a trade-off between modeling accuracy and

computational efficiency, only empirical recommendation is provided in [74];

2.2 Small-Signal Stability Analysis

•

increasing truncation order to ensure modeling accuracy results in more eigenvalues with

critical damping ratio, which can mislead the assessment of damping performance;

•

the truncation will always bring spurious eigenvalues lying outside vertical lines formed by

convergent eigenvalues. Such spurious eigenvalues are commonly distinguished graphically,

which requires unnecessarily large truncation orders and hinders automatic stability

analysis of LTP systems.

Moreover, the strength of the frequency coupling effect has not been quantitatively evaluated

to answer when the LTP theory must be applied.

2.2 Small-Signal Stability Analysis

Linear Time-Invariant Theory

According to the LTI control theory, there exist two types of stability evaluation methods,

namely eigenvalue analysis and Nyquist stability criterion, corresponding to the time-domain

state-space model and frequency-domain impedance model, respectively.

Real Axis

Imaginary Axis

Real Axis

Imaginary Axis

Figure 2.3: Relation between eigenvalues and time-domain free responses

Figure 2.3 qualitatively shows the relation between eigenvalues and time-domain dynamic

responses of LTI systems. Eigenvalues of the system matrix in the state-space model provide

the complete information of the system dynamics: the system is marginal stable when real

parts of all eigenvalues are negative, and the damping performance of each oscillation mode is

quantified by the damping ratio. Another useful tool of the eigenvalue-based stability analysis

framework is the participation factor analysis, which quantifies the relative contribution of each

oscillation mode to different state variables [17]. The participation factor analysis of critical

modes can facilitate efficient allocation of monitoring and protection devices. Additionally,

eigenvalue sensitivity analysis can be performed to quantify impact of changes of control and

physical parameters on the movement of eigenvalues, which provides guidance to the parameter

optimization [75, 76]. For example, using the eigenvalue analysis, the authors in [77] concluded

that the maximum power transfer limit (obtained from static voltage stability analysis) of a

VSC-HVDC system is achievable when the PLL gains become very small. In [18], the eigenvalue

analysis result reveals that the low-frequency oscillation modes of droop-controlled converter

2. State of the Art

Converter Grid

Vg(s)

Zg(s)

Ic(s)

Yc(s)

Io(s)

Converter Grid

Vg(s)

Zg(s)

Ic(s)

Yc(s)

Io(s)

Yc(s)

Zg(s)Yc(s)

Vg(s)

Ic(s)

Io(s)

Yc(s)

Zg(s)Yc(s)

Vg(s)

Ic(s)

Io(s)

Converter Grid

Vg(s)

Zg(s)

Ic(s)

Yc(s)

Io(s)

Yc(s)

Zg(s)Yc(s)

Vg(s)

Ic(s)

Io(s)

Figure 2.4: Principle of the impedance-based

stability analysis. Top: impedance model.

Bottom: equivalent control system

Real Axis

Imaginary Axis

(-1, 0)

Gain difference

before instability

Phase difference

before instability

Unit Circle

-1/a

Phase Margin = φ

Gain Margin = 20 log a

Real Axis

Imaginary Axis

(-1, 0)

Gain difference

before instability

Phase difference

before instability

Unit Circle

-1/a

Phase Margin = φ

Gain Margin = 20 log a

Figure 2.5: Nyquist diagram for single-input-

single-output system

systems are highly sensitive to the network configuration and droop coefficients, while the

high-frequency modes are largely influenced by the interaction between inner voltage-current

controller, load and network dynamics. The interaction between different types of converters

has been recently investigated in [19, 78] based on eigenvalue analysis.

In the frequency domain, the impedance-based stability criterion is a well-established

technique. It is originally developed for the design-oriented analysis of input filters of DC-DC

converters [79]. In [80], it was applied to the analysis of AC power systems, and has gained

wide application for the stability assessment of grid-tied converters since the publication of

[36]. This method is inspired by the similarity between equivalent electrical circuits and the

feedback control system, as shown in Figure 2.4. The converter is described with a Norton

equivalent circuit, which is connected to an ideal voltage source in series with a grid impedance.

The output current can be determined from the feedback control diagram

Io(s) = Yg(s)

Yc(s) + Yg(s)Ic(s)−1

Zg(s) + Zc(s)Vg(s)

= [Ic(s)−Yc(s)Vg(s)] 1

1 + Zg(s)Yc(s)

When the following conditions are satisfied

1. the voltage source is stable when unloaded (i.e., Vg(s)has no unstable poles);

2. the current source is stable when unloaded (i.e., Ic(s)has no unstable poles),

the load is stable when powered with an ideal voltage source (i.e.,

(

)has no right-half

plane zeros),

the system stability is determined by whether the impedance ratio

Zg(s)Yc(s)

, namely the

minor loop gain, satisfies the Nyquist criterion. The phase margin and gain margin shown in

Figure 2.5 can be used to quantify the stability robustness. However, these indicators loose

their effectiveness when the converter model becomes an impedance matrix, for which the

generalized Nyquist criterion [81] is required for the stability investigation.

An attractive feature of the impedance-based stability analysis method is the black-box

modeling capability, since the impedance profile can be measured with frequency scanning

2.2 Small-Signal Stability Analysis

technique. However, the equivalent impedance reflects only the input-output dynamics at the

converter terminal, which cannot reveal internal instability issues. The generalized Nyquist

criterion gives only the conclusion whether the system is stable or not. Indicators with

clear physical meaning still need to be developed to quantify the stability margin and guide

control optimization. The passivity theory (i.e., when every converter exhibits a non-negative

resistance in a certain frequency range, unstable oscillations will not occur in that frequency

range) could be a promising option [82, 83], though it is not a necessary condition for the

system stability.

In addition, it is commonly assumed that the impedance ratio has no right-half plane poles.

Then, the system stability is evaluated by checking whether the Nyquist plot encircles the

critical point (

−

0) in the complex plane. In practice, this assumption is not always satisfied

[84, 85, 86]. Determination of the right-half plane poles of the impedance ratio from the

impedance measurement results deserves further investigation. In contrast, eigenvalue-based

analysis yields highly detailed system dynamics assessments, and has been successfully used

for positioning and parameter optimization of power system stabilizers to damp low-frequency

oscillations in the bulk power systems [16].

Linear Time-Periodic Theory

In the frequency domain, the harmonic transfer matrix given by Eq. (2.5) reveals that the

finite dimensional LTP system is equivalent to an infinite-dimensional LTI system. Therefore,

the Nyquist stability criterion is also applicable to the stability analysis of the LTP system.

To name a few, harmonic impedance models of GFL converters operated under unbalanced

conditions are built in [87, 68, 67]. It is concluded that both voltage and current imbalances

can reduce the stability margin. In [88], the harmonic transfer matrix is used to address

the small-signal stability of voltage-controlled modular multilevel converters. Moreover, its

effectiveness for the stability analysis of single-phase converters is confirmed by experimental

validation in [64, 89].

In the time domain, stability analysis of LTP systems can be performed by using Floquet

theory [90] dating back to 1883. Consider the homogeneous state-space equation, i.e., the

free-response problem of Eq. (2.3),

∆x˙ (t) = A(t) ∆x(t)

the Floquet theory gives that the solution is determined by

∆x(t) = Φ(t, 0) ∆x(0) (2.6)

where ∆

(0) is the initial condition at

= 0.

(

0) is called the state transition matrix,

which is

-periodic, namely

Φ(t+T0, T0)

Φ(t, 0)

. It can be obtained by solving the matrix

differential equation

˙(t, 0) = A(t)Φ(t, 0) ,with Φ(0,0) = I

2. State of the Art

where

denotes the identity matrix. From Eq. (2.6), the state transition over

full periods is

given by

∆x(2T0) = Φ(2T0, T0) ∆x(T0) = Φ(2T0, T0)Φ(T0,0) ∆x(0) = (Φ(T0,0))2∆x(0)

⇓

∆x(kT0)=(Φ(T0,0))k∆x(0)

Inserting the eigendecomposition Φ(T0,0) = RΛR−1, yields

∆x(kT0) = RΛkR−1

⏞⏟⏟ ⏞

(Φ(T0,0))k

∆x(0)

where Λis the eigenvalue matrix,

is the eigenvector matrix. The state vector ∆

x(kT0)

asymptotically stable, only if eigenvalues of

Φ(T0,0)

are located within the open unit circle

in the complex plane. In the Floquet theory, matrix

Φ(T0,0)

and its eigenvalues are called

monodromy matrix and characteristic multipliers, respectively. It was used in [91] for stability

analysis of a single-phase asymmetric cascaded H-bridge multilevel inverter operated under

the standalone mode. Based on the Floquet theory, the authors of [92] figured out that the

stability of the decoupled double synchronous reference frame (DDSRF-PLL) is influenced by

both input voltage level and unbalance factors.

The Floquet theory can only predict whether the system is stable or not. It fails to

provide more insightful characterization of system dynamics, e.g., damping, sensitivity and

participation information. The generalization of the powerful eigenvalue-based analysis for

LTI systems to LTP systems remains an open topic. Though eigenvalues of HSS models are

calculated in [64, 65] to detect the precise stability boundary of a single-phase converter system,

the physical meaning of eigenvalues of the HSS model is still unclear, and the link between

these eigenvalues and time domain dynamics has not been revealed, since the HSS model is

inherently derived in the frequency domain for periodic inputs. The same problem exists when

the eigenvalues of the DP/EHD model is computed for stability analysis [93].

Moreover, some numerical difficulties in the stability analysis of LTP systems remain to be

solved. In the time domain, for large-scale converter-dominated systems, solving the numerical

matrix equation can be time-consuming. In the frequency domain, the infinite-dimensional

matrix must be truncated for the eigenvalue calculation. Though it has been rigorously proved

in [94] that eigenvalues of the truncated matrix converge to those of the non-truncated one

as the truncation order approaches infinity, selection of the truncation order

is a trade-off

between computational accuracy and efficiency. In [58, 94], the truncation order is increased

until

unchanged eigenvalues are found in the fundamental strip, where

denotes the number

of state variables. Yet this method ignores the rate of convergence of eigenvalues and can result

in unnecessarily high truncation orders. An iterative eigenvector sorting method is used in [95],

the basic idea behind is that eigenvectors, whose nonzero elements are symmetric about the

DC component and within the truncation order, are less influenced by the truncation. However,

no rigorous mathematical proof has been achieved. In [74], it is suggested that

should be

larger than the maximum harmonic order of nonzero Fourier coefficients of

(

). However, no

explicit formula has been established, and the impact of changing the truncation order has not

2.3 Harmonic Resonance Analysis

been quantitatively evaluated. An accurate and efficient truncation order selection method is

demanded to pave the way for easier application of the LTP theory.

2.3 Harmonic Resonance Analysis

Power systems were previously dominated by synchronous generators with significant

mechanical inertia. Since timescales of the electromechanical and electromagnetic dynamics

are sufficiently separated, the harmonic resonance analysis of conventional power systems

mainly focuses on the inherent characteristics of the passive network [96, 97, 98, 99, 100].

With the increasing penetration of renewable energy generation, the modern power system

is gradually evolving as power converter dominated. The multiple-timescale control loops of

power converters can cause cross couplings with dynamics of power networks, and frequency

coupling effects caused by asymmetric controllers and time-periodic operation trajectories

further increase the system complexity. This poses new challenges to the harmonic resonance

analysis. As reviewed above, efficient and accurate modeling of power converters has been

intensively investigated, however, an effective system-level analysis approach for resonance

frequency identification and resonance severity evaluation of converter-dominated systems

operating under different conditions is still an open topic.

In power system engineering, two types of harmonic resonances have to be distinguished:

parallel and series resonances. The parallel resonance refers to the phenomenon that a small

current injection can result in very large bus voltages. The series resonance refers to those cases

where a small bus voltage can cause large branch currents. In general, there are mainly two

techniques for harmonic resonance analysis. The first method is the frequency scanning analysis

(FSA) [96], which can confirm whether the resonance exists and identify the resonance frequency

by examining the bus voltage (branch current) responses to harmonic current (voltage) injection.

This is actually the same procedure as the impedance measurement. Although the FSA is

very straightforward, it fails to provide more insightful information for the design of resonance

mitigation strategies. The second method is the resonance mode analysis (RMA) [97, 98,

99, 100] which is based on the eigen-analysis of the network admittance and impedance

matrix. Compared with FSA, RMA can further reveal the excitability and observability of

a certain resonance mode at each bus by means of participation factor analysis. Meanwhile,

the sensitivity analysis in combination with the RMA technique can quantify the impact of

each parameter on the system harmonic resonance, which could provide guidelines for the

development of the resonance mitigation scheme. The RMA has been utilized in [101, 102] to

investigate the resonance interactions between converters and the grid by integrating converters’

small-signal equivalent impedance models into the system admittance matrix. However, the

converter impedance models used in these papers are obtained from the linearization around

the DC operation point. The influence of the frequency coupling effect, caused by the PWM,

the PLL and the non-ideal grid conditions (e.g., unbalanced grid voltages and impedance), on

the system resonance characteristics cannot be fully captured.

2. State of the Art

2.4 Thesis Contribution

Aiming to fill in the knowledge gap of recent research, the unique contributions of this thesis

can be summarized as follows:

•

A modular and flexible modeling, stability and resonance analysis framework is developed

in MATLAB/Simulink platform. By introducing the Wirtinger calculus, completely

complex-domain nonlinear and small-signal modeling of state-of-the-art converters is

achieved. A complex-valued power flow analysis method is developed for converter-

dominated power systems to improve the efficiency of development of the small-signal

model.

•

The conventional modal analysis for LTI systems is generalized to LTP systems.

Definitions of damping ratio, participation factor and eigenvalue/eigenvector sensitivity

are accordingly modified to evaluate the time-domain dynamic performance of GFL and

GFM converters.

•

To fully capture the impact of the frequency coupling effect on the harmonic resonance

behavior of modern grids, a time-periodic impedance matrix is proposed. A generalized

resonance mode analysis method is applied to the time-period impedance matrix to

investigate the influence of the converter integration and unbalanced operations on the

resonance frequency and severity.

•

Based on LTP sensitivity analysis and participation analysis, a design-oriented control

parameter and structure optimization method is proposed to improve the damping and

stability margin of GFL and GFM converters.

Modeling of Converter-Dominated

Power Systems

This chapter deals with the time-domain steady-state and small-signal modeling of converter-

dominated power systems considering both balanced and unbalanced operation conditions.

First, complex-valued description of the state-of-the-art grid-following and grid-forming

converters is proposed. Then, by introducing the complex partial derivative defined within

the Wirtinger calculus framework, the small-signal models of different types of converters

are directly derived in the complex domain. To improve the development efficiency of the

small-signal model, a complex-valued power flow analysis method is proposed to determine the

steady-state operation trajectory. Next, the component connection method is implemented

in the stationary reference frame for system-level modeling. Simulation and experimental

tests have been carried out on single-converter and multi-converter systems to validate the

analytical methodology.

3.1 Complex-Valued Large-Signal Model

In a three-phase system with no zero-sequence component, a real-valued three-phase

voltage/current vector [

xa, xb, xc

]

can be equivalently described with a two-phase Cartesian

vector [xα, xβ]Tby applying the Clarke transformation TClarke

[︄xα

xβ]︄=TClarke 









TClarke =2

3[︄1−1

2−1

0√3

2−√3

2]︄.

Assembling the in-quadrature entries of the Cartesian vector into a complex scalar, the number

of variables and equations for the system description can be halved [103, 104]. Figure 3.1

exemplarily shows the waveform of a three-phase quantity described with two real-valued

variables or one complex-valued variable.

3. Modeling of Converter-Dominated Power Systems

Figure 3.1: Waveform of a three-phase current or voltage. Blue: the complex-valued variable.

Red: the real part (αcomponent). Green: the imaginary part (βcomponent).

In the complex plane, fixing the

-axis in the direction of the real axis, the in-quadrature

axis may coincide with either the imaginary axis or the negative imaginary axis (see Figure 3.2).

In this thesis, the obtained two coordinates are defined as the original coordinate and the

conjugate coordinate, respectively. This transformation is given by

[︄xαβ

x∗

αβ ]︄=Tr2c[︄xα

xβ]︄Tr2c=[︄1j

1−j]︄Tc2r=T−1

r2c

where

xαβ

denotes the complex-valued voltage/current variable in the original coordinate, and

the associated variable x∗

αβ is in the conjugate coordinate.

xαβ

β ω

xβ

xα xα

xβ x*

αβ

Figure 3.2: Complex-domain description of three-phase variables

Physically, the information provided by

x∗

αβ

is redundant, since [

xα, xβ

]

can be fully

recovered from either

xαβ

x∗

αβ

by separating their real and imaginary parts. Mathematically,

it plays an essential role for the power flow analysis and the linearization in the complex

domain. For instance, the absolute value function used for the steady-state and dynamic

modeling of the power converters

f(xαβ) = f(xα, xβ) = fRe (xα, xβ) + j·fIm (xα, xβ) = |xαβ|=√︂x2

α+x2

3.1 Complex-Valued Large-Signal Model

does not have the classical complex partial derivative

∂f/∂xαβ

, since the function is non-

holomorphic, namely it does not fulfill the Cauchy-Riemann condition:



∂fRe

∂xα

=xα

√︂x2

α+x2

β

=(︄∂fIm

∂xβ

= 0)︄



∂fRe

∂xβ

=xβ

√︂x2

α+x2

β

=(︃−∂fIm

∂xα

= 0)︃

Therefore, the absolute value function cannot be directly linearized in the complex domain.

To solve this problem, the relaxed definition of the complex partial derivative, which is called

the Wirtinger calculus [105], is adopted in this thesis. The main idea behind the Wirtinger

calculus is to reformulate fas a function of both xαβ and x∗

αβ

f(xαβ) = |xαβ|=f(︂xαβ, x∗

αβ)︂=√︂xαβx∗

αβ

and the complex derivatives are calculated by treating

xαβ

and

x∗

αβ

as independent variables

∂f

∂xαβ

=x∗

αβ

2√︂xαβx∗

αβ

∂f

∂x∗

αβ

=xαβ

2√︂xαβx∗

αβ

Then, the linearized form is given by

∆f=∂f

∂xαβ Xαβ, X∗

αβ

∆xαβ +∂f

∂x∗

αβ Xαβ , X∗

αβ

∆x∗

αβ

=X∗

αβ

2√︂XαβX∗

αβ

∆xαβ +Xαβ

2√︂XαβX∗

αβ

∆x∗

αβ

where

Xαβ

and

X∗

αβ

denote the steady-state values of

xαβ

and

x∗

αβ

. The variables preceded by

∆denote small-signal ones.

It can be confirmed that the complex conjugate operator, as well as the real and imaginary

part extraction operators are also non-holomorphic. The Wirtinger calculus enables the direct

linearization in the complex domain. To make this thesis self-contained, details about the

Wirtinger calculus are provided in the Appendix A.

In this section, the nonlinear state-space model of the voltage source converter (VSC),

power networks and loads will be developed in the complex domain based on the fundamentals

introduced above. The notations used in this section follow the rules below:

The subscript in the notation defines the reference frame, specifically,

and

are for the

stationary

αβ

reference frame,

and

for the rotational

reference frame. Variables

with single character are in the real value domain

, e.g.,

xα

and

. The subscript with

double characters defines variables in the complex value domain C, e.g., vαβ and vdq.

3. Modeling of Converter-Dominated Power Systems

When the variable comprises a single sequence component, an extra symbol +for the

positive-sequence component or symbol

−

for the negative-sequence component is added

into the subscript, e.g., vdq+.

Other characters in the subscript are used to denote the physical meaning of variables,

e.g., cindicates that vαβc is the filter capacitor voltage.

Variables of different converters or buses are indexed with comma-separated numbers in

the subscript, e.g., vαβc,i stands for the filter capacitor voltage of the ith converter.

In addition, the complex conjugate operator is denoted by the symbol star

(·)∗

(·)T

and

(·)−1

denote the transpose and inverse operators respectively.

Im {·}

is the imaginary part

extraction operator, and

Re {·}

stands for the real part extraction operator. If not specified

otherwise, boldface is used for vectors and matrices.

3.1.1 Grid-Following Voltage Source Converters

Figure 3.3 depicts the schematic and control of a single GFL converter system. The converter

is assumed to be supplied by an ideal DC voltage

VDC

. Its output is connected to the grid

through an LC(L) filter. For the single converter system, the grid is modeled as an ideal voltage

source in series with an RL impedance. The grid strength quantified by the short-circuit ratio

(SCR)

SCR =V2

|Zg|S0

=V2

|Rg+jω0Lg|S0

(3.1)

can be modified by changing the grid impedance

and

stands for the rated voltage

of the converter, and

is the apparent output power of the converter.

ω0

= 2

π·

rad/s

the fundamental frequency.

vαβg

PLL

PWM Current Ref.

Calculation

Current

Control

vαβc

iαβf

iαβr

VDC

vαβr

6θ, ω

LgRg

L R

iαβg

vαβ+

vdq+

idqr

θ, ω

vαβg

PLL

PWM Current Ref.

Calculation

Current

Control

vαβc

iαβf

iαβr

VDC

vαβr

6θ, ω

LgRg

L R

iαβg

vαβ+

vdq+

idqr

θ, ω

Figure 3.3: Schematic and control of a single grid-following converter system

The control of the GFL converter can be generally divided into three parts. First, the

PLL serves as a synchronization unit for the detection of the magnitude, frequency and phase

angle of the filter capacitor voltage

vαβc

. To realize precise and fast grid synchronization even

for unbalanced or harmonically distorted grid voltages, two advanced PLLs are considered,

the dual synchronous reference frame PLL (DSRF-PLL) [106] and the dual second order

generalized integrator PLL (DSOGI-PLL) [107]. Next, based on the instantaneous power

theory [108], the current reference

iαβr

idqr

is calculated from the given power command

and the detected positive-sequence capacitor voltage

vαβ+

vdq+

. Third, the current

3.1 Complex-Valued Large-Signal Model

control is realized with either a proportional-integral (PI) controller in the

frame or a

frequency-adaptive proportional-resonant (PR) controller in the αβ frame [109].

Advanced Three-phase PLLs

Figure 3.4 shows the basic structure of the classical three-phase SRF-PLL. Let

vαβc =V+ejθ0=V+ejω0t

be the balanced three-phase input voltage. When the SRF-PLL is quasi synchronized, namely

the estimated phase angle

deviates only slightly from

θ0

(

θ≈θ0

), the phase angle estimation

error can be approximated by the Park transformation

TP ark(θ) = e−jθ

of the input voltage

vdq =TP ark(θ)vαβc =e−jθvαβc =V+ej(θ0−θ)

=V+cos (θ0−θ) + jV+sin (θ0−θ)

≈V+

⏞⏟⏟⏞

+j V+(θ0−θ)

⏞⏟⏟ ⏞

Tpark LPF PI I

vαβcθ

vdq ω

Tpark LPF PI I

vαβcθ

vdq

Figure 3.4: Basic scheme of SRF-PLL

To eliminate the phase angle detection error, a PI controller is applied to regulate the

imaginary part of

vdq

, namely

vq≈V+(θ0−θ)

, to zero. Under ideal three-phase balanced

and harmonic-free conditions, the SRF-PLL yields precise and fast frequency and phase angle

tracking performance. However, when the input voltage is unbalanced and/or harmonically

distorted, for instance

vαβc =V+ejω0t+V−e−jω0t+jθ−(3.2)

contains a negative-sequence voltage with magnitude and initial phase angle of

V−

and

θ−

the disturbance at the frequency of 2ω0in vqresulting from the Park transformation

vdq =TP ark(θ)vαβc =e−jθvαβc

=V+ej(θ0−θ)+e−jθV−e−jω0t+jθ−

≈V++jV+(θ0−θ) + V−e−j2ω0t+jθ−

will significantly degrade the synchronization accuracy of the SRF-PLL. Although the in-loop

low-pass filter (LPF) can be used to suppress harmonics, the low cut-off frequency required

to reject disturbances at 2

ω0

can result in an unacceptably slow response. To overcome the

3. Modeling of Converter-Dominated Power Systems

shortcoming of the SRF-PLL, two advanced PLL schemes, DSRF-PLL and DSOGI-PLL,

are commonly used for applications requiring high accuracy and fast dynamic response even

under adverse utility voltages. Their common idea is to employ a specific positive-sequence

component detector (PSCD) in front of the conventional SRF-PLL.

Dual Synchronous Reference Frame PLL, DSRF-PLL

As shown in Figure 3.5, by taking both

and

−θ

as the input of the Park transformation, the

DSRF-PLL implements two

frames rotating in opposite directions, namely

+and

dq−

frames. Assume that the PLL perfectly synchronizes to the positive-sequence voltage, i.e.,

θ=θ0, the unbalanced input voltage expressed in the dq+and dq−frames are

v¯dq+=TP ark (θ0)vαβ =V++V−ejθ−e−j2ω0t

v¯dq−=TP ark (−θ0)vαβ =V+ej2ω0t+V−ejθ−

.(3.3)

Tpark

LPFTpark

LPF PI I

vαβc

vdq+

vdq-

-θ

θ-2θ

2θ

Tpark

DSRF vd+

vq+

ηω

Tpark

LPFTpark

LPF PI I

vαβc

vdq+

vdq-

-θ

θ-2θ

2θ

Tpark

DSRF vd+

vq+

Figure 3.5: Basic scheme of DSRF-PLL

It is observed from Eq. (3.3) that the magnitude of the double fundamental frequency oscillation

term present in

+or

dq−

frame corresponds to the DC term in the other rotational frame.

By subtracting the oscillatory terms, the positive and negative sequence voltage components

can be separated. The 1st-oder LPF

GLP F (s) = ωf

s+ωf

with a bandwidth of

ωf

is used to improve extraction performance of the DC terms. The four

LPFs in Figure 3.5 are always designed to have the same cut-off frequency.

Combining the PSCD structure with the SRF-PLL, the complete state-space model of the

DSRF-PLL is

v˙dq+=−ωfvdq+−ωfe−j2θvdq−+ωfe−jθvαβc

v˙dq−=−ωfvdq−−ωfej2θvdq++ωfejθvαβc

η˙ = kivq+=kiIm {vdq+}

˙=ω=η+kpvq+=η+kpIm {vdq+}

(3.4)

where state variables

vdq+

and

vdq−

are for the low-pass filters.

and

are PI control

coefficients, and ηis associated with the integrator.

3.1 Complex-Valued Large-Signal Model

Dual Second Order Generalized Integrator PLL, DSOGI-PLL

The basic scheme of the DSOGI-PLL is illustrated in Figure 3.6. Instead of separating the

positive and negative sequence voltage components in different

frames, the DSOGI-PLL

directly realizes the sequence decomposition in the

αβ

frame. When the resonance frequency

of the DSOGI structure is adapted to that of the positive-sequence component of the input

voltage, the DSOGI structure will be able to precisely estimate the input voltage (Eq. (3.2))

and its quadrature signal, namely

Xαβ1=V+ejω0t+V−e−jω0t+jθ−

jω0Xαβ2=V+ejω0t−V−e−jω0t+jθ−

.(3.5)

Xαβ1

and

Xαβ2

stand for steady-state values of

xαβ1

and

xαβ2

, which are the state variables

related to the integrators in the DSOGI structure, as shown in Figure 3.6. The positive-sequence

component can be solved from Eq. (3.5), yielding

V+ejω0t=1

2(Xαβ1+jω0Xαβ2).(3.6)

xαβ1

xαβ2

DSOGI

PII

Sequ.

Dec.

Tpark

vd+

vq+ θ

vdq+

ωxαβ2

vαβ-

vαβ+

vαβc

xαβ1

ωxαβ2

xαβ1

xαβ2

DSOGI

PII

Sequ.

Dec.

Tpark

vd+

vq+ θ

vdq+

ωxαβ2

vαβ-

vαβ+

vαβc

xαβ1

ωxαβ2

Figure 3.6: Basic scheme of DSOGI-PLL

The complete state-space model of the DSOGI-PLL is

x˙αβ1=−kωxαβ1−ω2xαβ2+kωvαβc

x˙αβ2=xαβ1

η˙ = kivq+=kiIm {vdq+}

˙=ω=η+kpvq+=η+kpIm {vdq+}

(3.7)

along with the algebraic relation

vq+= Im {vdq+}= Im {︃e−jθ ·xαβ1+jωxαβ2

2}︃.

Equation (3.3) and (3.6) describe the basic principle of the DSRF and DSOGI structures by

assuming that the phase angle and frequency used by those two blocks are perfectly synchronized

to those of the positive-sequence component of the input voltage. This assumption is adopted

by all existing LTI models. However, as given in Eq. (3.4) and (3.7), the phase angle and

frequency are actually estimated by the PLL itself. It will be demonstrated in the next chapter

that neglecting the phase angle or frequency feedback coupling in the PSCD structure can

result in wrong stability assessment results.

3. Modeling of Converter-Dominated Power Systems

Current Reference Calculation

The power control of the GFL converter is based on the instantaneous power theory. Define

Sr=Pr+jQr

as the complex power reference, the current reference in the

frame and the

αβ

frame are

determined by

idqr =idr +jiqr =2

3(︄Sr

vdq+)︄∗(3.8)

and

iαβr =iαr +jiβr =2

3(︄Sr

vαβ+)︄∗(3.9)

respectively.

Current Control

Figure 3.7 shows block diagrams of the current controller in the

frame. The complex-valued

state-space equation of the PI current controller is given by

x˙dqC =kiC (idqr −idqf )(3.10)

along with the algebraic relation

vdqr =xdqC +kpC (idqr −idqf ) + jω0Lidqf (3.11)

where

kpC

and

kiC

are PI coefficients, and

xdqC

is the state variable of the integrator.

iαβf

the converter-side filter inductor current. Neglecting the switching process and the time delay

caused by the modulation and sampling, the voltage command

vdqr

(or

vαβr

) can be regarded

as the terminal voltage of the converter.

vαβr

ω0L

idr

iqr

Tpark

park

T−

iαβf

kiC

kiC I

kpC

vαβr

ω0L

idr

iqr

Tpark

park

T−

iαβf

kiC

kiC I

kpC

jω0L

idqr

vαβr

Tpark

park

T−

xdqC

iαβf

idqf vdqr

kpC

kiC

jω0L

idqr

vαβr

Tpark

park

T−

xdqC

iαβf

idqf vdqr

kpC

kiC

vαβr

ω0L

idr

iqr

Tpark

park

T−

iαβf

kiC

kiC I

kpC

jω0L

idqr

vαβr

Tpark

park

T−

xdqC

iαβf

idqf vdqr

kpC

kiC

Figure 3.7: Block diagrams of PI current controller. Left: real domain implementation. Right:

complex domain representation.

αβ

frame, the current reference

iαβr

is a time-periodic signal. To achieve zero steady-

state control error, the frequency-adaptive PR controller can be used for the current control.

To make these two categories of current controllers (PI and PR controllers) comparable,

the 1st-order PR controller shown in Figure 3.8 is adopted in this thesis. The state-space

3.1 Complex-Valued Large-Signal Model

description of the PR controller is given by

x˙αβC =krC (iαβr −iαβf ) + jωxαβC (3.12)

with the algebraic relation

vαβr =xαβC +kpC (iαβr −iαβf )(3.13)

where kpC and krC are PR coefficients, and xαβC is the state variable for the integrator.

jω

xαβC

kpC

iαβrvαβr

krC

krC I

kpC

krC

kpC

ωω

iαr

iβr

iαf

iβf

vαr

vβr

xαC

xβC

iαβf

jω

xαβC

kpC

iαβrvαβr

krC

krC I

kpC

krC

kpC

iαr

iβr

iαf

iβf

vαr

vβr

xαC

xβC

iαβf

Figure 3.8: Block diagrams of PR current controller. Left: real domain implementation. Right:

complex domain representation.

LC(L) Filter and Grid Impedance

Since the grid-side filter inductor shares the same state variable with the grid impedance, it

is treated as part of the grid impedance. The passive LC(L) filter and grid impedance are

assumed to be linear. In the αβ frame, their dynamics can be described by

dtiαβf =−R

Liαβf +1

Lvαβr −1

Lvαβc

dtvαβc =1

Ciαβf −1

Ciαβg

dtiαβg =−Rg

iαβg +1

vαβc−1

vαβg

(3.14)

where

iαβg

is the current flowing through the grid impedance,

vαβg

is the voltage at the grid

connection point, see Figure 3.3. State variables in different reference frames are linked by the

Park respectively inverse Park transformation:

idqf =e−jθiαβf idqg =e−jθiαβg

vdqc =e−jθvαβc vdqg =e−jθvαβg vαβr =ejθvdqr

(3.15)

Inserting the relation given by Eq. (3.15), the state-space model of the LC filter and grid

impedance in the dq frame is obtained

dtidqf =−jωidqf −R

Lidqf +1

Lvdqr −1

Lvdqc

dtvdqc =−jωvdqc +1

Cidqf −1

Cidqg

dtidqg =−jωidqg −Rg

idqg +1

vdqc −1

vdqg

.(3.16)

3. Modeling of Converter-Dominated Power Systems

Nonlinear terms indicated by the red color in Eq. (3.16) result from the transformation between

the αβ and dq frames given by Eq. (3.15).

Steady-State Analysis

Let

Vαβc =Vαβc+ejω0t+Vαβc−e−jω0t

and

Vαβg =Vαβg+ejω0t+Vαβg−e−jω0t

denote steady-state operation trajectories of the unbalanced filter capacitor voltage

vαβc

and

the grid voltage

vαβg

. The steady-state value of the converter-side filter inductor current

iαβf

will also be unbalanced, which can be written as

Iαβf =Iαβf+ejω0t+Iαβf−e−jω0t.

The current controller ensures that the positive-sequence current

Iαβf+

equals the reference

value, namely

Iαβf+=2

3(︄Sr

Vαβc+)︄∗.(3.17)

The signal flow graph of the negative-sequence current is shown in Figure 3.9, the algebraic

relation for PI and PR controller are given by

Iαβf−=1

2jω0L−R−kpC +kiC /(2jω0)Vαβc−(3.18)

and

Iαβf−=1

jω0L−R−kpC +krC/(2jω0)Vαβc−(3.19)

respectively.

jω0L





































j L R





j L R

















Figure 3.9: Signal flow graphs of the negative-sequence current. Top: the PI current controller.

Bottom: the PR current controller.

3.1 Complex-Valued Large-Signal Model

Based on Kirchhoff’s current law, two equations can be derived from the steady-state equivalent

circuit shown in Figure 3.10

fGF L+=Iαβf+−jω0CVαβc+−Vαβc+−Vαβg+

Rg+jω0Lg

= 0 (3.20)

fGF L−=Iαβf−+jω0CVαβc−−Vαβc−−Vαβg−

Rg−jω0Lg

= 0 (3.21)

which are sufficient for solving of the two unknown voltages

Vαβc+

and

Vαβc−

of the grid-

following converter.

Rg-jω0Lg

-1/jω0C

Vαβg-

Vαβc-

Iαβc-(Vαβc-)Rg-jω0Lg

-1/jω0C

Vαβg-

Vαβc-

Iαβc-(Vαβc-)

Vαβg+

Rg+jω0Lg

1/jω0C

Vαβc+



Vαβg+

Rg+jω0Lg

1/jω0C

Vαβc+



Figure 3.10: Steady-state positive (left) and negative (right) sequence equivalent circuits of the

grid-following converter.

3.1.2 Grid-Forming Voltage Source Converters

With increasing penetration of renewable energy sources, grid-connected converters are required

to provide voltage and frequency regulation capability instead of just injecting maximum

available power. To achieve this, the GFM control draws more and more attention. In this

subsection, the state-of-the-art virtual synchronous generator (VSG) algorithm emulating

both the stationary droop and dynamic inertia characteristics of the conventional synchronous

generator is adopted.

Figure 3.11 illustrates a VSG-controlled single converter system. The VSG algorithm

consists mainly of two parts. First, an outer power control loop generates the voltage phase

angle

and magnitude

, as shown in Figure 3.12. Second, an inner cascaded voltage-current

control loop shown in Figure 3.13 is designed to reject high frequency disturbances. Additionally,

a virtual impedance is introduced in the inner control to improve system stability and power

sharing accuracy.

VDC C

PWM Inner V-I

Control

LRg

iαβg

Outer

Control

vαβg

vαβr

iαβf

iαβg

vαβc

iαβg

Figure 3.11: Schematic and control of a single VSG converter system

3. Modeling of Converter-Dominated Power Systems

Outer Power Control

As shown in Figure 3.12, the instantaneous power sV SG obtained from

sV SG =p+jq =3

2vαβci∗

αβg

is passed through a 1st-order low-pass filter

GLP F (s) = ωs

s+ωs

with a cut-off frequency of ωs. The state-space description is given by

˙=−ωsS+ωssV SG (3.22)

where

is the state variable of the LPF. Different from the conventional droop

control, the swing equation is added in the active power loop of the VSG algorithm to provide

inertia support

ω˙ = 1

Jω0

(Re {Sr} − Re {S}+kP(ω0−ω))

Jω0(︃Sr+S∗

2−S+S∗

2+kP(ω0−ω))︃

˙=ω

(3.23)

where

jQr

is the power reference.

is the inertia constant.

denotes the active

power droop coefficient.

The reactive power loop determines the voltage magnitude according to the droop equation

E=E0+Im {Sr} − Im {S}

=E0+

Sr−S∗

2j−S−S∗

(3.24)

with E0the rated voltage magnitude. kQis the reactive power droop coefficient.

Power

Calculation

GLPF(s)

I1/Jω0s

Swing equation

LPF

1/kQ

ωθ

ω0

LPF Q

iαβgvαβc

Figure 3.12: Block diagram of the outer power control loop

Inner Cascaded Voltage-Current Control

As shown in Figure 3.13, the inner control consists of three parts, namely the virtual impedance

Zv(s)and voltage as well as current control implemented with standard PI controllers.

3.1 Complex-Valued Large-Signal Model

jω0C

Kffi

Tpark

idq g

park

T

jω0L

Tpark

idqr

Voltage Control Current Control

Zv(s)

idqg

Vir. Imp.

vαβr

vαβc

vdqc

vdqcr xdq V

iαβf

idqf

xdq C

Kffv

vdqc

vdqr

Figure 3.13: Block diagram of the inner cascaded voltage-current control loop with the virtual

impedance

ωv

Rv+jω0Lv

idqg

xdqvVirtual Impedance

vdqcr

Figure 3.14: Block diagram of the virtual impedance Zv(s)

The virtual impedance

Zv(s) = ωv

s+ωv

sLv+Rv+jω0Lv

shown in Figure 3.14 is described by the state-space equation

x˙dqv =−ωvxdqv +ωvidqg (3.25)

where ωvis the bandwidth of the 1st-order low-pass filter ωv

s+ωvin Zv(s).

The reference value of the filter capacitor voltage is synthesized from the outputs of the

reactive power loop and the virtual impedance unit

vdqcr =E−(Lvωv(idqg −xdqv)+(Rv+jω0Lv)idqg).(3.26)

The state-space equation of the voltage controller is

x˙dqV =kiV (vdqcr −vdqc)(3.27)

together with the algebraic relation for its output

idqr =xdqV +kpV (vdqcr −vdqc) + jω0Cvdqc +Kffiidqg (3.28)

where

kpV

and

kiV

are the proportional and integral coefficients of the voltage PI controller,

and the state variable xdqV is related to the integrator. Kffi is the current feedforward gain.

3. Modeling of Converter-Dominated Power Systems

The PI current controller takes the output of the voltage controller as the reference

command, and the state-space model is

x˙dqC =kiC (idqr −idqf ).(3.29)

Again, the output of the current controller

vdqr =xdqC +kpC (idqr −idqf ) + jω0Lidqf +Kffvvdqc (3.30)

is regarded as the terminal voltage of the converter.

kpC

and

kiC

are the proportional and

integral coefficients of the current controller, and

xdqC

is the state variable of the integrator.

Kffv is the voltage feedforward gain.

The state-space model of the LC filter and grid impedance of the grid-forming converter is

the same as that of the grid-following converter given by Eq. (3.14) and Eq. (3.16). Variables

in different reference frames share the relations described already by Eq. (3.15).

Steady-State Analysis

Let

Vαβc =Vαβc+ejω0t

and

Iαβg =Iαβg+ejω0t

denote steady-state values of the balanced filter capacitor voltage

vαβc

and grid-side current

iαβg. Then, for the fundamental frequency, the virtual impedance defines the relation

Vαβv+=Vαβc++ (Rv+jω0Lv)Iαβg+(3.31)

where Vαβv+is a virtual voltage constrained by the droop equation

fV SG+=Sr−3

2Vαβc+I∗

αβg+−kP(ω−ω0)−1j·kQ(︂√︂Vαβv+V∗

αβv+−E0)︂= 0.(3.32)

Under grid-connected operations, the steady-state value of

is fixed by the stiff grid to

ω0

Equation (3.32) is sufficient for solving the only unknown variable

Vαβc+

. The steady-state

equivalent circuit is shown in Figure 3.15. For islanded operations,

becomes another unknown

variable. Since no stiff grid bus exists, the capacitor voltage of the grid-forming converter

needs to be selected to provide the phase angle reference. If the phase angle of

Vαβc+

is set to

zero, yields

fV SGω =Vαβc+−V∗

αβc+= 0.(3.33)

This additional relation makes the number of equations and unknown variables the same.

3.1 Complex-Valued Large-Signal Model

Vαβg+

Rg+jω0Lg Rv+jω0Lv

Vαβv+

Iαβg+

Droop

Relation

Vαβc+

Figure 3.15: Steady-state positive sequence equivalent circuit of the grid-forming converter

3.1.3 Power Networks and Passive Loads

Power Networks

The power networks are modeled as three-phase RL branches. In the natural

abc

reference

frame, the state-space equation of the ith three-phase branch connected between the mth bus

and the nth bus is given by

Labc,i

dt 





ia,i

ib,i

ic,i





=−Rabc,i 





ia,i

ib,i

ic,i





+





va,m

vb,m

vc,m





−





va,n

vb,n

vc,n





(3.34)

where

[va,m, vb,m, vc,m]T

and

[va,n, vb,n, vc,n]T

denote three-phase nodal voltages at the mth

bus and the nth bus.

[ia,i, ib,i, ic,i]T

is the three-phase branch current. Diagonal matrices

Labc,i= diag ([La,i, Lb,i, Lc,i])

and

Rabc,i= diag ([Ra,i, Rb,i, Rc,i])

give the inductance and

resistance of each phase. The branch can be unbalanced, namely diagonal elements of

Labc,i

and Rabc,i can be different.

In the complex domain, the state-space equation of the branch can be derived from

Eq. (3.34) by using the Clarke transformation and the real-complex-domain transformation

Lmn

dt [︄iαβBranch,i

i∗

αβBranch,i ]︄=−Rmn [︄iαβBranch,i

i∗

αβBranch,i ]︄+[︄vαβ,m

v∗

αβ,m ]︄−[︄vαβ,n

v∗

αβ,n ]︄(3.35)

with

Lmn =Tr2cTClarkeLabc,iT−1

ClarkeTc2r

Rmn =Tr2cTClarkeRabc,iT−1

ClarkeTc2r

[︂iαβBranch,i, i∗

αβBranch,i]︂T=Tr2cTClarke [ia,i, ib,i, ic,i]T

[︂vαβ,m, v∗

αβ,m]︂T=Tr2cTClarke [va,m, vb,m, vc,m]T

[︂vαβ,n, v∗

αβ,n]︂T=Tr2cTClarke [va,n, vb,n, vc,n]T

.(3.36)

Let

IαβBranch,i =IαβBranch+,iejω0t+IαβBranch−,ie−jω0t

Vαβ,m =Vαβ+,mejω0t+Vαβ−,me−jω0t

Vαβ,n =Vαβ+,nejω0t+Vαβ−,ne−jω0t

(3.37)

3. Modeling of Converter-Dominated Power Systems

denote steady-state values of iαβBranch,i,vαβ,m and vαβ,n. Their complex conjugates are

I∗

αβBranch,i =I∗

αβBranch+,ie−jω0t+I∗

αβBranch−,iejω0t

V∗

αβ,m =V∗

αβ+,me−jω0t+V∗

αβ−,mejω0t

V∗

αβ,n =V∗

αβ+,ne−jω0t+V∗

αβ−,nejω0t

.(3.38)

Applying Fourier transform to Eq. (3.35), the red terms in Eq. (3.37) and Eq. (3.38) are linked

by [︄IαβBranch+,i

I∗

αβBranch−,i ]︄=Ymn (︄[︄ Vαβ+,m

V∗

αβ−,m ]︄−[︄Vαβ+,n

V∗

αβ−,n ]︄)︄ (3.39)

where the branch admittance matrix Ymn is given by

Ymn = (jω0Lmn +Rmn)−1=[︄Y++

mn Y+−

Y−+

mn Y−−

mn ]︄(3.40)

Resistive-Inductive Loads

There exist different types of loads in the electric power systems, RL loads are taken as an

example here. Mathematically, it can be treated as an RL branch connected between a bus and

the ground. When the ith load is connected to the mth bus, according to the aforementioned

modeling procedure of RL branches, the complex-domain state-space equation of the load can

be derived

dt [︄iαβLoad,i

i∗

αβLoad,i ]︄=−Ri[︄iαβLoad,i

i∗

αβLoad,i ]︄+[︄vαβ,m

v∗

αβ,m ]︄(3.41)

with

Li=Tr2cTClarkeLLoad,iT−1

ClarkeTc2r

Ri=Tr2cTClarkeRLoad,iT−1

ClarkeTc2r

(3.42)

where

LLoad,i

and

RLoad,i

are real-valued three-phase load inductance and resistance matrices.

Let

IαβLoad,i =IαβLoad+,iejω0t+IαβLoad−,ie−jω0t

denote the steady-state trajectory of the load current

iαβLoad,i

, it can be deduced from Eq. (3.41)

[︄IαβLoad+,i

I∗

αβLoad−,i ]︄=Yi[︄Vαβ+,m

V∗

αβ−,m ]︄(3.43)

where load admittance matrix Yiis given by

Yi= (jω0Li+Ri)−1=[︄Y++

iY+−

Y−+

iY−−

i]︄.(3.44)

3.1.4 Steady-State Power Flow Calculation

For a grid-connected converter-dominated power system with

•one reference bus,

•NLoad RL passive loads,

3.1 Complex-Valued Large-Signal Model

•NBus feeder buses and NGF L grid-following converters,

considering unbalanced operations, 2

NBus

unknown bus voltages and 2

NGF L

unknown filter

capacitor voltages need to be solved, half positive-sequence voltages and half negative-sequence

voltages.

According to Eq. (3.20) and (3.21), 2

NGF L

equations can be formulated for the grid-

following converters. The other 2

NBus

equations can be obtained by applying Kirchhoff’s

current law to each bus. Specifically, taking the mth bus as an example, following equations

can be deduced

fBus+,m =0=











∑︁NBus

i=1 δBranch

mi (︂Y++

mi (Vαβ+,m −Vαβ+,i) + Y+−

mi (︂V∗

αβ−,m −V∗

αβ−,i)︂)︂

+∑︁NGF L

i=1 δGF L

Vαβ+,m−Vαβc+,m

Rg,i+jω0Lg,i

+∑︁NLoad

i=1 δLoad

mi (︂Y++

Load,iVαβ+,m +Y+−

Load,iV∗

αβ−,m)︂(3.45)

fBus−,m =0=











∑︁NBus

i=1 δBranch

mi (︂Y−+

mi (Vαβ+,m −Vαβ+,i) + Y−−

mi (︂V∗

αβ−,m −V∗

αβ−,i)︂)︂

+∑︁NGF L

i=1 δGF L

V∗

αβ−,m−V∗

αβc−,m

Rg,i+jω0Lg,i

+∑︁NLoad

i=1 δL

mi (︂Y−+

Load,iVαβ+,m +Y−−

Load,iV∗

αβ−,m)︂(3.46)

where

Rg,i

and

Lg,i

denote parameters of the grid impedance of the ith grid-following converter.

δBranch

δGF L

and

δLoad

are connectivity indicators of branches, grid-following converters and

loads, respectively. They follow the definition:

•δBranch

equals one when there exists a branch between the mth bus and the ith bus,

otherwise it is zero.

•δGF L

equals one when the mth grid-following converter is connected to the mth bus,

otherwise it is zero.

•δLoad

mi equals one when the mth load is connected to the mth bus, otherwise it is zero.

For clarity, Eq. (3.45) is taken as an example to explain the physical meaning behind the

power flow equation:

•

The first term summarizes all positive-sequence currents flowing from the mth bus to

other buses.

•

The second term summarizes all positive-sequence currents flowing from the mth bus to

all grid-following converters.

•

The third term summarizes all positive-sequence currents flowing from the mth bus to

all loads.

According to Kirchhoff’s current law, the algebraic sum of all those currents flowing away from

the mth bus should be zero.

3. Modeling of Converter-Dominated Power Systems

Combine Eq. (3.20), (3.21), (3.45) and (3.46), the steady-state power flow problem can be

written in a compact form











fBus+,1, fBus−,1,· · · , fBus+,NBus , fBus−,NBus

⏞⏟⏟ ⏞

2NBus











fGF L+,1, fGF L−,1,· · · , fGF L+,NGF L , fGF L−,NGF L

⏞⏟⏟ ⏞

2NGF L













(3.47)

with the unknown voltage vector











Vαβ+,1, Vαβ−,1,· · · , Vαβ+,NBus , Vαβ−,NBus

⏞⏟⏟ ⏞

2NBus











Vαβc+,1, Vαβc−,1,· · · , Vαβc+,NGF L , Vαβc−,NGF L

⏞⏟⏟ ⏞

2NGF L













.(3.48)

Due to the appearance of the non-holomorphic complex conjugate operator and the

real/imaginary part extraction operator, the classical complex derivative

∂f/∂V

does not

exist, thus, the Newton-Raphson method cannot be directly utilized to solve the nonlinear

equations defined by Eq. (3.47) and (3.48). The Wirtinger calculus is used to overcome this

issue by extending Eq. (3.47) and (3.48) to

[︄f

f∗]︄4(NBus+NGF L)×1[︄V

V∗]︄4(NBus+NGF L)×1

(3.49)

which can now be solved with the Newton-Raphson method by using the Jacobian matrix

JP F defined within the Wirtinger calculus framework

JP F =[︄∂f

∂V

∂f

∂V∗

∂f∗

∂V

∂f∗

∂V∗]︄4(NBus+NGF L)×4(NBus+NGF L)

.(3.50)

It should be clarified that, by separating the real and imaginary parts of Eq. (3.47) and

(3.48), i.e.,

[︄Re {f}

Im {f}]︄4(NBus+NGF L)×1[︄Re {V}

Im {V}]︄4(NBus+NGF L)×1

,(3.51)

unknown bus voltages can also be solved with the classical real-valued Jacobian matrix.

However, the compactness of the complex variables is lost. Besides, compared to Eq. (3.51),

the number of independent variables and consequently of derivatives in Eq. (3.49) and (3.50)

is halved, requiring less computation time and exhibiting better convergence [110].

3.2 Small-Signal Model

Under balanced conditions, only positive-sequence voltages need to be solved. Moreover,

considering the existence of grid-forming converters, Eq. (3.45) needs to be modified as

fBus+,m =0=











∑︁NBus

i=1 δBranch

mi Y++

mi (Vαβ+,m −Vαβ+,i)

+∑︁NGF L

i=1 δGF L

Vαβ+,m−VGF L

αβc+,m

Rg,i+jω0Lg,i

+∑︁NLoad

i=1 δLoad

mi Y++

Load,iVαβ+,m

+∑︁NV SG

i=1 δV SG

Vαβ+,m−VV SG

αβc+,i

Rg,i+jω0Lg,i

(3.52)

where

NV SG

is the number of the grid-forming converters. The connectivity indicator

δV SG

equals one when the ith grid-forming converter is connected to the mth bus. The superscript

GFL

and

V SG

is added to the voltage variables to distinguish between the filter capacitor

voltages of the GFL converters and VSG converters.

Then, the power flow equations and the unknown voltage vector become











fBus+,1· · · , fBus+,NBus

⏞⏟⏟ ⏞

NBus











fGF L+,1,· · · , fGF L+,NGF L

⏞⏟⏟ ⏞

NGF L











fV SG+,1,· · · , fV SG+,NGF L

⏞⏟⏟ ⏞

NV SG













(3.53)

and











Vαβ+,1,· · · , Vαβ+,NBus

⏞⏟⏟ ⏞

NBus











VGF L

αβc+,1,· · · , V GF L

αβc+,NGF L

⏞⏟⏟ ⏞

NGF L











VV SG

αβc+,1,· · · , V V SG

αβc+,NV SG

⏞⏟⏟ ⏞

NV SG













.(3.54)

When the system is operated in islanded mode, one extra unknown variable, the steady-state

system frequency

, should be added into Eq. (3.54). Accordingly, Eq. (3.33) needs to be

included into Eq. (3.53).

3.2 Small-Signal Model

3.2.1 Small-Signal Model of Grid-Following Converters

For the implementation of the GFL converter, there exist different possibilities to combine the

PLL and the current controller. Among them, the DSRF-PLL plus the PI current controller

3. Modeling of Converter-Dominated Power Systems

and the DSOGI-PLL plus the PR current controller are two straightforward and widely adopted

variants, considering that the control is either implemented in the

frame or the

αβ

frame.

In this thesis, these two variants are defined as Type I and Type II GFL converter, respectively.

Without loss of generality, the Type I GFL converter is taken as an example to explain the

derivation of the small-signal model of the GFL converter.

DSRF-PLL

Since the imaginary part extraction operator

Im {·}

is non-holomorphic, the nonlinear state-

space model of the DSRF-PLL given by Eq. (3.4) must be linearized within Wirtinger calculus

framework, yielding

∆v˙dq+=ωf(︂−∆vdq+−e−j2θ0∆vdq−+e−jθ0∆vαβc +j(︂2e−j2θ0Vdq−−e−jθ0Vαβc)︂∆θ)︂

∆v˙dq−=ωf(︂−∆vdq−−e2jθ0∆vdq++ejθ0∆vαβc +j(︂−2ej2θ0Vdq++ejθ0Vαβc)︂∆θ)︂

∆v˙∗

dq+=ωf(︂−∆v∗

dq+−ej2θ0∆v∗

dq−+ejθ0∆v∗

αβc −j(︂2ej2θ0V∗

dq−−ejθ0V∗

αβc)︂∆θ)︂

∆v˙∗

dq−=ωf(︂−∆v∗

dq−−e−2jθ0∆v∗

dq++e−jθ0∆v∗

αβc −j(︂−2e−j2θ0V∗

dq++e−jθ0V∗

αβc)︂∆θ)︂

∆η˙ = kiIm {∆vdq+}=−1

2jki(︂∆vdq+−∆v∗

dq+)︂

∆θ

˙= ∆η+kpIm {∆vdq+}= ∆η−1

2jkp(︂∆vdq+−∆v∗

dq+)︂(3.55)

where

θ0

is the steady-state trajectory of

, and the steady-state trajectories of other state

variables are given by the symbols starting with uppercase letters (e.g.,

Vdq+

and

Vdq−

). It

is noted that the 3rd and 4th equations of Eq. (3.55) are the complex conjugate version of

the first two. As given in the 5th and 6th equations, the imaginary extraction operator

Im {·}

causes the coupling between variables in the original and conjugate coordinates.

The simplified small-signal model shown in Figure 3.16 is commonly used for the parameter

design of advanced PLLs [12]. Dynamics of the DSRF and DSOGI structures are approximated

by a first-order low-pass filter and a notch filter. The notch filter to remove the disturbance at

ω0

(caused by the Park transformation of the negative-sequence voltage) is ignored during

the parameter design. Then, the open-loop transfer function of the simplified model becomes

a type-2 control system given by

Gol (s) = V+·ωf

s+ωf

·(︃kp+ki

s)︃·1

s=V+kpωf(s+ωz)

s2(s+ωf)(3.56)

where

ωz

ki/kp

. Based on the extended symmetrical optimum method [45, 111], only the

designed bandwidth ωcneeds to be appropriately selected











ωf=gωc

kp=ωc/V+

ki=kpωc/g

3.2 Small-Signal Model

where the constant

is solved from Eq. (3.57) for a desired phase margin PM. Usually, a PM

within the range of 30°to 60°is selected [112], which corresponds to 1.732 ≤g≤3.732.

PM = tan−1g2−1

2g(3.57)

V+

θ0Notch filter

2ω0 LPF ω

PI I

(a) The DSRF and DSOGI structures are modeled as a notch filter

and an LPF

V+

θ0

LPF ω

PI I

(b) Neglecting the notch filter for parameter design

Figure 3.16: Simplified small-signal model of the advanced PLLs.

PI Current Control

Within the Wirtinger calculus framework, the linearization of the current reference calculation

unit Eq. (3.8) is given by

∆idqr =−2S∗

3(︂V∗

dq+)︂2∆v∗

dq++2

3V∗

dq+

∆S∗

∆i∗

dqr =−2Sr

3(Vdq+)2∆vdq++2

3Vdq+

∆Sr

.(3.58)

In the

frame, the PI current controller described by Eq. (3.10) is inherently linear, the

small-signal state-space equation can be directly obtained

∆x˙dqC =kiC (∆idqr −∆idqf )(3.59)

along with the algebraic equation

∆vdqr = ∆xdqC +kpC (∆idqr −∆idqf ) + jω0L∆idqf .(3.60)

3. Modeling of Converter-Dominated Power Systems

LC Filter and Grid Impedance

The state-space model of the LC filter and grid impedance in the

αβ

frame given by Eq. (3.14)

is linear, of which the small-signal model is

dt∆iαβf =−R

L∆iαβf +1

L∆vαβr −1

L∆vαβc

dt∆vαβc =1

C∆iαβf −1

C∆iαβg

dt∆iαβg =−Rg

∆iαβg +1

∆vαβc−1

∆vαβg

.(3.61)

The linearization of the frame transformation Eq. (3.15) is given by

∆idqf =e−jθ0∆iαβf −1je−jθ0Iαβf ∆θ

∆vdqc =e−jθ0∆vαβc −1je−jθ0Vαβc∆θ

∆vdqg =e−jθ0∆vαβg −1je−jθ0Vαβg∆θ

∆vαβr =ejθ0∆vdqr + 1jejθ0Vdqr∆θ

.(3.62)

Neglecting the synchronization dynamics of the PLL, in other words, the phase angle of

the filter capacitor voltage is assumed to be perfectly known, the block diagram shown in

Figure 3.17 can be used to describe the PI current controller and its plant (i.e., the converter-

side filter inductor). Based on the pole-zero cancellation principle [113, 114], the closed-loop

transfer function of the current control can be simplified as a standard PT1 element

GclC (s) = 1

1 + τs (3.63)

by choosing the control parameters

kpC =L

τkiC =R

where τis defined as the time constant of the current control.

Δidqr

kpCs+kiC

sL+R

Δidqf

jω0L

Δidqr

Plant

ΔxdqC

Δvdqr

kpC

kiC

kiC 1/(sL+R)

jω0L

Δvdqc

Δidqr

kpCs+kiC

sL+R

Δidqf

jω0L

Δidqr

Plant

ΔxdqC

Δvdqr

kpC

kiC 1/(sL+R)

jω0L

Δvdqc

Figure 3.17: Simplified block diagram of the current control loop

Complete Single Grid-Following Converter Model

Combine Eq. (3.55), (3.59) and (3.61), the complete small-signal model of the ith grid-following

converter in a power system with NBus buses can be formulated as

∆x˙GF L,i =AGF L,i∆xGF L,i +BGF LS,i∆SGF L,i +BGF LV,i∆vBus

∆yGF L,i =CGF L,i∆xGF L,i

(3.64)

3.2 Small-Signal Model

with a 14-dimensional state vector

∆xGF L,i =[︄∆vdq+,i,∆v∗

dq+,i,∆vdq−,i,∆v∗

dq−,i,∆ηi,∆θi,∆xdqC,i,∆x∗

dqC,i,

∆iαβf,i,∆i∗

αβf,i,∆vαβc,i,∆v∗

αβc,i,∆iαβg,i,∆i∗

αβg,i ]︄T

.(3.65)

Disturbances of the power reference command and the bus voltage are defined as input

variables

∆SGF L,i =[︂∆Sr,i,∆S∗

r,i]︂T

∆vBus =[︂∆vBus,1,∆v∗

Bus,1,· · · ,∆vBus,m,∆v∗

Bus,m,· · · ,∆vBus,NBus ,∆v∗

Bus,NBus ]︂T

(3.66)

associated with the input matrices

BGF LS,i =







06×2





02kiC

3V∗

dq+

2kiC

3Vdq+0



2×2

06×2







14×2

(3.67)

BGF LV,i =





014×2

⏞⏟⏟⏞

Bus 1

· · · 





012×2





−1

Lg,i 0

0−1

Lg,i









⏞⏟⏟ ⏞

Bus m

· · · 014×2

⏞⏟⏟⏞

Bus NBus







14×2NBus

(3.68)

where ∆

vBus,m

is the voltage at the mth bus, to which the ith grid-following converter is

connected.

The current injected into the grid is selected as the output signal, ∆

yGF L,i

[︂∆iαβg,∆i∗

αβg]︂T, by defining the output matrix

∆CGF L,i =

02×12 [︄1 0

0 1 ]︄2×2

(3.69)

Combination of Multiple Grid-Following Converters

Let

NGF L

denote the total number of grid-following converters, the full state-space model of

all grid-following converters can be summaried as

∆x˙GF L =AGF L∆xGF L +BGF LS ∆SGF L +BGF LV ∆vBus

∆yGF L =CGF L∆xGF L

(3.70)

with

∆xGF L =





∆xGF L,1

∆xGF L,2

∆xGF L,NGF L







AGF L =





AGF L,1

AGF L,2

...

AGF L,NGF L







3. Modeling of Converter-Dominated Power Systems

∆SGF L =





∆SGF L,1

∆SGF L,2

∆SGF L,NGF L







BGF LS =





BGF LS,1

BGF LS,2

...

BGF LS,NGF L







BGF LV =





BGF LV,1

BGF LV,2

BGF LV,NGF L







∆yGF L =





∆yGF L,1

∆yGF L,2

∆yGF L,NGF L







CGF L =





CGF L,1

CGF L,2

...

CGF L,NGF L







The small-signal model of the Type II grid-following converter can be derived following

the same procedure.

3.2.2 Small-Signal Model of Grid-Forming Converters

In the GFL converters, the PLLs, which aim to estimate the voltage phase angle by bringing

to zero, are the asymmetric control units causing couplings between original and conjugate

coordinates. An asymmetric control unit can also be found in the outer control loop of the

GFM converters, where active and reactive power are controlled separately. Therefore, the

Wirtinger calculus is also needed for the complex-domain modeling of GFM converters.

Outer Power Control

The instantaneous power calculation unit in Figure 3.12 is the only nonlinear part in the outer

control. Based on the Wirtinger calculus, the linear approximation of the instantaneous power

is given by

∆sV SG =3

2(︂Vαβc∆i∗

αβg +I∗

αβg∆vαβc)︂(3.71)

Since the rests of the outer power loop are linear, the small-signal model can be directly

obtained by replacing large-signal variables with corresponding small-signal ones

∆S

˙=−ωs∆S+ωs∆sV SG (3.72a)

∆ω˙ = 1

Jω0

(Re {∆Sr} − Re {∆S} − kP∆ω)(3.72b)

∆θ

˙= ∆ω(3.72c)

∆E=Im {∆Sr} − Im {∆S}

(3.72d)

3.2 Small-Signal Model

where Eq. (3.72a) describes the dynamic of the low-pass filter. Eq. (3.72b) and Eq. (3.72c) are

for the active power control loop. Eq. (3.72d) is related to the reactive power control loop.

The droop coefficients are selected according to the grid codes. For instance, according to

EN 50438, it is required that the change of 100% active power corresponds to the change of

2% grid frequency, while the change of 100% reactive power corresponds to the change of 2%

nominal voltage. Let

be the desired inertia time constant, the moment of inertia

can be

obtained from

J=2H|Sr|

ω2

The typical value of Hof the classical synchronous generator is between 2 and 12 seconds.

Inner Cascaded Voltage-Current Control

The inner control loop described with variables in the

frame is linear, the small-signal model

is summarized as

∆x˙dqv =−ωv∆xdqv +ωv∆idqg

∆x˙dqV =kiV (∆vdqcr −∆vdqc)

∆x˙dqC =kiC (∆idqr −∆idqf )

(3.73)

with the algebraic relation

∆vdqcr = ∆E−(Lvωv(∆idqg −∆xdqv)+(Rv+jω0Lv) ∆idqg)

∆idqr = ∆xdqV +kpV (∆vdqcr −∆vdqc) + jω0C∆vdqc +Kffi∆idqg

∆vdqr = ∆xdqC +kpC (∆idqr −∆idqf ) + jω0L∆idqf +Kffv∆vdqc

.(3.74)

The small-signal model of the LC filter and grid impedance is the same as that of the grid-

following converter given by Eq. (3.63). The linear approximation of the frame transformation

is also the same as Eq. (3.56).

The PI coefficients of the current control are designed by using the pole-zero cancellation

technique given by Eq. (3.63). Neglecting the dynamics of reference frame transformation,

the PI voltage controller and its plant can be described with the block diagram shown in

Figure 3.18. The open-loop transfer function

GolV (s) = kpV 1

τ(︂s+kiV

kpV )︂

s2C(︂s+1

τ)︂

has the same form as Eq. (3.56). Similarly, the extended symmetrical optimum method can be

used for the selection of the PI coefficients of the voltage controller.

Δidqf

Δvdqcr

Plant

ΔxdqV Δidqr

kpV

kiV

kiV 1/sC

Δidqg

jω0C

1/(1+sτ)

jω0C

Δvdqc

Δidqf

Δvdqcr

Plant

ΔxdqV Δidqr

kpV

kiV 1/sC

Δidqg

jω0C

1/(1+sτ)

jω0C

Δvdqc

Figure 3.18: Simplified block diagram of the voltage control loop

3. Modeling of Converter-Dominated Power Systems

Complete Single Grid-Forming Converter Model

The complete small-signal model of the ith grid-forming converter is obtained by combining

Eq. (3.72), (3.73) and (3.61)

∆x˙V SG,i =AV SG,i∆xV SG,i +BV SGS,i∆SV SG,i +BV SGV,i∆vBus

∆yV SG,i =CV SG,i∆xV SG,i

(3.75)

where the 16-dimensional state variable vector is

∆xV SG,i =[︄∆Si,∆S∗

i,∆ωi,∆θi,∆xdqv,i,∆x∗

dqv,i,∆xdqV,i,∆x∗

dqV,i,

∆xdqC,i,∆x∗

dqC,i,∆iαβf,i,∆i∗

αβf,i,∆vαβc,i,∆v∗

αβc,i,∆iαβg,i,∆i∗

αβg,i ]︄T

(3.76)

Disturbances of the power reference command ∆

SV SG,i

[︂∆Sr,i,∆S∗

r,i]︂T

and the bus

voltage ∆

vBus

are defined as input variables. The current injected into the grid is selected

as the output signal ∆

yV SG,i

[︂∆iαβg,i,∆i∗

αβg,i]︂T

. Corresponding state-space matrices are

given by

BV SGS,i =







02×2

[︂1

2Jω0

2Jω0]︂1×2

03×2





kiV

2j·kQ

kiV

−2j·kQ

kiV

−2j·kQ

kiV

2j·kQ





2×2

08×2







16×2

BV SGV,i =





016×2

⏞⏟⏟⏞

Bus 1

· · · 





014×2





−1

Lg,i 0

0−1

Lg,i









⏞⏟⏟ ⏞

Bus m

· · · 016×2

⏞⏟⏟⏞

Bus NBus







16×2NBus

∆CV SG,i =

02×14 [︄1 0

0 1 ]︄2×2



2×16

It is assumed that the ith VSG converter is connected to the mth bus.

Combination of Multiple Grid-Forming Converters

Let

NV SG

denote the total number of VSG converters, the full state-space model of all VSG

converters is summarized as

∆x˙V SG =AV SG∆xV SG +BV SGS∆SV SG +BV SGV ∆vBus

∆yV SG =CV SG∆xV SG

(3.77)

3.2 Small-Signal Model

with

∆xV SG =





∆xV SG,1

∆xV SG,2

∆xV SG,NV SG







AV SG =





AV SG,1

AV SG,2

...

AV SG,NV SG







∆SV SG =





∆SV SG,1

∆SV SG,2

∆SV SG,NV SG







BV SGS =





BV SGS,1

BV SGS,2

...

BV SGS,NV SG







BV SGV =





BV SGV,1

BV SGV,2

BV SGV,NV SG







∆yV SG =





∆yV SG,1

∆yV SG,2

∆yV SG,NV SG







CV SG =





CV SG,1

CV SG,2

...

CV SG,NV SG







3.2.3 Small-Signal Model of Power Networks and Loads

In the

αβ

frame, the small-signal model of the ith linear RL branch and the ith load described

by Eq. (3.35) and Eq. (3.41) can be written in a compact form

∆x˙Branch,i =ABranch,i∆xBranch,i +BBranch,i∆vBus

∆x˙Load,i =ALoad,i∆xLoad,i +BLoad,i∆vBus

(3.78)

with

ABranch,i =−L−1

mnRmn

BBranch,i =



02×2

⏞⏟⏟⏞

Bus 1

,02×2

⏞⏟⏟⏞

Bus 2

,· · · ,L−1

⏞⏟⏟⏞

Bus m

,· · · ,02×2,· · · ,−L−1

⏞ ⏟⏟ ⏞

Bus n

,· · · ,02×2

⏞⏟⏟⏞

Bus NBus







2×2NBus

ALoad,i =−L−1

iRi

BLoad,i =



02×2

⏞⏟⏟⏞

Bus 1

,· · · ,L−1

⏞⏟⏟⏞

Bus m

,· · · ,02×2

⏞⏟⏟⏞

Bus NBus







2×2NBus

where

∆xBranch,i =[︂∆iαβBranch,i,∆i∗

αβBranch,i]︂T

3. Modeling of Converter-Dominated Power Systems

denotes the current of the ith branch flowing from the mth bus to the nth bus.

∆xLoad,i =[︂∆iαβLoad,i,∆i∗

αβLoad,i]︂T

is the current of the ith load connected to the mth bus.

Define NBranch and NLoad as the total number of branches and loads, state-space models

of all branches and loads can be written in the compact form

∆x˙Branch =ABranch∆xBranch +BBranch∆vBus

∆x˙Load =ALoad∆xLoad +BLoad∆vBus

(3.79)

with state-space vectors and matrices given by

∆xBranch =





∆xBranch,1

∆xBranch,2

∆xBranch,NBranch







BBranch =





BBranch,1

BBranch,2

BBranch,NBranch







ABranch =





ABranch,1

ABranch,2

...

ABranch,NBranch







∆xLoad =





∆xLoad,1

∆xLoad,2

∆xLoad,NLoad







BLoad =





BLoad,1

BLoad,2

BLoad,NLoad







ALoad =





ALoad,1

ALoad,2

...

ALoad,NLoad







3.2.4

Complete Small-Signal Model of the Converter-Dominated Power

System

As shown in Figure 3.19, the converter-dominated power system can be generally decomposed

into four parts, grid-following converters, grid-forming converters, power networks and loads.

The state-space model of each part is given by Eq. (3.70), (3.75) and (3.79). It is observed that

the bus voltages ∆

vBus

are used as common inputs to each individual part. Since the power

networks are modeled as series connected RL branches instead of transmission lines described

with the

-model, the bus voltages do not represent state variables. To ensure the bus voltages

are well-defined, a virtual resistance

is assumed to be connected between each bus and the

3.2 Small-Signal Model

ground. According to Kirchhoff’s current law and Ohm’s law, the voltage of the mth bus is

[︄∆vBus,m

∆v∗

Bus,m ]︄=rV







MGF L (2m−1:2m, :) ∆yGF L

+MV SG (2m−1:2m, :) ∆yV SG

+MLoad (2m−1 : 2m, :) ∆xLoad

+MBranch (2m−1:2m, :) ∆xBranch







(3.80)

where

MGF L

((2

m−

1) : 2

:) gives the (2m-1)th and the 2mth rows of the connectivity

matrix of the grid-following converter, which follows the definition that

MGF L (2m−1 : m, 2i−1 : 2i) = [︄1 0

0 1 ]︄

when the ith grid-following converter is connected to the mth bus, otherwise the submatrix is

02×2. The same goes for the connectivity matrix of the grid-forming converter MV SG.

Network

vBus,1

vBus,2

vBus,m

vBus,m+1

vBus,m+2

vBus,NBus

ΔyGFL,1 ΔyGFL,2

ΔyGFL,i

ΔyVSG,i

ΔyLoad,1

ΔyLoad,i

Grid-Following Converter

Grid-Forming Converter

Load

Forming Converter

Network

vBus,1

vBus,2

vBus,m

vBus,m+1

vBus,m+2

vBus,NBus

ΔyGFL,1 ΔyGFL,2

ΔyGFL,i

ΔyVSG,i

ΔyLoad,1

ΔyLoad,i

Grid-Following Converter

Grid-Forming Converter

Load

Figure 3.19: Complete small-signal modeling of the converter-dominated power systems

The load connectivity matrix MLoad maps the load currents to each bus by defining

MLoad (2m−1 : m, 2i−1:2i) = [︄−1 0

0−1]︄

when the ith load is connected to the mth bus.

The branch connectivity matrix

MBranch

maps the current of the ith branch (flowing from

the mth bus to the nth bus) to each bus by defining

MBranch (2m−1 : m, 2i−1 : 2i) = [︄−1 0

0−1]︄

MBranch (2n−1 : n, 2i−1 : 2i) = [︄1 0

0 1 ]︄

Further, the bus voltage vector can be obtained

∆vBus =rV(︄MGF LCGF L∆xGF L +MV SGCV SG∆xV SG

+MLoad∆xLoad +MBranch∆xBranch )︄(3.81)

3. Modeling of Converter-Dominated Power Systems

Insert Eq. (3.81) into (3.70), (3.75) and (3.79), the complete small-signal state-space model

of a converter-dominated power system can be obtained by combining them together

∆x˙=A∆x+B∆S(3.82)

where

∆x=





∆xGF L

∆xV SG

∆xBranch

∆xLoad







∆S=[︄∆SGF L

∆SV SG ]︄B=





BGF L 0

0BV SG

0 0







A=





AGF L +BGF LV A1BGF LV A2BGF LV A3BGF LV A4

BV SGV A1AVSG +A2BV SGV A3BV SGV A4

BBranchA1BBranchA2ABranch +BBranchA3BBranchA4

BLoadA1BLoadA2BLoadA3ALoad +BLoadA4







with

A1=rVMGF LCGF L A2=rVMV SGCV SG A3=rVMLoad A4=rVMBranch

In Eq. (3.82), the state vector ∆

consists of the state variables of all grid-following and

grid-forming converters, the currents of all RL branches and the currents of all loads. The

input vector ∆

contains the power reference commands of all converters. The state space

matrices

and

in Eq. (3.82) are time periodic matrices with the fundamental frequency

ω0

. The stability and analytical solution of such LTP system will be investigated in the next

chapter.

3.3

General Description of the Simulation Platform and

Experimental Test Setup

Figure 3.20 shows the software framework for the numerical modeling and analytical stability

evaluation of converter-dominated power systems. This framework consists mainly of following

procedures:

The software Microsoft Excel is used for the management of input data, which specifies

the power system topology, control and physical parameters of the networks, loads and

different types of converters. A MATLAB script is generated from the Excel file.

A MATLAB/Simulink model is automatically built following the settings specified in

the MATLAB script. Meanwhile, the power flow analysis is performed to initialize the

state variables of the MATLAB/Simulink model, including currents of inductors, voltage

of capacitors and integrators in controllers. This enables that the numerical Simulink

model can be started from steady-state operation trajectories.

The linear time-periodic small-signal model is established by calling the jacobian function

provided by MATLAB Symbolic Math Toolbox. The LTP modal analysis (Chapter 4)

and resonance modal analysis (Chapter 5) are carried out for the assessment of the

system dynamic performance.

3.3 General Description of the Simulation Platform and Experimental Test Setup

The numerical MATLAB/Simulink model serves for the time-domain validation of the

small-signal analysis results. To further confirm the analytical and simulation results, a power-

hardware-in-the-loop (PHIL) test setup [115] shown in Figure 3.21 is used, which consists of

three two-level IGBT converters, a Cinergia grid emulator, a dSPACE MicroLabBox and an

OP5600 real-time simulator. The two-level IGBT converter is a scaled version of a low-voltage

high-power VSC product with a rated power of

500 kVA

and a rated peak phase voltage of

255 V

. The switching frequency is

3.2 kHz

. The laboratory prototype is down scaled to a

power rating of

2 kVA

and a rated voltage of

100 V

. Other parameters are accordingly scaled

to keep per unit values the same. The control of the converter is implemented in the dSPACE

MicroLabBox, which serves also as the recorder of all measurements. The grid emulator can

be operated as a power amplifier controlled by the OP5600 real-time simulator.

Specify control and physical parameters

Power Flow Analysis Automized Simulink

(Average) Model Generation

LTP state-space model

LTP eigenvalue-based

Analysis (modal space)

→ Stability

→ Damping

→ Sensitivity

→ Participation Factor

Linearization

Time-domain Simulation

(physical space)

Numerical

Simulink Library

IEEE 13-Bus System

Excel Input

Specify control and physical parameters

Power Flow Analysis Automized Simulink

(Average) Model Generation

LTP state-space model

LTP eigenvalue-based

Analysis (modal space)

→ Stability

→ Damping

→ Sensitivity

→ Participation Factor

Linearization

Time-domain Simulation

(physical space)

Numerical

Simulink Library

IEEE 13-Bus System

Excel Input

Figure 3.20: General framework for the modeling and stability analysis of converter-dominated

power systems.

vcvg

Host PC

DC Source

2-level Converter

with LC filter Line impedance Grid Emulator

Converter

Controller

dSPACE

RT Simulator

OPAL-RT

Measur.

Control

Signals

vcvg

Host PC

DC Source

2-level Converter

with LC filter Line impedance Grid Emulator

Converter

Controller

dSPACE

RT Simulator

OPAL-RT

Measur.

Control

Signals

OPAL-RT & Grid Emulator

Two-level Converters

Passive Loads

Host PC

vcvg

Host PC

DC Source

2-level Converter

with LC filter Line impedance Grid Emulator

Converter

Controller

dSPACE

RT Simulator

OPAL-RT

Measur.

Control

Signals

OPAL-RT & Grid Emulator

Two-level Converters

Passive Loads

Host PC

Figure 3.21: Configuration of the laboratory power-hardware-in-the-loop test setup.

3. Modeling of Converter-Dominated Power Systems

3.4 Case Study

3.4.1 Single-Converter System

The proposed power flow analysis and small-signal modeling method is first tested with

single-converter scenarios. Following three cases are considered:

•Case 1: single GFL converter connected to a balanced grid;

•

Case 2: single GFL converter connected to an unbalanced grid with

0.8 pu

negative-

sequence voltage;

•Case 3: single GFM converter connected to a balanced grid.

Physical parameters of the analytical model and MATLAB/Simulink model are adapted to

those of the physical test bench, given in Table 3.1.

Table 3.1: Physical Parameters of the Two-Level IGBT Converter System

Symbol Description Value

L/R converter-side filter inductance 5.6 mH /0.1 Ω

Cfilter capacitor 16 µF

Lg/Rggrid impedance 17.4 mH /0.53 Ω

VDC voltage of the DC source 300 V

fsw switching frequency 3200 Hz

Vαβg+positive-sequence grid voltage (peak value) 100 V

Vαβg−negative-sequence grid voltage (peak value) for Case 2 80 V

For each case, power flow analysis is performed to compute the steady-state value of the

filter capacitor voltage

vαβc

, the results are listed in Table 3.2. The voltage magnitude and

phase angle obtained from the power flow analysis are exactly the same as those obtained

from the numerical simulation in MATLAB/Simulink. This confirms the accuracy of the

proposed steady-state model and the power flow analysis method. For Case 1, both types of

GFL converters share the same steady-state capacitor voltage, while the negative-sequence

component becomes different as the grid becomes unbalanced in Case 2. In Case 3, the use of

the nonzero virtual impedance Zvchanges the steady-state capacitor voltage.

To verify the accuracy of the small-signal models, time-domain free and forced responses

obtained from the analytical small-signal model, the nonlinear average model in MAT-

LAB/Simulink and experimental tests will be compared. In this thesis, if not specified

otherwise, experimental results are plotted as solid gray lines. Dashed lines and solid lines

in other colors are for the analytical small-signal model and the nonlinear average Simulink

model.

Table 3.2: Power Flow Analysis Results for Single-Converter Cases

Case 1

Case 2 Case 3

Type I GFL Type II GFL Zv= 0 Zv= 0

Vαβc+Vαβc+Vαβc−Vαβc+Vαβc−Vαβc+Vαβc+

Magnitude (V) 96.24 96.24 65.51 96.24 68.11 97.96 99.62

Angle (rad) 0.4688 0.4688 0.3695 0.4688 0.4159 0.3795 0.3715

3.4 Case Study

Grid-Following Converter

The power reference of the GFL converters is set to

1.2 kW

. The bandwidth of the PLLs is

designed to be 20 Hz. The current controller time constant is chosen to be 0.5 ms.

For Case 1, a disturbance of

2 Hz

is first given to the state variable

1.0 s

to excite

the free response. The dynamic evolution of the estimated frequency of the DSRF-PLL

the filter capacitor voltage and the grid-side current are plotted in Figure 3.22. It should be

clarified that a rotational transformation

e−jω0t

is applied to the voltage and current variables

to remove the fundamental oscillation at

ω0

for a clear visualization of the dynamic responses.

As will be illustrated in the next chapter, such rotational transformation with fixed frequency

is linear, which does not change the system stability characteristic. Without loss of generality

only the real part of the complex-valued voltage and current variables are plotted. It can be

observed that all state variables return to their original steady state trajectory within about

0.2 s

after the disturbance. The agreement between results obtained from the small-signal

model, the average model and the measurement confirms the correctness of the proposed

small-signal modeling method.

In addition, to excite the forced response, a

100 W

power reference step is given to the

converter at

1.0 s

, the waveforms of the three aforementioned state variables are shown in

Figure 3.23. The slight deviation between the small-signal model and the nonlinear average

model is caused by the changing of the steady-state operation trajectory.

Figure 3.22: Free responses of the Type I GFL converter

2 Hz

frequency disturbance at

1.0 s

Figure 3.23: Forced responses of the Type I GFL converter to

100 W

power reference step at

t= 1.0 s

The same free and forced response tests have also been carried out on the Type II GFL

converter, and the results are plotted in Figure 3.24 and Figure 3.25. Comparing the dynamic

responses shown in Figure 3.23 and Figure 3.25 (or Figure 3.22 and Figure 3.24), it is seen that

the Type II GFL converter exhibits a better damping performance. This difference cannot be

3. Modeling of Converter-Dominated Power Systems

explained by the existing LTI models in literature, according to which the two implementations

of grid-following converters should be equivalent.

Figure 3.24: Free responses of the Type II GFL converter to

2 Hz

frequency disturbance at

t= 1.0 s

Figure 3.25: Forced responses of the Type II GFL converter to

100 W

power reference step at

t= 1.0 s

To test the modeling accuracy under unbalanced conditions, the grid voltage is set to

contain a negative-sequence component with a magnitude of

0.8 pu

in Case 2. The designed

PLL bandwidth is increased to

30 Hz

. Free responses to the

2 Hz

disturbances are shown

in Figure 3.26. It can be observed that more oscillation components appear in the dynamic

responses of the two types of GFL converters. As the designed PLL bandwidth increases,

the Type II GFL converter is more robust against the appearance of the negative-sequence

voltage. Moreover, sustained oscillations can be observed in the filter-capacitor voltage and

grid-side current in Figure 3.26, because the system is under unbalanced operation. The

steady-state negative-sequence components are transformed into 100-Hz oscillations by the

rotational transformation e−jω0t.

Grid-Forming Converter

Case 3 is aimed to verify the accuracy of the proposed small-signal model of the VSG converter.

The physical and control parameters are listed in Table 3.3. Forced responses of the frequency

as well as the output active and reactive power to a

100 W

power reference jump are plotted

in Figure 3.27. It can be observed that the adoption of the virtual impedance can improve the

dynamic performance of the grid-connected VSG converter. Moreover, a large inertia constant

can degrade the damping performance of the grid-connected single VSG converter system,

as shown in Figure 3.27c.

3.4 Case Study

(a) Type I GFL converter

(b) Type II GFL converter

Figure 3.26: Free responses of GFL converters to 2 Hz disturbances at t= 1.0 s

3.4.2 Multiple-Converter System

Three multiple-converter scenarios are designed to test the accuracy and effectiveness of the

proposed modeling framework for larger converter-dominated power systems:

•Case 4: three GFM converters operated in islanded mode;

•

Case 5: parallel connected GFM converter and GFL converter supply an induction

machine;

•Case 6: modified IEEE 13-bus system with four GFL converters.

In Case 4, three GFM converters are operated in islanded mode to supply a common

resistive load (see Figure 3.28). In practice, except for the passive loads, induction machine

loads exhibiting highly nonlinear couplings between dynamics of power, voltage and frequency,

are important units in power systems. Completely neglecting the interaction between dynamic

loads and power sources can result in unrealistic stability evaluation results. Therefore, Case 5

is designed to study the interaction between different types of converters and dynamic loads.

To this end, the PHIL test setup is reconfigured as shown in Figure 3.29. Two converters

are controlled in grid-following mode and VSG mode, respectively. For the sake of flexibility,

3. Modeling of Converter-Dominated Power Systems

Table 3.3: Control Parameters of the single GFM Converter

Symbol Description Value

SrPower reference 1 kW

kPActive power droop coefficient 320 W ·s/rad

kQReactive power droop coefficient 207 Var/V

τCurrent control time constant 0.5 ms

PM Desired phase margin of the voltage control 45°

Kffv Voltage feedforward gain 0.75 A/V

(a) J= 0.0132 kg ·m2and Zv= 5.6 mH + 0.1 Ω

(b) J= 0.0132 kg ·m2and virtual impedance Zvis disabled

Figure 3.27: Forced responses of the VSG converter to 100 W power reference step at t= 1.0 s

a third converter is used as an emulator of the induction machine, which makes it possible

to easily change electrical and mechanical constants of the induction machine for parameter

studies. In Case 6, the modified IEEE 13-bus system shown in Figure 3.30 is adopted to further

validate the scalability of the proposed power flow and small-signal modeling methodology.

Multiple Grid-Forming Converters in Islanded Mode

The three VSG converters in Figure 3.28 share the same control parameters listed in Table 3.3.

Parameters of the load and branches are given in Figure 3.28. The steady-state values of

the system frequency and the filter capacitor voltages of converters are calculated by using

3.4 Case Study

VSG 1

Network

LBranch,1

LBranch,3

DC+

VSG 2

DC-

LBranch,2

RLoad

VSG 3

1.46 mH

+ 0.05Ω

1.46 mH

+ 0.05Ω

2.93 mH

+ 0.05Ω

9.4 Ω

Figure 3.28: Configuration of the multiple-

converter test system for Case 4

Grid Following

Network

LBranch,1

LBranch,3

DC+VSG

IM Emulator

DC-

LBranch,2

1.46 mH

+ 0.05Ω

5.9 mH

+ 0.15Ω

5.6 mH

+ 0.1 Ω

Figure 3.29: Configuration of the multiple-

converter test system for Case 5

1432

5 6 9

VSC 3

VSC 4

VSC 2

VSC 1

Stiff Grid

1432

5 6 9

VSC 3

VSC 4

VSC 2

VSC 1

Stiff Grid

Figure 3.30: Topology of the modified IEEE 13-bus system

the proposed power flow analysis. As given in Table 3.4, the maximum percentage deviation

between the results obtained from the power flow analysis and the numerical simulation in

MATLAB/Simulink is smaller than 1

1%, which confirms the accuracy of the analytical method.

The error comes from the assumption that the impedance

jω0L

is calculated with

the nominal frequency ω0.

Around the steady-state operation trajectory, the small-signal model is developed following

the systematic procedure described in Section 3.2. After the system reaches steady state,

an active power reference step of

200 W

is given to VSG 1 to excite forced responses. The

overlap of the waveforms obtained from the small-signal model, the average model and the

measurement can be observed in Figure 3.31. Similar to the single-GFM-converter case shown

in Figure 3.27, it is seen that poorly damped low-frequency oscillations between the VSG

converters can appear as the inertia constant increases. Deviations between the reactive power

responses obtained from the average model and experimental tests may result from errors in

the nominal value and nonlinearities of the physical line impedance.

Table 3.4: Steady-state Values of the Frequency and Filter Capacitor Voltages

Simulink Simulation Power Flow Analysis Percent Deviation (%)

Mag. (V) Angle (rad) Mag. (V) Angle (rad) Mag. Angle

Vαβc,197.0733 0 96.0233 0 1.0816 0

Vαβc,297.0733 0 96.0233 0 1.0816 0

Vαβc,397.0700 0.0163 96.0201 0.0164 1.0817 0.1867

f50.2486 N/A 50.2540 N/A 0.0106 N/A

3. Modeling of Converter-Dominated Power Systems

VSG 3, nonlinear modelVSG 3, nonlinear model

VSG 2, nonlinear modelVSG 2, nonlinear model

VSG 1, nonlinear modelVSG 1, nonlinear model

VSG 3, nonlinear model

VSG 2, nonlinear model

VSG 1, nonlinear model

VSG 3, nonlinear model

VSG 2, nonlinear model

VSG 1, nonlinear model

VSG 1, small-signal modelVSG 1, small-signal model

VSG 3, small-signal modelVSG 3, small-signal model

VSG 2, small-signal modelVSG 2, small-signal model

VSG 1, small-signal model

VSG 3, small-signal model

VSG 2, small-signal model

VSG 1, small-signal model

VSG 3, small-signal model

VSG 2, small-signal model

VSG 3, nonlinear model

VSG 2, nonlinear model

VSG 1, nonlinear model

VSG 1, small-signal model

VSG 3, small-signal model

VSG 2, small-signal model

(a) J= 0.0132 kg ·m2

VSG 3, nonlinear modelVSG 3, nonlinear model

VSG 2, nonlinear modelVSG 2, nonlinear model

VSG 1, nonlinear modelVSG 1, nonlinear model

VSG 3, nonlinear model

VSG 2, nonlinear model

VSG 1, nonlinear model

VSG 3, nonlinear model

VSG 2, nonlinear model

VSG 1, nonlinear model

VSG 1, small-signal modelVSG 1, small-signal model

VSG 3, small-signal modelVSG 3, small-signal model

VSG 2, small-signal modelVSG 2, small-signal model

VSG 1, small-signal model

VSG 3, small-signal model

VSG 2, small-signal model

VSG 1, small-signal model

VSG 3, small-signal model

VSG 2, small-signal model

VSG 3, nonlinear model

VSG 2, nonlinear model

VSG 1, nonlinear model

VSG 1, small-signal model

VSG 3, small-signal model

VSG 2, small-signal model

(b) J= 0.066 kg ·m2

Figure 3.31: Forced responses of three VSG converters to

200 W

power reference step given to

VSG 1 at t= 1.0 s

Interaction between Dynamic Loads and Different Types of Converters

In Figure 3.29, power references of the GFL converter and the GFM converter are set to be

600 W

and

400 W

, respectively. Inertia constants of the VSG converter and the induction

machine are

0.00132 kg ·m2

and

JIM

0.00686 kg ·m2

. Other control parameters are

the same as those of Case 1 and Case 3. The steady-state operation trajectory is obtained

from the numerical simulation in MATLAB/Simulink. The small-signal model of the induction

machine is given in the Appendix B. After the system reaches steady state, a

−

20% induction

machine load torque jump is applied to excite forced responses. Dynamic evolution of the

grid-following converter frequency, VSG converter frequency, induction machine stator current

and rotor speed obtained from the small-signal model, the nonlinear model and experimental

measurements are plotted in Figure 3.32. The overlap between theoretical and experimental

results confirms again the effectiveness and accuracy of the proposed modeling framework.

Figure 3.32: Forced responses of converters and induction motor to 0.2 pu load torque jump

3.5 Summary

Modified IEEE 13-bus System

In this subsection, the power flow and small-signal modeling framework are tested with the

modified IEEE 13-bus system. The system is scaled to a voltage rating of

400 V

(phase-

phase root-mean-square value), and the stiff grid is assumed to contain

0.5 pu

negative-

sequence voltage. As shown in Figure 3.30, four Type I GFL converters are integrated into

the distribution network. Parameters of branches and loads are listed in Table 3.5 and

Table 3.6. The comparison between simulation (indicated by "Sim") and power flow (indicated

by "PF") calculation results shown in Figure 3.33 confirms the accuracy of the proposed method.

Figure 3.34 and Figure 3.35 show time-domain dynamic responses of the PLL frequency and

the grid-side filter inductor current of all VSCs to a 5% active power reference step of VSC 1

1.0 s

. Dynamic evolutions of the small-signal model agrees well with the nonlinear average

model. It is noted that the system becomes unstable as the PLL bandwidth increases to

40 Hz

Nevertheless, by initializing all state variables with the results of the power flow analysis, the

simulation can be started from the equilibrium, allowing the distinction between instability

due to the absence of equilibrium and small-signal instability.

Table 3.5: Branch Parameters of the Modified IEEE 13-Bus System

From To Value From To Value

1 4 0.1313 Ω + 0.4178 mH 7 8 0.0755 Ω + 0.2404 mH

2 3 0.0755 Ω + 0.2404 mH 8 9 0.0752 Ω + 0.2394 mH

3 4 0.1259 Ω + 0.4007 mH 9 10 0.0756 Ω + 0.2406 mH

4 5 0.0713 Ω + 0.2269 mH 10 11 0.0756 Ω + 0.2406 mH

5 6 0.01 Ω + 0.03183 mH 8 12 0.2034 Ω + 0.6475 mH

4 9 0.1313 Ω + 0.4178 mH 9 13 0.0656 Ω + 0.2089 mH

Table 3.6: Load Parameters of the Modified IEEE 13-Bus System

Connection Bus Value Connection Bus Value

366.07 Ω + 154.6 mH 10 56.90 Ω + 160.9 mH

628.36 Ω + 65.44 mH 11 15.79 Ω + 27.54 mH

783.34 Ω + 124.8 mH 12 93.15 Ω + 199.2 mH

911.30 Ω + 20.54 mH - -

3.5 Summary

In this chapter, a systematic procedure is developed to build the small-signal model of converter-

dominated power systems considering balanced and unbalanced operations. The Wirtinger

calculus is introduced for the fully complex-domain modeling. The stationary reference frame

is selected as the common basis for the implementation of the component connection method.

Compared to the classical real-domain modeling in the rotational reference frame, following

unique benefits can be observed

The complex variable in the original coordinate inherently contains the sequence

information, i.e., positive (negative) frequency component refers to positive (negative)

sequence component. In contrast, the

frame is a fictitious coordinate, it is not

3. Modeling of Converter-Dominated Power Systems

(a) Voltage Magnitude

(b) Voltage Phase Angle

Figure 3.33: Steady-state bus voltages of the modified IEEE 13-Bus system

VSC 1 VSC 2

VSC 3 VSC 4

Figure 3.34: Dynamic responses of the PLL estimated frequency to a 5% active power reference

step of VSC 1 (Solid lines – average model. Dashed lines – small signal model) Left: PLL bandwidth

is 30 Hz. Right: PLL bandwidth is 40 Hz

VSC 1 VSC 2

VSC 3 VSC 4

Figure 3.35: Dynamic responses of the

-component of the grid-side inductor current to a 5%

active power reference step of VSC 1 (Solid lines – average model. Dashed lines – small signal

model) Left: PLL bandwidth is 30 Hz. Right: PLL bandwidth is 40 Hz

straightforward to link a single real-valued component to the description of three-phase

systems.

3.5 Summary

For the derivation of the small-signal model, the complex-valued description can simplify

the calculation. For instance, the linearization of the current reference calculation and

the frame transformation in the complex domain is simpler than that in the real domain.

To build small-signal models in the

frame, one converter needs to be selected to

provide a common rotational reference frame for the Park respectively inverse Park

transformation. This becomes nonintuitive when the system is operated under unbalanced

or harmonically distorted conditions. This shortcoming is overcome by selecting the

stationary reference frame as the common reference frame.

To verify the accuracy of the proposed modeling method, a systematic software framework

is developed based on the MATLAB/Simulink platform. A PHIL test setup is used for the

experimental validation. The effectiveness of the proposed modeling methodology is confirmed

by tests performed on both single-converter and multi-converter systems.

Small-Signal Stability Analysis

This chapter focuses on the stability analysis of the complex-valued LTP small-signal model

derived in Chapter 3. First, the widely used eigenanalysis/modal analysis framework for LTI

systems is reviewed. A time-domain physical interpretation is proposed to link the dynamic

characteristics of LTI and LTP systems. Then, the modal analysis is generalized for the LTP

systems. Definitions of the damping ratio, participation factor and eigenvalue sensitivity are

accordingly modified for the evaluation of dynamic performance of the LTP systems. Moreover,

two accurate and efficient truncation order selection methods are developed for the eigenvalue

and eigenvector calculation of the LTP systems. The chapter closes with applications of the

above techniques to the stability analysis of both grid-following and grid-forming converter

systems.

4.1 Modal Analysis for the Linear Time-Invariant System

The modal analysis is proved to be a useful tool for the stability analysis of a general N-

dimensional LTI system

∆x˙=A∆x+B∆u

∆y=C∆x+D∆u(4.1)

where ∆

,∆

and ∆

are state, input and output vectors.

and

are defined as

the system, input, output and feedforward matrices respectively, which are all time invariant.

Since the inputs and outputs do not change the stability of the LTI system, solving the free

response problem

∆x˙=A∆x,with ∆x(t0)=∆x0(4.2)

is sufficient for the stability evaluation. t0denotes the start time instance.

4. Small-Signal Stability Analysis

The mathematical foundation of the modal analysis is the eigen/spectral decomposition of

the system matrix

A=RΛR−1=[︂r1r2· · · rN]︂

⏞⏟⏟ ⏞







λ1

λ2

...

λN







⏞⏟⏟ ⏞













⏞⏟⏟ ⏞

R−1

.(4.3)

Here, the system matrix

is assumed to be diagonalizable, which is a common case for the

small-signal models of power systems. The diagonal matrix

is defined as the eigenvalue

matrix.

and

R−1

are the right eigenvector and left eigenvector matrices. The ith eigenvalue

(or mode)

λi

, the ith right eigenvector

(the ith column of

) and the ith left eigenvector

(the ith row of R−1) satisfy the relation

Ari=λiri(4.4a)

liA=λili(4.4b)

lirj=∑︂N

k=1 likrkj ={︄1,if i=j

0,if i=j(4.4c)

where

lik

is the kth element of

, that is the element in the ith row and kth column of

R−1

rkj is the kth element of rj, namely the element in the kth row and jth column of R.

Stability and Damping Ratio

Applying the space transformation

∆x=R∆z,(4.5)

the physical system Eq. (4.2) is transformed into the so-called modal space

dt (R∆z) = A(R∆z)⇒∆z˙=R−1AR

⏞ ⏟⏟ ⏞

∆z

⇓







∆z˙1

∆z˙2

∆z˙N







⏞⏟⏟ ⏞

∆z˙

=





λ1

λ2

...

λN







⏞⏟⏟ ⏞

Λ=R−1AR







∆z1

∆z2

∆zN







⏞⏟⏟ ⏞

∆z

(4.6)

It is seen that the state variables

{∆zi, i = 1, ..., N}

are decoupled, and the ith state variable

∆zican be directly solved

∆zi= ∆zi0eλi(t−t0), t ∈[t0,∞)(4.7)

where ∆zi0= ∆zi(t0)is the initial condition of ∆zi.

4.1 Modal Analysis for the Linear Time-Invariant System

Let the initial condition of ∆xbe the unit vector along the kth axis,

∆x0=ek=[︃0

⏞⏟⏟⏞

1st

· · · 0 1

⏞⏟⏟⏞

kth

0· · · 0

⏞⏟⏟⏞

Nth ]︃T

,(4.8)

namely the kth element of ∆

is one and others are zero. Then, the initial value of ∆

can

be obtained from







∆z1

∆zi

∆zN







⏞⏟⏟ ⏞

∆z













⏞⏟⏟ ⏞

R−1







∆x1

∆xi

∆xN







⏞⏟⏟ ⏞

∆x







l11 · · · l1i· · · l1N

li1· · · lii · · · liN

lN1· · · lNi · · · lNN







⏞⏟⏟ ⏞

R−1







∆x1

∆xi

∆xN







⏞⏟⏟ ⏞

∆x

⇓

∆zi(t0)=∆zi0=li∆x0=lik

.(4.9)

Returning to Eq. (4.5), the response of the kth physical state variable is given by







∆x1

∆xk

∆xN







⏞⏟⏟ ⏞

∆x

=[︂r1r2· · · rN]︂

⏞⏟⏟ ⏞







∆z1

∆zk

∆zN







⏞⏟⏟ ⏞

∆z







r11 · · · r1k· · · r1N

rk1· · · rkk · · · rkN

rN1· · · rNk · · · rNN







⏞⏟⏟ ⏞







∆z1

∆zk

∆zN







⏞⏟⏟ ⏞

∆z

⇓

∆xk=∑︁N

i=1 rki∆zi=∑︁N

i=1 rkilikeλi(t−t0)

(4.10)

It is noted that the free response of ∆

is the superposition of

oscillation modes

{λi, i = 1, ..., N}

. The system is stable if the real parts of all eigenvalues are negative. When

the system is stable, defining

λi=σi+jωi,(4.11)

the damping performance of the ith oscillation mode

λi

is quantified by the damping factor

σi

and the damping ratio

ξi=−σi

√︂σ2

i+ω2

.(4.12)

Specifically, the amplitude of the state decays to 1

or 37% of the initial value in 1

/|σi|

seconds or 1/(2πξi)cycles of oscillation.

Participation Factor

From Eq. (4.9) and (4.10),

lik

quantifies the significance of the initial condition of the kth state

∆xkto excite the ith mode λi.rki reflects the observability of the ith mode in the kth state.

Their product

pki =rkilik (4.13)

4. Small-Signal Stability Analysis

is defined as the participation factor of the kth state in the ith mode, and conversely. Physically

it is consistent with the initial value of the ith oscillation mode observed in the dynamic

evolution of ∆xk, as given in Eq. (4.10).

Eigenvalue Sensitivity

Another important tool within the classical modal analysis framework is the eigenvalue

sensitivity analysis. Let

be an arbitrary parameter in the system matrix

, the partial

derivative of Eq. (4.4a) with respect to φis

∂(Ari)

∂φ =∂(λiri)

∂φ

⇓

∂A

∂φ ri+A∂ri

∂φ =∂λi

∂φ ri+λi

∂ri

∂φ

.(4.14)

Multiplying both sides of Eq. (4.14) with

and inserting Eq. (4.4b) and (4.4c), the sensitivity

of λiwith respect to φis obtained

∂A

∂φ ri+liA∂ri

∂φ =∂λi

∂φ liri

⏞⏟⏟⏞

+λili

∂ri

∂φ

⇓

∂λi

∂φ =li

∂A

∂φ ri

.(4.15)

Let

be the element in the ith row and kth column of

, namely

Aik

, the sensitivity of

λi

with respect to Aik is

kth

↓

∂λi

∂Aik =li∂A

∂Aik ri=[︂li1· · · lik · · · liN ]︂

⏞ ⏟⏟ ⏞







0· · · 0· · · 0

0· · · 1· · · 0

0· · · 0· · · 0







⏞⏟⏟ ⏞

∂A

∂Aik

←ith







r1i

rki

rNi







⏞⏟⏟ ⏞

⇓

∂λi

∂Aik =likrki

(4.16)

which is the same as the participation factor given by Eq. (4.13).

Eigenvalue sensitivity information can be used for the control parameter tuning. Specifically,

when the feedback loop gain between the ith and the kth state variables is changed by ∆

Aik

as shown in Figure 4.1, its impact on the change of the eigenvalue λican be predicted by

∆λi= ∆Aik

∂λi

∂Aik

.(4.17)

4.2 Time-Domain Physical Interpretation of the LTP System

1/s

Aik

Rest of the

System

Aik

x

Figure 4.1: Physical meaning of the eigenvalue sensitivity

This provides guidance for control parameter optimization to move critical eigenvalues towards

the left-half of the complex plane, as will be presented in Chapter 6.

4.2 Time-Domain Physical Interpretation of the LTP System

The classical modal analysis reviewed above can reveal the dynamic behavior of an LTI system

in detail, however, it cannot be directly applied to the stability analysis of the LTP model

of converter-dominated power systems. To generalize the modal analysis to LTP systems, a

time-domain physical interpretation is proposed for comparison of the dynamic characteristics

of LTI and LTP systems.

Linear Time-Invariant System

Applying the rotational transformation

∆xh=e−jhω0t∆x,(4.18)

in other words observing the system in a new coordinate frame rotating the original one by

e−jhω0t, the state-space description of the LTI system given by Eq. (4.2) becomes

dt 

ejhω0t∆xh

⏞⏟⏟ ⏞

∆x



=A

ejhω0t∆xh

⏞ ⏟⏟ ⏞

∆x





⇓

∆x˙h= (A−jhω0I)∆xh

(4.19)

where

ω0

is constant,

is an integer and

is the identity matrix. It is seen that the

rotational transformation with fixed frequency

hω0

is linear. Equation (4.2) and (4.19) are

fully decoupled, and both can independently describe the LTI physical system. Obviously,

eigenvalues of Eq. (4.19),

{λi−jhω0, i = 1, ..., N}

, are frequency-shifted versions of those of

Eq. (4.2).

Linear Time-Periodic System

For the free response problem of a general N-dimensional LTP system

∆x˙=A(t)∆x,with ∆x(t0)=∆x0(4.20)

4. Small-Signal Stability Analysis

the system matrix

(

)becomes time periodic with the fundamental period

= 2

π/ω0

. If

not specified otherwise, the time variable (

)is used to distinguish the time-periodic matrices

and vectors from time-invariant ones.

Inserting the Fourier series expansion

A(t) = ∑︂∞

k=−∞ Akejkω0t

and using the notation defined in Eq. (4.18), the state-space model Eq. (4.20) can be

reformulated as

∆x˙=∑︁∞

k=−∞ Akejkω0t∆x=A0∆x+∑︁k=0 Akejkω0t∆x

⏞⏟⏟ ⏞

∆x−k

⇓

∆x˙=A0∆x+∑︁k=0 Ak∆x−k

(4.21)

and in the rotated coordinate frame

∆x˙h= (A0−jhω0I)∆xh+∑︂k=0 Ak∆xh−k.(4.22)

Writing Eq. (4.21) and (4.22) in the matrix form given by Eq. (4.23), the N-dimensional LTP

system is equivalently described as an infinite-dimensional LTI system with couplings between

observations in the original and rotated coordinate frames, i.e., ∆xand {∆xh, h ∈Z}.







∆x˙−1

∆x˙

∆x˙1













....

...

· · · A0+jω0IA−1A−2· · ·

· · · A1A0A−1· · ·

· · · A2A1A0−jω0I· · ·

...

....













∆x−1

∆x

∆x1







(4.23)

The same as the LTI system, both ∆

and

{∆xh, h ∈Z}

can fully capture dynamic

evolutions of the LTP physical system. If

λi

is predominantly related to ∆

, the frequency-

shifted eigenvalue

λi−jhω0

will dominate dynamics of ∆

. Therefore, it can be expected that

eigenvalues of Eq. (4.23) will formulate Nvertical lines in the complex plane. Different from

the LTI systems, due to the frequency coupling effect, oscillations associated with

λi−jhω0

can also appear in ∆

. Considering symmetry, the contribution of

λi−jhω0

in ∆

is the

same as that of

λi

in ∆

. These intuitive inferences derived from the proposed time-domain

physical interpretation will be mathematically proved in the next section. For the evaluation

and improvement of the damping performance of the LTP system, major concerns are to

calculate eigenvalues λimost relevant to ∆xand quantify the contribution of λi−jhω0.

4.3

Generalized Modal Analysis for Linear Time-Periodic

Systems

4.3 Generalized Modal Analysis for Linear Time-Periodic Systems

4.3.1 Eigenvalues and Eigenvectors of LTP Systems

To evaluate the stability and damping performance of the LTP system, the analytical solution

of the free response problem of the LTP system Eq. (4.20) is derived in this subsection.

According to the Floquet theory [90], there exists a time-periodic transformation matrix

R(t)

∆x=R(t)∆z(4.24)

which transforms Eq. (4.20) into a new state space where the system matrix

is a time-

invariant diagonal matrix

dt (R(t)∆z) = R(t)∆z˙+R

˙(t)∆z=A(t) (R(t)∆z)

⇓







∆z˙1

∆z˙2

∆z˙N







⏞⏟⏟ ⏞

∆z˙

=





λ1

λ2

...

λN







⏞⏟⏟ ⏞

Q=R−1(t)(A(t)R(t)−R

˙(t))







∆z1

∆z2

∆zN







⏞ ⏟⏟ ⏞

∆z

.(4.25)

Similar to Eq. (4.6), this new state space can be regarded as the modal space of the LTP

system, where ∆zican be directly solved

∆zi= ∆zi0eλi(t−t0).(4.26)

The initial condition ∆zi0is obtained from the transformation

∆z0=R−1(t0) ∆x0.(4.27)

Defining

R(t)=[r1(t),r2(t), . . . , rN(t)]

and considering the transformation given by Eq. (4.24), the free response of Eq. (4.20) is

formulated by the superposition of Nindependent fundamental solutions

∆xi, i = 1, ..., N

related to the Ndiagonal elements in Q

∆x=[︂r1(t)r2(t)· · · rN(t)]︂

⏞⏟⏟ ⏞

R(t)







∆z1

∆z2

∆zN







⏞⏟⏟ ⏞

∆z

=∑︂N

i=1 ∆zi0eλi(t−t0)ri(t)

⏞⏟⏟ ⏞

∆xi

(4.28)

where ri(t)is the ith column of R(t).

4. Small-Signal Stability Analysis

Each fundamental solution ∆xisatisfies Eq. (4.20), which means

dt 



∆zi0eλi(t−t0)ri(t)

⏞⏟⏟ ⏞

∆xi





=A(t)



∆zi0eλi(t−t0)ri(t)

⏞ ⏟⏟ ⏞

∆xi







⇓

∆zi0e−λit0(︂eλitr˙i(t) + λieλitri(t))︂=∆zi0e−λit0A(t)eλitri(t)

(4.29)

so that

λiri(t) = A(t)ri(t)−r

˙i(t).(4.30)

The time-periodic matrix A(t)and the vector ri(t)have the Fourier series expansion:

A(t) = ∑︂∞

k=−∞ Akejkω0t

ri(t) = ∑︂∞

h=−∞ rh

iejhω0t

r˙i(t) = ∑︂∞

h=−∞ jhω0·rh

iejhω0t

(4.31)

where the Fourier series coefficient

is associated with the kth order harmonic of

(

and the Fourier series coefficient

is associated with the hth order harmonic of

(

). The

subscript

(

)indicates that

(

)is the ith LTP eigenvector, namely the ith column of

R(t).

Inserting Eq. (4.31) into Eq. (4.30), yields

∑︂∞

h=−∞ (︂(λi+jhω0)rh

i−∑︂∞

k=−∞ Akrh−k

i)︂ejhω0t= 0 (4.32)

which is valid for any time instance t, therefore, items in the bracket must be zero, namely

Eq. (4.33) holds for all h∈Z.

(λi+jhω0)rh

i=∑︂∞

k=−∞ Akrh−k

i(4.33)

Equation (4.33) can be formulated into the matrix form

λi







r−1













....

...

· · · A0+jω0I A−1A−2· · ·

· · · A1A0A−1· · ·

· · · A2A1A0−jω0I· · ·

...

....













r−1







(4.34)

which can be compactly written as

λiRi= (A−N)Ri(4.35)

4.3 Generalized Modal Analysis for Linear Time-Periodic Systems

by defining

Ri=







r−1













....

...

· · · A0A−1A−2· · ·

· · · A1A0A−1· · ·

· · · A2A1A0· · ·

...

....













...

N−1

...







(4.36)

is an N-dimensional diagonal square matrix with diagonal elements equal to

jkω0

. It is

seen that Eq. (4.34) and (4.35) define a standard eigenvalue problem. Specifically,

λi

is the

eigenvalue of the infinite matrix

A−N

associated with the eigenvector

consisting of Fourier

series coefficients of the vector ri(t).

It can be easily confirmed that the pair of the complex value

λi

jkω0

and the vector

e−jkω0tri

(

)satisfy the relation Eq. (4.30), therefore, the complex value

λi

jkω0

is also the

eigenvalue of

A−N

, and the corresponding eigenvector is the Fourier series coefficient vector

of e−jkω0tri(t). Following the same procedure, it can be identified that the other elements in

the matrix

and their frequency-shifted copies are also the eigenvalues of

A−N

. This leads

to the conclusion that the eigenvalue plot of the infinite matrix

A − N

consists of

vertical

lines, as qualitatively demonstrated in Figure 4.2.

λ1

λ2

λ3

λN

λ1 + j(k)ω0

λ1 + j(k+1)ω0

(a) (b) (c)

Figure 4.2: Qualitative eigenvalue loci of

A−N

. Each element of the diagonal matrix

(a)

corresponds to one vertical line in the Eigenvalue map (b). (c) Eigenvectors of

A − N

associated

with λ1+jkω0(left) and λi+j(k+ 1)ω0(right)

Returning to the ith fundamental solution

∆xi= ∆zi0·eλi(t−t0)·ri(t),(4.37)

it can also be expressed by the frequency-shifted eigenvalue

λi

jkω0

and the associated

eigenvector e−jkω0tri(t)

∆xi=ejkω0t0∆zi0·e(λi+jkω0)(t−t0)·e−jkω0tri(t).(4.38)

The blue term in Eq. (4.38) results from the frequency shifting of the left eigenvector from

to ejkω0tli.

Defining

λi=σi+jωi,(4.39)

4. Small-Signal Stability Analysis

and employing the Fourier series expansion of

(

)in Eq. (4.31), the ith fundamental solution

∆xibecomes

∆xi= ∆zi0∑︂∞

h=−∞ rh

ieσi(t−t0)e−jωit0ej(ωi+hω0)t(4.40)

Two conclusions can be drawn from Eq. (4.38) and (4.40):

The eigenvalues in each vertical line shown in Figure 4.2 provide redundant information.

Together with the eigenvector, one arbitrary eigenvalue on the vertical line is enough to

determine the corresponding fundamental solution.

Each fundamental solution can contain multiple oscillation components sharing the

same damping factor

σi

, while their oscillation frequencies differ by integer multiples

of the fundamental frequency

ω0

. For the LTI system, only one oscillation frequency is

associated with each damping factor (see Eq. (4.10)), since the transformation matrix

of the LTI system is constant.

It is noted that the matrix

A−N

is exactly the same as the system matrix of the state-space

model given by Eq. (4.23). The conclusions stated above prove the intuitive guess derived

based on the physical explanation in Section 4.2.

In this thesis,

λi

is defined as LTP eigenvalue or LTP mode, and the associated time-periodic

eigenvector ri(t)is called LTP eigenvector.

Analysis Example - Lossy Mathieu equation

In order to illustrate the usefulness of the analysis method described above, the lossy Mathieu

equation is taken as an example. A two-dimensional lossy Mathieu equation can be generally

described by

dt [︄x1

x2]︄

⏞⏟⏟ ⏞

x˙

=[︄0 1

−5 + βcos ω0t−2ζ]︄

⏞ ⏟⏟ ⏞

A(t)

[︄x1

x2]︄

⏞ ⏟⏟ ⏞

.(4.41)

The non-zero Fourier series coefficients of A(t)are

A−1=[︄0 0

β/2 0 ]︄,A0=[︄0 1

−5−2ζ]︄,A1=[︄0 0

β/2 0 ]︄.(4.42)

In this example, the parameter values are selected to be

ζ= 0.8, β = 8, ω0= 2 .

For this two-dimensional system, two LTP eigenvalues can be determined by using the

method that will be introduced in Section 4.4

Q=[︄λ1

λ2]︄=[︄−0.1782

−1.4218 ]︄(4.43)

4.3 Generalized Modal Analysis for Linear Time-Periodic Systems

together with two LTP eigenvectors

r1(t) =











[︃−0.0005 −0.0005j

−0.0041 + 0.0041j]︃

⏞⏟⏟ ⏞

r−4

e−j4ω0t+[︃0.0061 + 0.0091j

0.0533 −0.0381j]︃

⏞ ⏟⏟ ⏞

r−3

e−j3ω0t+[︃−0.0301 −0.0817j

−0.3213 + 0.1349j]︃

⏞ ⏟⏟ ⏞

r−2

e−j2ω0t

+[︃−0.0230 + 0.2581j

0.5203 + 0.0000j]︃

⏞ ⏟⏟ ⏞

r−1

e−j1ω0t+[︃0.1863 + 0.1442j

−0.0332 −0.0257j]︃

⏞ ⏟⏟ ⏞

+[︃0.2441 −0.0871j

0.1306 + 0.5037j]︃

⏞ ⏟⏟ ⏞

ej1ω0t

+[︃−0.0866 −0.0086j

0.0499 −0.3449j]︃

⏞⏟⏟ ⏞

ej2ω0t+[︃0.0103 + 0.0036j

−0.0235 + 0.0612j]︃

⏞ ⏟⏟ ⏞

ej3ω0t+[︃−0.0006 −0.0004j

0.0029 −0.0050j]︃

⏞ ⏟⏟ ⏞

ej4ω0t











r2(t) =











[︃0.0006 −0.0001j

−0.0018 −0.0046j]︃

⏞⏟⏟ ⏞

r−4

e−j4ω0t+[︃−0.0086 + 0.0033j

0.0322 + 0.0467j]︃

⏞ ⏟⏟ ⏞

r−3

e−j3ω0t+[︃0.0602 −0.0419j

−0.2531 −0.1813j]︃

⏞ ⏟⏟ ⏞

r−2

e−j2ω0t

+[︃−0.1088 + 0.1894j

0.5333 −0.0517j]︃

⏞⏟⏟ ⏞

r−1

e−j1ω0t+[︃−0.1983 + 0.0096j

0.2819 −0.0136j]︃

⏞ ⏟⏟ ⏞

+[︃−0.1265 −0.1780j

0.5359 + 0.0000j]︃

⏞ ⏟⏟ ⏞

ej1ω0t

+[︃0.0640 + 0.0359j

−0.2344 + 0.2049j]︃

⏞⏟⏟ ⏞

ej2ω0t+[︃−0.0089 −0.0025j

0.0275 −0.0496j]︃

⏞ ⏟⏟ ⏞

ej3ω0t+[︃0.0006 + 0.0001j

−0.0014 + 0.0048j]︃

⏞ ⏟⏟ ⏞

ej4ω0t











Let the initial condition at t0= 0 be

x0=[︄x10

x20 ]︄=[︄1

0]︄.

The free response of the state variable x1is shown in Figure 4.3. It is seen that the dynamic

evolution of

is the superposition of two LTP modes, namely

λ1

(see Figure 4.3(a)) and

λ2

(see Figure 4.3(b)). Each LTP mode contains five oscillation components with oscillation

frequencies differing by integer multiples of ω0.

4.3.2 Stability and Damping Ratio

From the analytical solution of the free response of the LTP system given by Eq. (4.40), it can

be concluded that the system stability is determined by the real part of the LTP eigenvalue:

•

σi<

0holds for all

, all LTP oscillation modes are exponentially damped, thus, the

system is asymptotically stable.

•

σi

= 0, the

th LTP mode is an undamped oscillation, the system is marginally stable.

•

σi>

0, the corresponding LTP mode increases exponentially, so the system is unstable.

When the ith LTP eigenvector

(

)has non-zero Fourier series coefficients up to the Hth

order, the ith fundamental solution ∆xibecomes

∆xi= ∆zi0∑︂H

h=−Hrh

ieσi(t−t0)e−jωit0ej(ωi+hω0)t(4.44)

which contains (2H+1) oscillation components. The damping ratio of ith LTP mode is

generalized as a vector

ξi=[︂ξ−H

i· · · ξ0

i· · · ξH

i]︂T.(4.45)

4. Small-Signal Stability Analysis

(c)

(b)

λ2±j4ω0

λ2±j3ω0

λ2±j2ω0

λ2

λ2±jω0

(a)

λ1

λ1±jω0

λ1±j2ω0

λ1±j3ω0

λ1±j4ω0

Superposition

Figure 4.3: Free response of

to an initial condition of

[︁1 0 ]︁T

0 s

. The dynamic

evolution of

shown in (c) is the superposition of two LTP modes, namely

λ1

in (a) and

λ2

(b). Each LTP mode corresponds to five oscillation components.

ξh

igiven by

ξh

i=−σi

√︂σ2

i+ (ωi+hω0)2(4.46)

is the damping ratio of the hth oscillation component of the ith LTP mode.

For LTI systems,

ξh

equals zero unless

= 0, since the eigenvector

is time invariant.

This makes the LTI system a special case of the LTP system.

4.3.3 Participation Factor Analysis

To determine the relative participation of the ith LTP mode in the kth state, similar to the

case of the LTI systems, the initial condition ∆

is chosen to be the unit vector along the

kth axis, namely

in Eq. (4.8). Let

(

)be the ith row of

R−1

(

), the initial condition of

4.3 Generalized Modal Analysis for Linear Time-Periodic Systems

∆ziis obtained from







∆z1

∆zi

∆zN







⏞⏟⏟ ⏞

∆z







l1(t)

li(t)

lN(t)







⏞⏟⏟ ⏞

R−1(t)







∆x1

∆xi

∆xN







⏞⏟⏟ ⏞

∆x







l11(t)· · · l1i(t)· · · l1N(t)

li1(t)· · · lii(t)· · · liN (t)

lN1(t)· · · lNi(t)· · · lNN (t)







⏞⏟⏟ ⏞

R−1(t)







∆x1

∆xi

∆xN







⏞⏟⏟ ⏞

∆x

⇓

∆zi0= ∆zi(t0) = li(t0)ek=lik (t0) = ∑︁∞

h=−∞ lh

ikejhω0t0

(4.47)

where

lik

(

)represents the kth element of

(

), and gives the contribution of the initial

condition of ∆

to the ith LTP mode.

lik

(

)varies in a periodic manner on the start time

instance t0, and lh

ik denotes its hth order Fourier series coefficient.

Then, the dynamic response of the kth state of ∆xin Eq. (4.28) becomes

∆xk=∑︂N

i=1 lik (t0)rki (t)eλi(t−t0)(4.48)

where

rki

(

)is the kth element of the column vector

(

). Inserting the Fourier series expansion

rki (t) = ∑︂∞

h=−∞ rh

kiejhω0t

yields

∆xk=∑︂N

i=1 ∑︂∞

h=−∞ lik (t0)rh

kiejhω0t0

⏞⏟⏟ ⏞

∆xh

ki(t0)

e(λi+jhω0)(t−t0)(4.49)

It is seen that the initial value of the hth oscillation component of the ith LTP mode

∆xh

ki (t0) = lik (t0)rh

kiejhω0t0

is a time-period function of

. Formulating the Fourier series coefficients of

lik

(

)in vector

form

lik =[︂· · · l−h

ik · · · l−1

ik l0

ik l1

ik · · · lh

ik · · · ]︂T,

the upper bound of

lik

(

)is given by the

-norm

|lik|1

. Using the upper limit of ∆

ki (t0)

the participation factor of the hth oscillation component of the ith LTP mode in the kth state

is defined as

ki =|lik|1·rh

ki

∑︁N

i=1 ∑︁∞

h=−∞ |lik|1·rh

ki.(4.50)

The entire contribution of the ith LTP mode in the kth state is defined as

pki =∑︁∞

h=−∞ |lik|1·rh

ki

∑︁N

i=1 ∑︁∞

h=−∞ |lik|1·rh

ki.(4.51)

4. Small-Signal Stability Analysis

An alternative definition of the participation factor can be developed by considering the

average value of ∆xh

ki (t0)over one fundamental period T0, namely

ki =∫︂T0

t0=0

∆xh

ki(t0)dt0=∫︂T0

t0=0

lik(t0)rh

kiejhω0t0dt0=l−h

ik rh

ki.(4.52)

Accordingly, the total participation factor of the ith LTP mode in the kth state is defined as

pki =∑︂∞

h=−∞ ph

ki =∑︂∞

h=−∞ l−h

ik rh

ki.(4.53)

It can be confirmed that the sum of participation factors of the ith LTP mode (state) in all

states (modes) is one.

Analysis Example - Type I grid-following converter (DSRF-PLL plus PI current

control)

For the sake of clarity, the free response of the Type I GFL converter is solved to demonstrate

the application of the participation factor given by Eq. (4.50) and (4.51). Key control and

physical parameters are listed in Table 4.1. Non-zero Fourier coefficients of

(

)are indicated

by colored dots in Figure 4.4. Extra non-zero elements caused by the negative-sequence grid

voltage are marked with black dots. In other words, those terms indicated by black dots

become zeros when the three-phase grid voltage is balanced.

Table 4.1: Parameters of the Type I Grid-Following Converter for Participation Analysis

Symbol Description Value

SCR Short-circuit ratio 3

BW Bandwidth of the DSRF-PLL 20 Hz

τTime constant of the PI current controller 1 ms

SrPower reference 1.2 kW

VpMagnitude of the positive-sequence grid voltage 100 V

VnMagnitude of the negative-sequence grid voltage 50 V

State index

Fourier coefficient

State index

Fourier coefficient

Figure 4.4: Distribution of non-zero Fourier coefficients of the 14-dimensional time-periodic

matrix

(

). Each colored surface corresponds to a 14

14 Fourier series coefficient of

(

), and

their non-zero elements are denoted by dots. Extra black dots result from the negative-sequence

grid voltage.

4.3 Generalized Modal Analysis for Linear Time-Periodic Systems

After the system reaches steady state, a

1 Hz

disturbance is applied to the state variable

∆

of the DSRF-PLL. The free responses obtained from the nonlinear average model and the

small-signal model are shown in Figure 4.5. It can be observed that the evolution of ∆

dominated by an oscillation component at the frequency of

12.99 Hz

(

0.077 s

) for a low PLL

bandwidth

20 Hz

. The influence of the starting time instance is negligible. As the PLL

bandwidth BW increases to

33 Hz

, the time-periodic impact of the starting time instance

can be observed in the response of ∆η. Additionally, more oscillation frequencies appear.

0.077 s

(a) t0= 2.000 s (b) t0= 2.005 s

SSM Model, BW=20 Hz

Average Model, BW=20 Hz

SSM Model, BW=33 Hz

Average Model, BW=33 Hz

Figure 4.5: Free response of PLL estimated frequency ∆

to initial condition ∆

η0

= 2

different starting time instances

Figure 4.6(a) and Figure 4.7(a) show the LTP eigenvalue maps for different PLL bandwidths.

The 14 LTP eigenvalues are marked with red dots. According to Eq. (4.44), each LTP

eigenvalue/mode is related to multiple oscillation components. Those oscillation components

corresponding to nonzero

are marked with dots and stars in Figure 4.6(b) and Figure 4.7(b).

In Figure 4.6(b), only LTP mode 1 and mode 3 are plotted, since mode 2 and mode 4 have the

same damping factor (i.e., real part) and conjugate frequencies (i.e., imaginary part), the same

goes for Figure 4.7(b). Based on Eq. (4.50), the participation factor of each LTP mode in the

state ∆

is computed and given by the stacked bar plots in Figure 4.6(c) and Figure 4.7(c). In

Figure 4.6(b), LTP mode 1 has eight oscillation components (marked with one red dot and

seven different colored stars), the participation factor of each oscillation component of mode 1

is given by the first stacked bar in Figure 4.6(c). The color of the stacked bar is consistent with

that of the dots and stars in Figure 4.6(b). It can be observed that only one pair of oscillation

components in LTP mode 1 and 2 (

λ1,2

−

±j

5) has critical damping ratio (

ξ0

1,2

= 0

20).

The oscillation frequency is consistent with the free response shown in Figure 4.5. As BW

increases to

33 Hz

, more oscillation components having critical damping ratio appear in LTP

mode 1/2 (

−

±j

346

4and

−

±j

281

9) and mode 3/4 (

−

±j

101

5), as shown in

Figure 4.7(b). This explains the more oscillatory free response for

33 Hz

in Figure 4.5.

4. Small-Signal Stability Analysis

(c)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Participation factor

λ1,2

λ3,4

λ5,6

λ7,8

λ9

λ10

λ11

λ12

λ13,14

-44.6+j17.61

(a) (b)

×103

Mode 1

×103

Mode 3

-17+j81.5

λ1

λ3

Figure 4.6: Small-signal analysis for

20 Hz

. (a) LTP eigenvalue map. (b) Oscillation

components of each LTP mode. (c) PF of each LTP mode in state ∆η.

(c)

(a) (b)

λ1 ~λ6

λ9

λ10

λ11

λ12

λ7,8

λ13,14

Mode 1

Mode 5

Mode 3

-21.8+j346.4

-21.8-j281.9

λ1

-36.8+j101.5

λ3

λ5

-54.8+j19.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Participation factor

×103×103

Figure 4.7: Small-signal analysis for

33 Hz

. (a) LTP eigenvalue map. (b) Oscillation

components of each LTP mode. (c) PF of each LTP mode in state ∆η.

4.3.4 Sensitivity Analysis

This subsection aims to compute the sensitivity of the LTP eigenvalue with respect to an

arbitrary parameter. From Eq. (4.25), the system matrix A(t)of the LTP system satisfies

A(t)R(t)−R

˙(t) = R(t)Q(4.54)

Considering

R(t)R−1(t) = I

⇓

dt (︂R(t)R−1(t))︂=R

˙(t)R−1(t) + R(t)R

˙−1(t) = 0

4.4 LTP Eigenvalue Calculation

the adjoint equation can be derived from Eq. (4.54)

R−1(t)A(t) + R

˙−1(t) = QR−1(t)(4.55)

Equation (4.54) and (4.55) give that the ith diagonal element of

, the ith column of

(

)

and the ith row of R−1(t), namely λi,ri(t)and li(t), satisfy the relation

A(t)ri(t)−r˙i(t) = λiri(t)(4.56a)

li(t)A(t) + l

˙i(t) = λili(t)(4.56b)

Taking partial derivatives of both sides of Eq. (4.56a) with respect to an arbitrary parameter

φ, yields

∂A(t)

∂φ ri(t) + A(t)∂ri(t)

∂φ −∂r˙i(t)

∂φ =∂λi

∂φ ri(t) + λi

∂ri(t)

∂φ .(4.57)

Multiplying both sides of Eq. (4.57) by

(

)and substituting the relation given by Eq. (4.56b),

it becomes ∂λi

∂φ =li(t)∂A(t)

∂φ ri(t)−(︃l

˙i(t)∂ri(t)

∂φ +li(t)∂r˙i(t)

∂φ )︃

=li(t)∂A(t)

∂φ ri(t)−d

dt (︃li(t)∂ri(t)

∂φ )︃.(4.58)

The term d

dt (︃li(t)∂ri(t)

∂φ )︃

contains no time-invariant elements, therefore, the time-independent eigenvalue sensitivity is

determined by

∂λi

∂φ ={︃li(t)∂A(t)

∂φ ri(t)}︃0

(4.59)

where the operator

{ }0

extracts the DC component, namely the average value of

li(t)∂A(t)

∂φ ri(t)

over one fundamental period

. It is seen that the LTP eigenvalue sensitivity is the same

as the participation factor given by Eq. (4.52) and (4.53). The physical meaning of the LTP

eigenvalue sensitivity is the same as that of the LTI system.

4.4 LTP Eigenvalue Calculation

The proposed generalized modal analysis can provide insightful characterization of the dynamic

performance of the LTP system. However, the calculation of the LTP eigenvalue matrix

and the transformation matrix

(

)in Eq. (4.25) remains an open topic. Currently, there are

mainly two categories of LTP eigenvalue computation methods.

The first category is to calculate the N-dimensional monodromy matrix

(

T0,

0) [58] by

solving a matrix differential equation

˙(t, 0) = A(t)Φ(t, 0) with Φ(0,0) = I(4.60)

There exists the following relation between the LTP eigenvalues of

(

)and eigenvalues of

Φ(T0,0)

4. Small-Signal Stability Analysis







eλ1T0

eλ2T0

...

eλNT0







⏞⏟⏟ ⏞

eQT0

=





λΦ,1

λΦ,2

...

λΦ, N







⏞ ⏟⏟ ⏞

eig{Φ(T0,0)}

(4.61)

which gives another interpretation of the stability criterion, that is the LTP system is

asymptotically stable when magnitudes of all eigenvalues of Φ(T0,0) are smaller than unity.

According to Eq. (4.61), the LTP eigenvalue λican be solved from

eλiT0=λΦ, i.(4.62)

By choosing an appropriate ordinary differential equation solver, eigenvalue calculation results

to any required degree of accuracy can be derived from

(

T0,

0). Yet, it fails to provide more

insightful quantification of the system dynamics, e.g., damping, sensitivity and participation

information. Additionally, for large-scale converter-dominated power systems, the numerical

integration of the high-dimensional system matrix can be time-consuming.

The second category is to solve the eigenvalue problem of Eq. (4.34). Since numerical

implementation and calculation of the eigenvalues of an infinite matrix is impossible, the

matrix A−N must be truncated. Two questions should be answered:

whether the eigenvalues of the truncated matrix converge to those of the infinite matrix;

how to choose the minimum truncation order

Hmin

to balance the trade-off between

numerical accuracy and efficiency.

Practical power electronic systems have limited bandwidth and energy, thus,

(

)and

(

)

given in Eq. (4.31) have finite non-zero Fourier series terms. For such systems, it has been

rigorously proved in [94] that eigenvalues of the truncated matrix converge to those of the

non-truncated one as the truncation order approaches infinity.

Eigenvalue Sorting Method

To determine the minimum truncation order

Hmin

, the eigenvalue sorting method is used

in most references. The principle is to increase the truncation order

until

unchanged

eigenvalues are found in the fundamental strip

−ω0/

<Im{λi}< ω0/

2. For the single Type I

grid-following converter system with control and physical parameters specified in Table 4.2,

the eigenvalues of

A − N

for different truncation orders are plotted in Figure 4.8. A high

truncation order of 50 is used as a benchmark, of which the eigenvalues form eight vertical

lines in the complex plane. Further increasing

does not change the location of the lines. It

can be observed from Figure 4.8 that

•

Within the fundamental strip, more and more eigenvalues converge to the vertical lines

as the truncation order increases.

•

No matter how large

is, the truncation will always result in eigenvalues away from

the vertical lines (see the zoomed in plot). Such eigenvalues should not be considered for

4.4 LTP Eigenvalue Calculation

the stability evaluation. With the eigenvalue sorting method those eigenvalues can only

be graphically distinguished.

•

On each vertical line, either one pure real-valued eigenvalue or two complex-conjugate

eigenvalues within the fundamental strip should be selected as the LTP eigenvalues. The

truncation order is considered to be sufficient when

LTP eigenvalues are determined.

For the test case, a truncation order of 14 is needed, which results in a square matrix

with the dimension of 14 ·(2 ·14 + 1) = 406.

Table 4.2: Parameters of the Type I Grid-Following Converter for the Study of the LTP Eigenvalue

Calculation Algorithm

Symbol Description Value

SCR Short-circuit ratio 3

BW Bandwidth of the DSRF-PLL 20 Hz

τTime constant of the PI current controller 0.5 ms

SrPower reference 1.2 kW

VpMagnitude of the positive-sequence grid voltage 100 V

VnMagnitude of the negative-sequence grid voltage 50 V

Line 1

Line 2

Line 3

Line 4Line 5

Line 6

Line 7

Line 8

Zoom in

H=14

H=5

H=2

H=50

H=14

H=5

H=2

H=50

H=14

H=5

H=2

H=50

Line 1

Line 2

Line 3

Line 4Line 5

Line 6

Line 7

Line 8

Zoom in

H=14

H=5

H=2

H=50

Figure 4.8: Convergence process of the eigenvalue sorting method. All eigenvalues of

A − N

for

different truncation orders (top). Eigenvalues located in the fundamental strip (bottom).

Eigenvector Sorting Method

The eigenvalue sorting method ignores the rate of convergence of eigenvalues and can result

in unnecessarily high truncation orders. Specifically, in Figure 4.8, a high truncation order

of 14 is needed to see the converged eigenvalues on Line 6 and 7 in the fundamental strip

(

−ω0/

<Im{λi}< ω0/

2). However, in the frequency range from 3000 to

5000 rad/s

= 5

is already sufficient to determine the converged eigenvalues (see Figure 4.9). To tackle the

issue of the eigenvalue sorting methods, one eigenvector sorting method is proposed in [95].

The basic idea behind is that the Neigenvectors, whose nonzero terms are symmetric about

4. Small-Signal Stability Analysis

the DC component and within the truncation order, are less influenced by the truncation.

Thus, the associated Neigenvalues converge faster. For a truncation order H, to select those N

eigenvectors from the

+ 1) eigenvectors of the truncated matrix

A − N

, the

+ 1)

weighted mean values

wi=∑︁H

h=−Hhrh

i1

∑︁H

h=−Hrh

i1

(4.63)

are calculated. The weighted means of

(

)and

ejkω0tri

(

), namely

and

wi+k

, satisfy

wi+k

. The Neigenvectors whose weighted means are closest to zero are selected.

More details of the eigenvector sorting method are given in Algorithm 1.

Algorithm 1 Eigenvector Sorting Method

1: Initialization of the truncation order: Hmin = 1

2: repeat

3: Calculate eigenvalues and eigenvectors of truncated matrix A − N

4: Calculate weighted means of all eigenvectors based on Eq. (4.63)

Determine the

eigenvectors with the weighted means closest to zero, store the

associated Neigenvalues

6: Update the truncation order Hmin =Hmin + 1

7: until changes of the Neigenvalues between two iterations (land l+ 1) satisfy

|λl+1

i−λl

|λl+1

i|·100% ≤δ, i ∈[1, N]

Output: minimum truncation order Hmin, converged Neigenvectors and eigenvalues

Figure 4.9 shows the convergence process of the eigenvector sorting method. The high

truncation order

= 50 is again used as the benchmark. In each iteration step, the selected

LTP eigenvalues are marked with circles. A truncation order

= 5 is found to be sufficient

to determine the LTP eigenvalues, considering a tolerance of

= 0

01%. Compared with the

eigenvalue sorting method, the eigenvector sorting method shows faster convergence speed.

H=2H=2

H=3H=3

H=5H=5

H=50H=50

H=2

H=3

H=5

H=50

Line 1

Line 2

Line 3

Line 4Line 5

Line 6

Line 7

Line 8

H=2

H=3

H=5

H=50

Line 1

Line 2

Line 3

Line 4Line 5

Line 6

Line 7

Line 8

Figure 4.9: Eigenvalue map of

A − N

for different truncation orders

. The selected eigenvalues

in each step of the eigenvector sorting method are marked with circles.

4.4 LTP Eigenvalue Calculation

Adaptive LTP Eigenvalue Computation Methods

Though the eigenvector sorting method exhibits better convergence performance, no rigorous

mathematical proof has been achieved yet, and the impact of changing the truncation order

has not been quantitatively evaluated. To bridge the knowledge gap, two adaptive truncation

order selection methods are developed to pave the way for easier application of the LTP theory.

The first method is inspired by the fact that Eq. (4.30) is difficult to solve but easy to

validate. Assuming that a truncation order of

is selected, a convergence error index for the

ith eigenvalue λiand the corresponding eigenvector

Ri=







r−H







(4.64)

of the truncated matrix A−N is defined as

εi=∥λiri(t)−(A(t)ri(t)−r˙i(t))∥1(4.65)

where

(

)is an

-dimensional time-periodic vector derived from Fourier synthesis of the

Fourier series coefficient vector Ri

ri(t) = ∑︂H

k=−Hrk

iejkω0t.(4.66)

The operator

∥x(t)∥1

gives the

norm of the Fourier series coefficient vector of the time-

periodic signal

(

). The index

εi

quantifies the absolute error between the actual (frequency-

shifted) LTP eigenvalue/eigenvector and their approximation obtained from the truncated

matrix. The eigenvalue/eigenvector whose convergence error index does not exceed a pre-

defined threshold value

is regarded as converged. It can be expected that, with the increasing

of the truncation order

, more eigenvalues converge and form several vertical lines in the

complex plane. The eigenvalue with the smallest convergence error in each line is selected as

LTP eigenvalue, and the Fourier synthesis of the corresponding eigenvector is chosen as LTP

eigenvector. For the

-dimensional matrix

(

), the truncation order

stops increasing until

NLTP eigenvalues are determined.

The same setting given in Table 4.2 is used to demonstrate the convergence process of the

proposed method. Figure 4.10 plots all eigenvalues of

A − N

for different truncation orders.

Converged eigenvalues are indicated by markers with higher visibility, and the selected LTP

eigenvalues (i.e., eigenvalues with the smallest convergence error in each line) are indicated by

circles. Four LTP eigenvalues (marked with red circles) have already converged for

= 5,

while the other ten need a higher truncation order of seven. It is seen that the proposed

adaptive truncation order selection method can deal with the difference between rates of

convergence of LTP eigenvalues. The wrong eigenvalues caused by the truncation effect can be

distinguished by using the proposed convergence error index.

4. Small-Signal Stability Analysis

H=7

H=5

H=3

Figure 4.10: Convergence process of the truncation order selection method

To validate the accuracy of the proposed adaptive truncation order selection method, more

test scenarios are generated by choosing different PLL bandwidths and current controller time

constants. The balanced grid voltage is also considered for comparison. The monodromy

matrix

(

T0,

0) obtained by using MATLAB ode45 function with a simulation time step

MaxStep

0.0001 s

is used as benchmark, considering the relation between LTP eigenvalues

(

)and eigenvalues of

Φ(T0,0)

given by Eq. (4.61). For different combinations of PLL

bandwidth BW and current controller time constant

, the maximum percentage deviation

between eQT0and eigenvalues of Φ(T0,0) is calculated

max {︄

eλiT0−λΦ, i

λΦ, i 

×100% |i= 1,· · · ,14}︄.(4.67)

The result is plotted in Figure 4.11 for a threshold value of

= 0

001. It should be clarified

that

eQT0

and eigenvalues of

(

T0,

0) are sorted in descending order of magnitude before the

calculation of the percentage deviation.

As shown in Figure 4.11, the maximum percentage deviation is smaller than 0.03 %, which

confirms the accuracy of the proposed methodology. Figure 4.12 shows maps of required

truncation orders for the determination of the 14 LTP eigenvalues and eigenvectors. It is seen

that

= 2 is sufficient for the balanced scenario, while higher truncation orders (up to seven)

are needed for the unbalanced scenario, where the required truncation order is predominantly

influenced by the PLL bandwidth.

0.02

0.03

Percentage Devi. (%)

0.02

0.03

Percentage Devi. (%)

0.02

0.03

Percentage Devi. (%)

Figure 4.11: Maximum percentage deviation between

eQT0

and eigenvalues of

(

T0,

0). Left:

Balanced scenario. Right: Unbalanced scenario.

To test the scalability of the proposed method, a second case study is carried out on the

modified IEEE 13-bus system with four Type I grid-following converters shown in Figure 3.30.

4.4 LTP Eigenvalue Calculation

Figure 4.12: Maps of required truncation order for different parameter combinations. Left:

Balanced scenario. Right: Unbalanced scenario.

The LTP small-signal model with 94 states is obtained by using the modeling framework

presented in Chapter 3. Four cases listed in Table 4.3 are considered, and the required

truncation order is determined using the proposed method.

Table 4.3: Parameters of Different Test Cases of the IEEE 13-bus System

Index Grid Voltage BW (Hz) τ(ms) Truncation Order

Case 1 balanced 20 1 3

Case 2 unbalanced with 0.5 pu 30 1 3

negative-sequence component

Case 3 balanced 30 1 7

Case 4 unbalanced with 0.5 pu 40 1 8

negative-sequence component

The agreement between diagonal elements of

eQT0

and eigenvalues of the monodromy

matrix

(

T0,

0) can be observed in Figure 4.13. For all cases, the maximum percentage

deviation is smaller than 0.03%, which validates the correctness of the proposed method. It is

noted that four eigenvalues of

(

T0,

0) of Case 4 locate outside the unit circle, which indicates

that the system is unstable. It is consistent with the time-domain simulation results shown in

Figure 3.34 and Figure 3.35.

The second adaptive LTP eigenvalue calculation method is inspired by the proposed

time-domain physical interpretation in Section 4.2. According to the classical LTI theory,

oscillation modes commonly exhibit different degrees of controllability and observability in

state variables of the LTI system. As for the equivalent representation of the LTP system given

by Eq. (4.23), if

λi

can only be excited and examined in ∆

, it will be one of the eigenvalues of

, namely

λi

is immune to the frequency coupling effect. When the impact of the frequency

coupling becomes not negligible,

λi

can be obtained from the modified selective modal analysis

given in Algorithm 2.

At the core of Algorithm 2 are an iteration aimed at computing the eigenvalue

λi

and an

adaptive truncation order selection method. First, grouping state observations in Eq. (4.23)

into two categories

∆xr= [∆x]

4. Small-Signal Stability Analysis

Case 2

Unit circleUnit circle Case 2

Unit circle

Case 1

Unit circleUnit circle

Case 4Case 4

Unit circleUnit circle Case 4

Unit circle

Case 3Case 3

Unit circleUnit circle Case 3

Unit circle

Case 2

Unit circle

Case 1

Unit circle

Case 4

Unit circle

Case 3

Unit circle

Figure 4.13: Distribution of diagonal elements of

eQT0

(blue circles) and eigenvalues of

(

T0,

(orange crosses).

and

∆xz= [∆x−H;. . . ; ∆x−2; ∆x−1; ∆x1; ∆x2;. . . ; ∆xH]

Correspondingly, Eq. (4.23) can be reformulated as

[︄∆x˙r

∆x˙z]︄=[︄A0A12

A21 A22 ]︄[︄ ∆xr

∆xz]︄(4.68)

The color of the partitioned matrices is consistent with that in Eq. (4.23). At the kth iteration,

∆

x˙z

is approximated by

λk

∆

, where

λk

is the estimation of

λi

at the kth iteration step.

This makes the second row of Eq. (4.68) algebraic, yielding an N-dimensional reduced-order

system [︄∆x˙r

λk

i∆xz]︄=[︄A0A12

A21 A22 ]︄[︄ ∆xr

∆xz]︄

⇓

∆x˙r=



A0+A12(︂λk

iI−A22)︂−1A21

⏞⏟⏟ ⏞





∆xr

(4.69)

The truncation order

, i.e., the number of rotated coordinate frames in ∆

providing

redundant observations of the physical LTP system, is adaptively increased until

converges.

Then, the next estimation

λk+1

λi

is determined from the eigenvalue analysis of Eq. (4.69).

4.4 LTP Eigenvalue Calculation

Algorithm 2

1: Group observations in different coordinate frames:

e.g., ∆xr= [∆x]and ∆xz= [∆x−H;. . . ; ∆x−1; ∆x1;. . . ; ∆xH]

2: Determine initial guess of LTP eigenvalues:

Eigenvalue calculation of A0,{︁λ0

i, i ∈[1, N]}︁

3: for i←1 : Ndo

4: Initialize the iteration index k= 0

5: repeat

6: Increase the truncation order Huntil Gin Eq. (4.69) does not change

7: Compute eigenvalues of A0+G, set λk+1

ias the one closest to λk

8: Update the iteration index k=k+ 1

9: until the value of λk

iconverges

10: end for

Output: converged Neigenvalues and eigenvectors

The local convergence of Algorithm 2 for

λi

is guaranteed when the magnitude of the

coefficient ρidefined as

ρi=liri

li∂G

∂λiri

=liri

−liA12(λiI−A22)−2A21ri

(4.70)

is larger than unity.

ρi

is the ratio between participation factors of

λi

in ∆

and ∆

Considering symmetry, it is also the ratio of participation factors of

λi

and

{λi−jhω0}

in ∆

and

are the N-dimensional left and right eigenvectors of Eq. (4.69), associated with

λi

Actually, the magnitude of

ρi

quantifies the strength of the frequency coupling effect. Smaller

|ρi|

indicates comparable contribution of

λi

and

{λi−jhω0}

in ∆

(

), namely stronger

frequency coupling effect. When

|ρi|

approaches unity, observations in rotated coordinate

frames should be incrementally moved from ∆xzto ∆xrto make the algorithm converge.

Again the Type I grid-following converter with the parameters listed in Table 4.2 is used

to test Algorithm 2. The result is demonstrated in Figure 4.14. Blue crosses indicate the LTP

eigenvalues obtained by using the first adaptive method. Orange dots are the initial guess

of the LTP eigenvalues. Red circles are the converged LTP eigenvalues determined by using

Algorithm 2.

To quantify the impact of the frequency coupling effect on the relevant LTP eigenvalues, the

participation ratio given by Eq. (4.70) and the percentage deviation between the initial guess

and the true LTP eigenvalue are calculated. It is observed from Table 4.4 that comparably

accurate prediction of

λ1−4

and

λ10−14

can be obtained from eigenvalue calculation of

while

λ5,6

λ7,8

and

λ9

are highly influenced by the frequency coupling effect. Observations

in rotated coordinates, ∆

x−2

and ∆

, need to be added into ∆

of Algorithm 2 for the

computation of

λ5−9

. In other words, the contribution of

λ5−9±

jω0

in the dynamics of ∆

is not negligible.

4. Small-Signal Stability Analysis

λ1

λ2

λ3

λ4

λ5

λ6

λ8

λ9

λ7

λ14

λ10

λ11

λ12

λ13

(a) ∆xr= [∆x],λ5,6,λ7,8and λ9cannot be obtained

λ1

λ2

λ3

λ4

λ5

λ6

λ8

λ9

λ7

λ14

λ10

λ11

λ12

λ13

λ1

λ2

λ3

λ4

λ5

λ6

λ8

λ9

λ7

λ14

λ10

λ11

λ12

λ13

(b) ∆xr= [∆x−2; ∆x; ∆x2], all LTP eigenvalues can be determined

Figure 4.14: LTP eigenvalue map of the Type I grid-following converter obtained by using

Algorithm 2 with different groupings of the state observation

Table 4.4: Coupling Effect Quantification of LTP Eigenvalue

Eigenvalue Value Participation Factor Ratio Percentage Deviation (%)

λ1,2−25.23 ±j3.031 6851 0.74

λ3,4−54.14 ±j65.23 48.61 22.1

λ5,6−120.4±j231.0 2.384 N/A

λ7,8−365.9±j216.8 2.313 N/A

λ9−457.8 2.389 N/A

λ10,11 −600.2±j4120 1387 0.07

λ12,13 −730.8±j3727 734.0 0.13

λ14 −1026 42.15 10.8

4.5 Case Study

4.5.1 Grid-Following Converter

Based on the generalized modal analysis method, the dynamic characteristics of the two

types of grid-following converters are investigated and compared in this subsection. As a base

case, the time constant of the current controller is chosen to be

0.5 ms

. A bandwidth of

20 Hz

is designed for the DSRF-PLL and DSOGI-PLL. The grid strength quantified

by the short-circuit ratio is initially set to be

SCR

= 3

6. These three variables build the

parameter space for the stability investigation.

LTP Eigenvalue-Based Stability Analysis

Figure 4.15 shows the eigenvalue map of the two different grid-following converter systems

operated under balanced conditions. Considering that oscillations related to left half-plane

eigenvalues far away from the imaginary axis decay rapidly and do not threaten the system

4.5 Case Study

stability, only the three pairs of complex conjugate LTP eigenvalues with real parts larger

than

−200 rad/s

are considered. Participation factors of those six eigenvalues are shown in

Figure 4.16. Participation factors of state variables belonging to the same control or physical

unit (PLL, current controller as well as filter and grid) are summarized.

The participation factor analysis reveals that, under weak grid conditions, the system

stability is influenced by the strong interaction between dynamics of the PLL, current control

and power networks, instead of being dominated by one single unit. This highlights the necessity

of full-order-model-based analysis. Moreover, the Type II grid-following converter is immune

to the frequency coupling effect. In other words, there exists a rotational transformation with

fixed frequency which can transform the LTP small-signal model of the Type II grid-following

converter to an LTI one. In contrast, the small-signal model of the Type I grid-following

converter is time periodic, this is an intrinsic difference between them.

λ1

λ2

λ3

λ4

λ5

λ6

(a) Type I grid-following converter

λ1

λ2

λ3

λ4

λ5

λ6

(b) Type II grid-following converter

Figure 4.15: LTP eigenvalue map of the grid-following converter

The small-signal analysis results given by Figure 4.15 and Figure 4.16 are only valid in the

vicinity of the steady-state operation trajectory determined by the initial parameters. To gain

a global overview of the impact of the interaction between the PLL, the current control and

the network on the system stability, the PLL bandwidth BW is swept from

20 Hz

50 Hz

, the

current control time constant

from

0.5 ms

3.5 ms

, and SCR from 3 to 2. As summarized

in Table 4.5, following conclusions can be drawn from the movement of eigenvalues shown in

Figure 4.17 and Figure 4.18:

As the SCR decreases from 3 to 2,

λ3,4

of both types of converters approximate the

imaginary axis from the left side (see Zoom 2 in Figure 4.17(a) and Figure 4.17(b)), i.e.,

the system stability margin becomes smaller, while

λ1,2

and

λ5,6

are comparably less

influenced.

4. Small-Signal Stability Analysis

(a) Type I grid-following converter

(b) Type II grid-following converter

Figure 4.16: Magnitude of participation factors of the grid-following converter

Slower current controllers make

λ3,4

of both types of converters migrate towards the right-

half plane (see Figure 4.18), and even makes the Type I grid-following converter unstable

(see Zoom 2 in Figure 4.18(a)). Additionally, larger

can result in poorly damped

low-frequency (1-2 Hz) oscillations of Type II converter, indicated by the movement of

λ1,2shown in Zoom 1 in Figure 4.18(b).

The PLL bandwidth BW has large impact on

λ3,4

and

λ5,6

. Specifically, a higher BW

can help to counteract the impact of weaker grids and slower current controls on

λ3,4

(see

Zoom 2 in Figure 4.17 and Figure 4.18). However, increasing BW moves

λ5,6

towards

the right-half plane (see Zoom 3 in Figure 4.17 and Figure 4.18), and can even lead to

the instability of the Type I grid-following converter.

Table 4.5: Impact of Parameter Changes on the System Stability

Eigenvalue Oscillation Frequency Type I Grid- Type II Grid-

Following Converter Following Converter

λ1,2<4 Hz -τ(→)

λ3,46∼20 Hz

SCR (←) SCR (←)

τ(→,instability)τ(→)

BW (←) BW (←)

λ5,6about 50 Hz BW (→,instability) BW(→)

Arrows in Table 4.5 indicate the direction of movement of eigenvalues, as corresponding

parameters increase.

Influence of the Unbalanced Operation

The influence of the grid imbalance is investigated by considering following two scenarios:

4.5 Case Study

Zoom 1Zoom 1

Zoom 1Zoom 2

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Zoom 2Zoom 2

Zoom 1

Zoom 1Zoom 2

Zoom 3

Zoom 2 SCR=2.8SCR=2.8

SCR=2.2SCR=2.2

SCR=3.0SCR=3.0

SCR=2.4SCR=2.4

SCR=2.6SCR=2.6

SCR=2.0SCR=2.0

SCR=2.8

SCR=2.2

SCR=3.0

SCR=2.4

SCR=2.6

SCR=2.0

BW (Hz)

SCR=2.8

SCR=2.2

SCR=3.0

SCR=2.4

SCR=2.6

SCR=2.0

BW (Hz)

Zoom 1

Zoom 1Zoom 2

Zoom 3

Zoom 2 SCR=2.8

SCR=2.2

SCR=3.0

SCR=2.4

SCR=2.6

SCR=2.0

BW (Hz)

λ1,2

λ3 λ5

Zoom 1

Zoom 1Zoom 2

Zoom 3

Zoom 2 SCR=2.8

SCR=2.2

SCR=3.0

SCR=2.4

SCR=2.6

SCR=2.0

BW (Hz)

λ1,2

λ3 λ5

(a) Type I grid-following converter

Zoom 1

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Zoom 1

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Zoom 3

Zoom 1Zoom 1

Zoom 2Zoom 2 Zoom 3Zoom 3

Zoom 1

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Zoom 3

Zoom 1

Zoom 2 Zoom 3

SCR=2.8SCR=2.8

SCR=2.2SCR=2.2

SCR=3.0SCR=3.0

SCR=2.4SCR=2.4

SCR=2.6SCR=2.6

SCR=2.0SCR=2.0

SCR=2.8

SCR=2.2

SCR=3.0

SCR=2.4

SCR=2.6

SCR=2.0

BW (Hz)

SCR=2.8

SCR=2.2

SCR=3.0

SCR=2.4

SCR=2.6

SCR=2.0

BW (Hz)

λ1

λ2

λ3 λ5

Zoom 1

Zoom 2

Zoom 3

Zoom 1

Zoom 2 Zoom 3

SCR=2.8

SCR=2.2

SCR=3.0

SCR=2.4

SCR=2.6

SCR=2.0

BW (Hz)

λ1

λ2

λ3 λ5

(b) Type II grid-following converter

Figure 4.17: Eigenvalue locus of the grid-following converter for different combinations of SCR

and BW

Unbalanced stiff grid voltage: A negative-sequence voltage is assumed to exist in the

grid, and the ratio

Vg−/Vg+

is swept in the range of 0 to 1, where

Vg+

and

Vg−

refer to magnitudes of the positive and negative sequence grid voltages.

Unbalanced grid impedance: The inductance of phase a of the grid line impedance is

increased by a multiple kZvarying from 1 to 3.

4. Small-Signal Stability Analysis

Zoom 2 Zoom 3

Zoom 1

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Zoom 3

Zoom 2

BW (Hz)

Ti=1.0msTi=1.0ms

Ti=2.5msTi=2.5ms

Ti=0.5msTi=0.5ms

Ti=2.0msTi=2.0ms

Ti=1.5msTi=1.5ms

Ti=3.0msTi=3.0ms

Ti=1.0ms

Ti=2.5ms

Ti=0.5ms

Ti=2.0ms

Ti=1.5ms

Ti=3.0ms

BW (Hz)

Ti=1.0ms

Ti=2.5ms

Ti=0.5ms

Ti=2.0ms

Ti=1.5ms

Ti=3.0ms

Zoom 2 Zoom 3

Zoom 1

Zoom 3

Zoom 2

BW (Hz)

Ti=1.0ms

Ti=2.5ms

Ti=0.5ms

Ti=2.0ms

Ti=1.5ms

Ti=3.0ms

λ1,2

λ3

λ5

Zoom 2 Zoom 3

Zoom 1

Zoom 3

Zoom 2

BW (Hz)

Ti=1.0ms

Ti=2.5ms

Ti=0.5ms

Ti=2.0ms

Ti=1.5ms

Ti=3.0ms

λ1,2

λ3

λ5

(a) Type I grid-following converter

Ti increase

Zoom 2 Zoom 3

Zoom 1

Zoom 3

Zoom 2

Ti increase

Zoom 2 Zoom 3

Zoom 1

Zoom 3

Zoom 2

BW (Hz)

Ti=1.0msTi=1.0ms

Ti=2.5msTi=2.5ms

Ti=0.5msTi=0.5ms

Ti=2.0msTi=2.0ms

Ti=1.5msTi=1.5ms

Ti=3.0msTi=3.0ms

Ti=1.0ms

Ti=2.5ms

Ti=0.5ms

Ti=2.0ms

Ti=1.5ms

Ti=3.0ms

BW (Hz)

Ti=1.0ms

Ti=2.5ms

Ti=0.5ms

Ti=2.0ms

Ti=1.5ms

Ti=3.0ms

Ti increase

Zoom 2 Zoom 3

Zoom 1

Zoom 3

Zoom 2

BW (Hz)

Ti=1.0ms

Ti=2.5ms

Ti=0.5ms

Ti=2.0ms

Ti=1.5ms

Ti=3.0ms

λ1

λ2

λ3

λ5

Ti increase

Zoom 2 Zoom 3

Zoom 1

Zoom 3

Zoom 2

BW (Hz)

Ti=1.0ms

Ti=2.5ms

Ti=0.5ms

Ti=2.0ms

Ti=1.5ms

Ti=3.0ms

λ1

λ2

λ3

λ5

(b) Type II grid-following converter

Figure 4.18: Eigenvalue locus of the grid-following converter for different combinations of

and

In the presence of the grid imbalance, small-signal models of both types of converters

become LTP. Figure 4.19 presents the eigenvalue locus with the varying parameter

. It

is observed that the increasing of

causes the bifurcation of

λ5,6

, which threatens the

stability of the Type I grid-following converter with higher PLL bandwidth.

λ1,2

and

λ3,4

are immune to the presence of negative sequence grid voltage. The parameter sweep result

is demonstrated in Figure 4.20.

λ1,2

and

λ5,6

remain almost unchanged, while

λ3,4

4.5 Case Study

travel towards the right-half plane as

increases, which is similar to the eigenvalue movement

pattern of reducing the SCR shown in Figure 4.17.

kV increasekV increase kV increasekV increase

kV increase kV increase

Figure 4.19: Eigenvalue locus for

30 Hz

and different

. Left: Type I grid-following

converter. Right: Type II grid-following converter

kZ increasekZ increase kZ increasekZ increase

kZ increase kZ increase

Figure 4.20: Eigenvalue locus for different

. Left: Type I grid-following converter. Right:

Type II grid-following converter

The real-life grid imbalance caused by faults or asymmetric loads can be treated as the

combination of voltage and impedance imbalance. Systematic investigations on the practical

imbalanced scenarios considering also the dual-sequence current control [116] deserve further

investigation, but are out of the scope of this thesis.

Comparison between LTI and LTP Stability Analysis Results

To further confirm the necessity of the application of the LTP theory, the LTI models developed

in [29, 31, 32] which neglect the angle and frequency feedback dynamics of the DSRF-PLL

and DSOGI-PLL are used for comparison in a stability analysis.

For two different grid strengths,

SCR

= 3 and

SCR

= 5, eigenvalues of both types of

grid-following converters are calculated for various combinations of PLL bandwidth

and current control time constant

. Real parts of the right-most eigenvalues are plotted in

Figure 4.21 and Figure 4.22. It can be observed that neglecting the phase angle feedback of the

DSRF-PLL can result in wrong stability assessment results, neglecting the frequency feedback

of the DSOGI-PLL gives overoptimistic stability assessment results. Additionally, only the

LTP models can be used for the stability analysis under unbalanced conditions.

4. Small-Signal Stability Analysis

Unstable Region

(a) SCR=5, balanced grid

-60 -50 -40 -30 -20 -10 0

Real part of the right-most eigenvalues

(b) SCR=3, balanced grid

Figure 4.21: Real part of the right-most

eigenvalues of the Type I grid-following con-

verter for different parameter combinations.

White areas are the unstable region. Dashed

black lines indicate stability boundaries deter-

mined based on conventional LTI models.

Unstable Region

-60 -50 -40 -30 -20 -10 0

Real part of the right-most eigenvalues

(a) SCR=5, balanced grid (b) SCR=3, balanced grid

Figure 4.22: Real part of the right-most

eigenvalues of the Type II grid-following con-

verter for different parameter combinations.

White areas are the unstable region. Dashed

black lines indicate stability boundaries deter-

mined based on conventional LTI models.

Time-Domain Validation

To validate the eigenvalue-based stability analysis results, time-domain dynamic responses

obtained from the analytical small-signal model, the numerical nonlinear average model and

experimental measurements are compared. After the system reaches steady state, a power

reference step of

−200 W

is given to the converter. Forced responses of ∆

, real parts of the

grid-side current and the capacitor voltage for different control parameter combinations are

shown in Figure 4.23 to Figure 4.25. The consistency between dynamic responses obtained

from the nonlinear average MATLAB/Simulink model, the analytical small-signal model and

experimental measurements confirms the accuracy of the linear small-signal model. For the

base case (

0.5 ms

and

20 Hz

), the two types of converters demonstrate similar

dynamic characteristics, as shown in Figure 4.23 and Figure 4.25. As

increases to

3 ms

the Type I GFL converter becomes unstable (see Figure 4.26). In contrast, the Type II grid-

following converter remains stable, and the low-frequency oscillation corresponding to

λ1,2

can be observed in Figure 4.24.

To verify the instability caused by a higher DSRF-PLL bandwidth, the bandwidth BW

of the Type I grid-following converter is increased from

20 Hz

35 Hz

1.0 s

, the

exponentially amplified evolution of and the real part of the grid-side current can be observed

in Figure 4.27.

4.5 Case Study

Figure 4.23: Forced responses of the Type II GFL converter to a

−200 W

power reference step at

1.0 s

for

20 Hz

and

0.5 ms

. Grey lines: measurements. Solid lines:nonlinear average

model. Dashed lines:small-signal model

Figure 4.24: Forced responses of the Type II GFL converter to a

−200 W

power reference step at

1.0 s

for

20 Hz

and

3 ms

. Grey lines: measurements. Solid lines:nonlinear average

model. Dashed lines:small-signal model

4.5.2 Grid-Forming Converter

In this subsection, the eigenvalue-based stability analysis is carried out for the single grid-

forming converter system shown in Figure 3.11. The control and physical parameters of the

base case are given in Table 4.6. The 16 eigenvalues shown in Figure 4.28 can be generally

divided into four groups according to their locations in the complex plane. Participation

factors of the twelve oscillation modes are given in Figure 4.29. The other four over damped

modes do not threaten the system stability.

It can be observed from Figure 4.29 that the high-frequency eigenvalues in Group IV are

related to the LCL resonance. Three pairs of oscillations modes exist in Group I.

λ1,2

merely related to the outer power loop.

λ3,4

and

λ5,6

mainly contribute to the dynamics of

the cascaded voltage-current loop. Moreover,

λ7,8

in Group II is influenced by the interaction

between the inner control loop and the LCL passive components.

To study the impact of parameter changes on the system stability, the inertia constant

droop coefficients

and

, the virtual impedance

jω0Lv

, the current controller

4. Small-Signal Stability Analysis

Figure 4.25: Forced responses of the Type I GFL converter to a

−200 W

power reference step at

1.0 s

for

20 Hz

and

0.5 ms

. Grey lines: measurements. Solid lines:nonlinear average

model. Dashed lines:small-signal model

Table 4.6: Base Case Parameters of the GFM Converter

Symbol Description Value

SCR Short-circuit ratio 5

SrPower reference 1 kW

kPActive power droop coefficient 320 W ·s/rad

kQReactive power droop coefficient 207 Var/V

JInertia constant 0.132 kg ·m2

τCurrent control time constant 0.5 ms

PM Desired phase margin of the voltage control 45°

ZvVirtual impedance 5.6 mH + 0.1 Ω

time constant

and the desired voltage control phase margin

are changed in the following

ranges:

•J

is swept from

0.08 kg ·m2

0.41 kg ·m2

, corresponding to the inertia time constant

Hchanges from 2 s to 9.5 s.

•kP

is swept from 127

W·s/rad

to 637

W·s/rad

, which means that 100% active power

change corresponds to the change from 5% to 1% of the nominal frequency.

•kQ

is swept from 200

Var/V

to 2000

Var/V

, which means that 100% reactive power

change corresponds to the change from 10% to 1% of the nominal voltage.

•SCR is swept from 2 to 17.

•Zvis swept from 20% to 100% of the converter-side filter inductance.

•τis swept from 0.5 ms to 3 ms.

•PM is swept from 30°to 60°.

Figure 4.30 and Figure 4.31 show the eigenvalue loci for different combinations of the

inertia constant

and the droop coefficients

and

. It can be observed that

λ1,2

migrate

towards the right half plane with decreasing oscillation frequency and damping ratio as the

4.5 Case Study

Ti changes from 0.5 to 3ms

Figure 4.26: Measurements of the Type I

grid-following converter for the increase of τ

BW changes from 20 to 35Hz

Figure 4.27: Measurements of the Type I

grid-following converter for the increase of BW

λ1

λ2

λ3

λ4

λ5

λ6

λ9

λ10

λ11

λ12

λ7

λ8

Figure 4.28: LTP eigenvalue map of the grid-forming converter

Figure 4.29: Magnitude of participation factors of the grid-forming converter

inertia constant

increases. Large active power droop coefficient

can improve the damping

performance of those modes. The impact of kQon λ1,2is negligible.

Figure 4.32 shows the eigenvalue loci for different combinations of SCR and virtual

impedance

. As SCR increases (the grid becomes stronger),

λ1,2

migrate towards the

right half plane with increasing oscillation frequency and decreasing damping ratio. To ensure

the system stability, large virtual impedances should be selected. Moreover, the increasing of

the SCR also makes

λ3,4

approach the imaginary axis from the left side, which degrades the

damping performance of the system.

4. Small-Signal Stability Analysis

λ1

λ2

J increase

kP (W s/rad)

600100 H=3.5s

H=2.0s

H=6.5s

H=5.0s H=8.0s

H=9.5s

Figure 4.30: Eigenvalue locus of the GFM converter for different combinations of kPand J

kQ (Var/V)

2000200 H=3.5s

H=2.0s

H=6.5s

H=5.0s H=8.0s

H=9.5s

J increase

kQ increase

λ1

λ2

λ5

λ6

λ3

λ4

Figure 4.31: Eigenvalue locus of the GFM converter for different combinations of kQand J

λ1

λ2

SCR increase

λ7

λ8

λ3

λ4

λ5

λ6

SCR increase

Zv increase

(Zv/R+jω0L )

10.2 SCR=5

SCR=2

SCR=11

SCR=8 SCR=14

SCR=17

Figure 4.32: Eigenvalue locus of the GFM converter for different combinations of SCR and Zv

Figure 4.33 shows the movement of the eigenvalues for different inner loop control

parameters. It is seen that the inner loop can interact with the low-frequency power loop

mode

λ1,2

as the inner current control loop becomes slower. Fast current control is desired to

guarantee the system stability and damping of the high-frequency oscillation modes λ9−12.

4.5 Case Study

λ1

λ2

λ5

λ6

λ3

λ4

λ9

λ10

λ11

λ12

Current Control Time Constant τ (ms)

30.5 PM=40⁰

PM=35⁰

PM=50⁰

PM=45⁰ PM=55⁰

PM=60⁰

Figure 4.33: Eigenvalue locus of the GFM converter for different combinations of τand PM

Time-Domain Validation

To verify the eigenvalue-based stability analysis results, time-domain forced responses have

been performed by applying

200 W

active power reference step to the VSG converter for

different control parameters. Figure 4.34 - Figure 4.36 show the forced responses for different

outer power loop control parameters. It is seen that the dynamic evolution becomes more

oscillatory as the inertia constant increases. Larger active power droop coefficient is beneficial

for improving the damping, while the influence of the reactive power droop coefficient is

negligible. Moreover, it is seen from Figure 4.37 and Figure 4.38 that reducing the virtual

impedance can cause the instability. The negative impact of a slower inner current controller

on the system stability can be confirmed by the dynamic responses shown in Figure 4.39.

J=0.132 kg m2

J=0.264 kg m2

J=0.396 kg m2

J=0.528 kg m2

(a) Active Power

J=0.132 kg m2

J=0.264 kg m2

J=0.396 kg m2

J=0.528 kg m2

(b) Frequency

Figure 4.34: Forced responses of the VSG converter for different inertia constants

4. Small-Signal Stability Analysis

kP=320 W s/rad

kP=400 W s/rad

kP=500 W s/rad

kP=600 W s/rad

(a) Active Power

kP=320 W s/rad

kP=400 W s/rad

kP=500 W s/rad

kP=600 W s/rad

(b) Frequency

Figure 4.35: Forced responses of the VSG converter for

0.264 kg ·m2

different active power

droop coefficients

kQ=207 Var/V

kQ=500 Var/V

kQ=800 Var/V

kQ=1000 Var/V

(a) Active Power

kQ=207 Var/V

kQ=500 Var/V

kQ=800 Var/V

kQ=1000 Var/V

(b) Frequency

Figure 4.36: Forced responses of the VSG converter for

0.264 kg ·m2

and different reactive

power droop coefficients

4.6 Summary

This chapter generalizes the classical modal analysis to the linear time-periodic systems by

deriving the analytical free-response solution. The proposed indices, including damping ratio,

4.6 Summary

Lv=11.2 mH

Lv=7.8 mH

Lv=5.6 mH

Lv=3 mH

(a) Active Power

Lv=11.2 mH

Lv=7.8 mH

Lv=5.6 mH

Lv=3 mH

(b) Frequency

Figure 4.37: Forced responses of the VSG converter for different virtual impedances

participation factor and eigenvalue sensitivity, can provide insightful description and assessment

of dynamic performance of LTP systems. Two iterative LTP eigenvalue calculation algorithms

are developed to balance the trade-off between computational accuracy and efficiency. The

participation factor ratio is defined as a measure of the strength of the frequency coupling

effect, which answers when the LTP theory must be applied. Based on the proposed LTP

modal analysis, the stability of both grid-following and grid-forming converters are examined,

and the major findings are:

•

Higher bandwidth of DSRF-PLL and DSOGI-PLL can counteract the negative impact

of weaker grids and slower current controls on the system stability. However, it brings

poorly damped or even amplified oscillations around the fundamental frequency.

•

The Type II grid-following converter (DSOGI-PLL plus PR current control) has a larger

stability margin and is more robust than Type I GFL converter (DSRF-PLL plus PI

current control) against grid voltage imbalances.

•

Under strong grid conditions, the grid-forming converter with inner cascaded voltage-

current control can encounter instability problems. A large virtual impedance needs

to be used to ensure the system stability. Moreover, fast current control is generally

desired to guarantee the damping of high-frequency oscillation modes and weaken the

interaction between outer and inner loops.

All theoretical analysis has been validated by simulation results and experimental tests.

4. Small-Signal Stability Analysis

Lv changes from 3.0 to 1.5 mH

Overcurrent protection

Figure 4.38: Instability of the VSG converter

caused by small virtual impedance

τ changes from 0.5 to 1.5 ms

Figure 4.39: Instability of the VSG converter

caused by slow inner current control

Harmonic Resonance Analysis

This chapter deals with the harmonic resonance investigation of the converter-dominated power

systems considering unbalanced operations. Initially, the closed-form analytical solution of

the forced response of the LTP system is deduced. According to this, the impedance model

of the grid-following converter and the unbalanced network are derived to fully capture the

frequency coupling effect. Then, the LTP resonance mode analysis is developed by describing

the system with a time-periodic impedance matrix to identify the resonance frequencies. Based

on the eigenanalysis of the time-periodic impedance matrix, the definition of the participation

factor is modified to determine the propagation areas and corresponding severity of a certain

resonance. In addition, the impact of the converter controller and the grid imbalance on the

resonance characteristics are evaluated by using a sensitivity analysis method. The proposed

methodology is tested on an exemplary multiple-converter system.

5.1 Forced Response of the LTP System

As discussed in Section 4.3, the LTP eigenvalues of the time-periodic system matrix

(

)

provide the intrinsic stability characteristic of the LTP system. For the resonance analysis, we

are interested in the terminal equivalent impedance or admittance of converters, namely the

steady-state transfer functions between voltages and currents. To obtain the relevant transfer

functions, the forced response of the N-dimensional LTP system

∆x˙=A(t) ∆x+B(t) ∆u(5.1a)

∆y=C(t) ∆x+D(t) ∆u(5.1b)

is investigated in this section.

Based on the generalized modal analysis explained in Section 4.3, the forced response of

the LTP system can be derived following two steps shown in Figure 5.1:

In the first step, the physical input ∆

is transformed into an LTI modal space, and the

steady-state response of state variables in the modal space, namely ∆z, is derived.

5. Harmonic Resonance Analysis

In the second step, ∆

is transformed back into the physical space to obtain the physical

output ∆y.

STEP 1

Applying the space transformation described with the LTP eigenvector matrix

(

), Eq. (5.1a)

is transformed into the modal space

∆x˙=A(t) ∆x+B(t) ∆u

⇓∆x=R(t)∆z

∆z˙=Q∆z+f(t) ∆u=R−1(t)(︂A(t)R(t)−R

˙(t))︂

⏞⏟⏟ ⏞

∆z+R−1(t)B(t)

⏞⏟⏟ ⏞

f(t)

∆u

.(5.2)

In the modal space, Eq. (5.2) describes an LTI system, since the diagonal LTP eigenvalue

matrix Qis constant.

Without loss of generality, let the physical input ∆

contain a single component at the

frequency of ω, namely

∆u=Uejωt (5.3)

where

is a constant vector, which gives the magnitude and phase angle of the input vector

∆u. The time-periodic matrix

f(t) = ∑︂∞

p=−∞ fpejpω0t

transforms the single-frequency physical input ∆

to a multiple-frequency input in the modal

space

∆umodal =f(t) ∆u=∑︂∞

p=−∞ fpUej(ω+pω0)t.(5.4)

The Fourier series coefficient fpis associated with the pth order harmonic of f(t).

H2,0( jω )

LTI State-Space Model in the Modal Space

∆z = Q∆z + ∆umodal

∆Z s = sI - Q -1∆Umodal s

Δz

ω ω +ω0ω +2ω0ω +3ω0

ΔyΔu

ω ω +ω0ω +2ω0ω +3ω0

Physical Space

Modal Space

Δumodal

f t = f pejpω0t

∞

p=-∞

g t = gqe

qω0t

∞

q=-∞

STEP 1 STEP 2

H2,0( jω )U

Δumodal

ω ω +ω0ω +2ω0ω +3ω0

Figure 5.1: Qualitative forced response of the LTP system

100

5.1 Forced Response of the LTP System

In the LTI modal space, the steady-state relation between ∆

and ∆

umodal

can be easily

derived

∆Z(s)=(sI−Q)−1∆Umodal (s)(5.5)

where ∆

Z(s)

and ∆

Umodal (s)

are the Laplace transformation of ∆

and ∆

umodal

stands

for the N-dimensional identity matrix. To obtain the steady-state response of ∆

to ∆

Eq. (5.4) needs to be inserted into Eq. (5.5), and yields

∆z=∑︂∞

p=−∞ (j(ω+pω0)I−Q)−1fpUej(ω+pω0)t(5.6)

STEP 2

In STEP 1, the steady-state response of the state vector ∆

in the modal space to the input

vector ∆

in the physical space is deduced. STEP 2 aims to further derive the output vector

∆yin the physical space based on Eq. (5.1b) and the transformation ∆x=R(t)∆z.

Since both

C(t)

and

R(t)

are time-periodic matrices, their product defined as

g(t)

will

also be time-periodic, which has the Fourier series expansion

g(t) = C(t)R(t) = ∑︂∞

q=−∞ gqejqω0t.(5.7)

Then, the steady-state forced response ∆

in Eq. (5.1b) can be obtained with the inverse

transformation from the modal space to the physical space

∆y=C(t) ∆x=C(t)R(t) ∆z=g(t) ∆z

=∑︂∞

q=−∞ gqejqω0t∑︂∞

p=−∞ (j(ω+pω0)I−Q)−1fpUej(ω+pω0)t

⏞⏟⏟ ⏞

∆z

=∑︂∞

q=−∞ (︃∑︂∞

p=−∞ gp(j(ω+ (q−p)ω0)I−Q)−1f(q−p))︃

⏞⏟⏟ ⏞

Hq,0(jω)

Uej(ω+qω0)t

=∑︂∞

q=−∞ Hq,0(jω)Uej(ω+qω0)t

=∑︂∞

q=−∞ Yq(jω)ej(ω+qω0)t

(5.8)

The zero output matrix D(t)in Eq. (5.1b) is dropped, since, for the LTP small-signal model

of the converter-dominated power systems, the output variables are part of the state variables.

Equation (5.8) reveals the unique characteristic of the LTP system, that the input ∆

at a

single frequency

will excite the steady-state responses of ∆

with multiple components at the

frequencies of

{ω+qω0, q ∈Z}

. The transfer function

Hq,0(jω)

maps the input ∆

Uejωt

to the harmonic component Yqej(ω+qω0)tin the output ∆y, namely

Yq(jω)ej(ω+qω0)t=Hq,0(jω)Uej(ω+qω0)t=Hq,0(jω)Uejωt (5.9)

101

5. Harmonic Resonance Analysis

Discussion on Transfer function of LTP System

In Eq. (5.8), the inverse calculation of the diagonal matrix can be easily obtained

(j(ω+ (q−p)ω0)I−Q)−1=





j(ω+(q−p)ω0)−λ1

j(ω+(q−p)ω0)−λ2

...

j(ω+(q−p)ω0)−λN







Then, Hq,0(jω)can be expressed as the superposition of the NLTP eigenvalues/modes

Hq,0(jω) = ∑︂N

i=1 (︃∑︂∞

p=−∞

j(ω+ (q−p)ω0)−λi

gipfiq−p)︃.(5.10)

It gives that the transfer functions of the LTP system consist of Nclusters of first-order

transfer functions, and the ith cluster is

{︃1

j(ω+ (q−p)ω0)−λi

, q, p ∈Z}︃.(5.11)

The residue of 1

j(ω+ (q−p)ω0)−λi

(5.12)

ifq−p

, where

is the ith column of

, and

fq−p

is the ith row of

fq−p

. The LTI system

can be regarded as a special case of the LTP system, where

if0

is the only non-zero residue.

The input and output of the classical LTI system share the same frequency. However, for

the LTP systems, it can be concluded from Eq. (5.8) that the only possibility to make the

input ∆

and the output ∆

of the LTP system share the same harmonic space is to set the

input as

∆u=∑︂∞

p=−∞ Upej(ω+pω0)t(5.13)

which contains infinite harmonic components, of which the frequencies are differing by integer

multiples of ω0. Then, the spectra of ∆uand the output

∆y=∑︂∞

q=−∞ Yqej(ω+qω0)t

are linked by







Y−1

Y+1













....

...

· · · H−1,−1(jω)H−1,0(jω)H−1,+1 (jω)· · ·

· · · H0,−1(jω)H0,0(jω)H0,+1 (jω)· · ·

· · · H+1,−1(jω)H+1,0(jω)H+1,+1 (jω)· · ·

...

....







⏞ ⏟⏟ ⏞

HHT M (jω)







U−1

U+1







.(5.14)

The elementary transfer function matrix Hq,p (jω)in HHT M (jω)gives the relation

Yq=Hq,p (jω)Up.(5.15)

102

5.2 Impedance Modeling of Converter-Dominated Power Systems

Actually, the information stored in the transfer function matrix

HHT M

(

jω

)is highly

redundant. To illustrate this, similar to Eq. (5.3), the input ∆

is again assumed to contain a

single component while the frequency is shifted to ω+hω0, namely

∆u=Uej(ω+hω0)t.(5.16)

Then, the steady-state output can be derived by replacing ωwith ω+hω0in Eq. (5.8)

∆y=∑︂∞

q=−∞ (︃∑︂∞

p=−∞ gp(j(ω+ (q−p+h)ω0)I−Q)−1fq−p)︃

⏞⏟⏟ ⏞

Hq+h,h(jω)

Uej(ω+(q+h)ω0)t(5.17)

where the transfer function

Hq+h,h (jω)

maps the input ∆

Uej(ω+hω0)t

to the output

component Yq+hej(ω+(q+h)ω0)tand satisfies the relation

Hq+h,h (jω) = Hq,0(j(ω+hω0)) (5.18)

It can be concluded that transfer functions on diagonals of

HHT M

(

jω

)are frequency-shifted

versions of each other. Knowing one column or one row of

HHT M

(

jω

)is sufficient for the

determination of the harmonic transfer relation.

Formally

HHT M

(

jω

)in Eq. (5.14) is the same as that in Eq. (2.5) obtained by applying

the harmonic balance principle. The vital contribution of this section is that the analytical

closed-form expression of

HHT M

(

jω

)is deduced by using the generalized modal analysis, and

the impractical inverse calculation of the infinite transfer function matrix is avoided.

5.2

Impedance Modeling of Converter-Dominated Power Sys-

tems

5.2.1 Impedance Modeling of the Voltage Source Converters

The LTP state-space model of the grid-following and grid-forming converters have been

derived in Section 3.2. Based on the fundamentals introduced in Section 5.1, the equivalent

admittance/impedance of the single-converter system observed at the grid connection point is

derived in this subsection.

Assuming that the voltage at the grid connection point ∆vαβg contains infinite harmonic

components differing by integer multiples of ω0

∆vαβg =∑︂∞

p=−∞ Vp

αβgej(ω+pω0)t

∆v∗

αβg =∑︂∞

p=−∞ (︂Vp

αβg)︂∗ej(−ω−pω0)t,

the steady-state response of the grid-side inductor current ∆iαβg will have the same form

∆iαβg =∑︂∞

p=−∞ Ip

αβgej(ω+pω0)t

∆i∗

αβg =∑︂∞

p=−∞ (︂Ip

αβg)︂∗ej(−ω−pω0)t.

103

5. Harmonic Resonance Analysis

Defining the voltage and current Fourier series coefficient vectors

V=

· · · V−1

αβg V0

αβg V1

αβg · · ·

⏞⏟⏟ ⏞

original coordinate

· · · (︂V1

αβg)︂∗(︂V0

αβg)︂∗(︂V−1

αβg)︂∗· · ·

⏞⏟⏟ ⏞

conjugate coordinate 



I=

· · · I−1

αβg I0

αβg I1

αβg · · ·

⏞⏟⏟ ⏞

original coordinate

· · · (︂I1

αβg)︂∗(︂I0

αβg)︂∗(︂I−1

αβg)︂∗· · ·

⏞⏟⏟ ⏞

conjugate coordinate 



the harmonic transfer matrix of the single-converter system can be defined as

I=[︄HO,O (jω)HO,C (jω)

HC,O (jω)HC,C (jω)]︄

⏞⏟⏟ ⏞

HV SC (jω)

V.(5.19)

HV SC (jω)

is the equivalent admittance of the single-converter system observed at the grid

connected point. It is divided into four blocks, i.e.,

HO,O

(

jω

HO,C

(

jω

HC,O

(

jω

)and

HC,C

(

jω

). The subscript

and

indicate the original and conjugate coordinates.

HO,C

(

jω

)

and

HC,O

(

jω

)reflect the coupling between the original and conjugate coordinates caused by

the asymmetric control of the PLLs in the grid-following converter and the power control loop

in the grid-forming converter.

Theoretically the four blocks in

HV SC (jω)

are all infinite-dimensional matrices, for

instance, HO,C(jω)is given by

HO,C (jω) =







....

...

· · · H−1,−1

O,C (jω)H−1,0

O,C (jω)H−1,1

O,C (jω)· · ·

· · · H0,−1

O,C (jω)H0,0

O,C (jω)H0,1

O,C (jω)· · ·

· · · H1,−1

O,C (jω)H1,0

O,C (jω)H1,1

O,C (jω)· · ·

...

....







where Hq, p

O,C (−jω)maps the input voltage harmonic

(︂V−p

αβg)︂∗ej(−ω+pω0)t

in the conjugate coordinate to the output current harmonic

αβgej(−ω+qω0)t

in the original coordinate. Specifically, let the input voltage contain only

1 V

positive-sequence

component at the frequency ω. Except for

∆vαβg = 1ejωt,

the output current is also excited by

∆v∗

αβg = 1e−jωt.

104

5.2 Impedance Modeling of Converter-Dominated Power Systems

This is defined as the sequence coupling effect. Additionally, due to the time-periodic

operation trajectory, the current response contains multiple harmonics at the frequencies of

{ω+qω0, q ∈Z}and {−ω+qω0, q ∈Z}, quantified by

{︂Hn, 0

O,O(jω), n ∈Z}︂and {︂Hn, 0

O,C(−jω), n ∈Z}︂,

respectively. This is defined as the frequency shift effect.

Table 5.1: Parameters of the Single Type I Grid-Following Converter System

Symbol Description Value

L/R Converter-side filter inductance 5.6 mH /0.1 Ω

CFilter capacitor 16 µF

BW PLL bandwidth 25 Hz

τTime constant of the current control 1 ms

Lg/ RgGrid-side line impedance 1.5 mH /0.001 Ω

SrPower reference 2 kW

vαβg Voltage at the grid connection point (100ejω0t+ 50e−jω0t)V

+10% τ Base Case +10% BW +10% C+10% L1+10% L2

HO,O

0,0 (jω)

HO,O

2,0 (jω)

HO,C

0,0 (jω)

HO,C

2,0 (jω)

Figure 5.2: Equivalent admittances of the Type I grid-following converter for changes of different

control and physical parameters. Solid lines: analytical results. Crosses: frequency sweep results.

For the sake of clarity, the equivalent admittance of the Type I grid-following converter is

derived and evaluated. Table 5.1 lists initial values of the key control and physical parameters.

The unbalanced grid voltage contains 0.5 pu negative-sequence component. The nonlinear

large-signal model is implemented in MATLAB/Simulink. The frequency sweep method is used

to evaluate the accuracy of the harmonic transfer matrix

HV SC (jω)

given in Eq. (5.19). The

basic principle of the frequency sweep method is that a voltage disturbance ∆

vαβg

at different

frequencies (one frequency at a time) is injected into

vαβg

, then, the spectrum of the steady-

105

5. Harmonic Resonance Analysis

state current response ∆

iαβg

is obtained to calculate equivalent admittances. The equivalent

admittances with maximum magnitudes larger than

0.1 S

are shown in Figure 5.2 with solid

lines. The crosses in Figure 5.2 denote frequency sweep simulation results. The maximum

relative percentage error between analytical and frequency sweeping results is smaller than

0.05%, which verifies the correctness and accuracy of the proposed harmonic transfer matrix

model. Additionally, control parameters and LCL filter parameters are respectively increased

by 10% of the initial values to investigate the individual influence of the current control time

constant

, PLL bandwidth

, and LCL parameters on the equivalent admittances, as

shown in Figure 5.2. It is seen that:

Parameters of the LCL filter have larger impact on resonance peaks and resonance

frequencies of

H0,0

O,O

(

jω

)and

H2,0

O,C

(

jω

)in the high frequency range (around

1000 Hz

while their influence in the low frequency range (below 300 Hz) is comparably small.

Time constant

of the current controller mainly influences the damping of resonance

peaks. The system gets less damped as the time constant increases.

The bandwidth BW of the DSRF-PLL influences all four impedances merely in the low

frequency range.

5.2.2 Impedance Modeling of Power Networks and Loads

In the frequency domain, a three-phase branch (or load) can be described with







Ia(jω)

Ib(jω)

Ic(jω)





=





Ya(jω)

Yb(jω)

Yc(jω)







⏞⏟⏟ ⏞

Yabc(jω)







Va(jω)

Vb(jω)

Vc(jω)





(5.20)

where

[Ia(jω), Ib(jω), Ic(jω)]T

and

[Va(jω), Vb(jω), Vc(jω)]T

are the Fourier transformation

of the branch current and nodal voltage vectors,

[ia, ib, ic]T

and

[va, vb, vc]T

(

jω

(

jω

)

and

(

jω

)denote the admittance of each phase. The corresponding two-phase description

given by [︄Iα(jω)

Iβ(jω)]︄=[︄Yαα (jω)Yαβ (jω)

Yβα (jω)Yββ (jω)]︄

⏞⏟⏟ ⏞

Yαβ(jω)

[︄Vα(jω)

Vβ(jω)]︄

can be obtained by applying Clarke respectively inverse Clarke transformation

Yαβ (jω) = TClarkeYabc (jω)T−1

Clarke

[︄Iα(jω)

Iβ(jω)]︄=TClarke 





Ia(jω)

Ib(jω)

Ic(jω)





;[︄Vα(jω)

Vβ(jω)]︄=TClarke 





Va(jω)

Vb(jω)

Vc(jω)







106

5.3 LTP-Theory Based Harmonic Resonance Analysis

Then, the complex-domain impedance model of the branch can be derived

[︄I(jω)

I∗(jω)]︄=[︄YO,O (jω)YO,C (jω)

YC,O (jω)YC,C (jω)]︄

⏞⏟⏟ ⏞

YOC (jω)=Tr2cYαβ (jω)T−1

r2c

[︄V(jω)

V∗(jω)]︄(5.21)

where

{V(jω), I(jω)}

and

{V∗(jω), I∗(jω)}

denote the branch voltage and current in the

original and conjugate coordinates, respectively. When the branch is symmetric, i.e.,

(

jω

) =

(

jω

) =

(

jω

), it can be described by merely using variables in either the original coordinate

or the conjugate coordinate, since

YO,C

(

jω

) =

YC,O

(

jω

)=0. However, when the branch

becomes unbalanced, non-zero

YO,C

(

jω

)and

YC,O

(

jω

)cause the same sequence coupling effect

as the unbalanced control units of the voltage source converters. It can be expected that the

frequency shift effect can also appear when a nonlinearity of the branch (e.g., saturation) is

considered.

For the sake of consistency, the harmonic transfer matrix of the branch can be obtained by

extending each entry of

YOC

(

jω

)into an infinite matrix according to Eq. (5.18), for instance

Hq,p

O,O (jω) = {︄YO,O (jω +jpω0) if p=q

0 if p=q

Hq,p

O,C (jω) = {︄YO,C (jω +jpω0) if p=q

0 if p=q

.(5.22)

So far, the harmonic transfer matrix of basic elements in the converter-dominated grid

have been deduced, which can precisely capture the frequency coupling effect. In the next

section, these impedance models are used to investigate the resonance characteristics.

5.3 LTP-Theory Based Harmonic Resonance Analysis

5.3.1 Review of the Conventional Resonance Mode Analysis

Grid

L21 R21

L11 R11

VSC 1

L22 R22

L12 R12

VSC 2

L23 R23

L13 R13

VSC 3

BUS 2

BUS 1

BUS 3

BUS 4

y1(Lli ne1, Rli ne1)

y12(Lli ne2, Rli ne2)

y13(Lli ne3, Rli ne3)

y14(Lli ne4, Rli ne4)

Figure 5.3: Topology of a grid-connected multiple-converter system

107

5. Harmonic Resonance Analysis

Figure 5.3 shows an exemplary grid-connected three-converter system. When the three-

phase branches are symmetric, the conventional resonance modal analysis method can be used

to investigate the parallel resonance of the passive network. The process is given as follows:







I1(jω)

I2(jω)

I3(jω)

I4(jω)







⏞⏟⏟ ⏞

I(jω)

=





Y11 (jω)Y12 (jω)Y13 (jω)Y14 (jω)

Y21 (jω)Y22 (jω)Y23 (jω)Y24 (jω)

Y31 (jω)Y32 (jω)Y33 (jω)Y34 (jω)

Y41 (jω)Y42 (jω)Y43 (jω)Y44 (jω)







⏞⏟⏟ ⏞

Y(jω)







V1(jω)

V2(jω)

V3(jω)

V4(jω)







⏞⏟⏟ ⏞

V(jω)

⇓

V(jω) = Y−1(jω)I(jω) = Z(jω)I(jω)

(5.23)

where

(

jω

)gives the nodal voltage at the frequency of

, and

(

jω

)is the nodal current

injection vector.

(

jω

)and

(

jω

)are the network admittance and impedance matrices, which

are frequency dependent. The element of Y(jω)is formulated by

Ymn ={︄ym+∑︁n=1,...,NBus, n=mymn, if m =n

−ymn, if m =n

where

ymn

denotes the admittance between the mth bus and the nth bus, of course, it is non-

zero only when a physical connection exists between the two buses. The term

accounts for

the admittance of linear loads connected to the mth bus as well as the admittance-to-stiff-grid

in Figure 5.3.

Applying eigen decomposition to the network impedance matrix

(

jω

)in Eq. (5.23), yields

Z(jω) = R(jω)Λ(jω)R−1(jω)

=R(jω)





λ1(jω)

λ2(jω)

λ3(jω)

λ4(jω)







⏞⏟⏟ ⏞

Λ(jω)

L(jω)

⏞⏟⏟ ⏞

∆

=R−1(jω)

(

jω

)is the eigenvalue matrix of

(

jω

R(jω)

and

L(jω)

are the corresponding right and

left eigenvector matrices. The four eigenvalues/modes are named the modal impedances. A

sharp parallel resonance will only occur when

(

jω

)has eigenvalues/modes with very large

magnitude, i.e.,

(

jω

)approaches singular. The peak of the magnitude of the eigenvalue

is defined as the resonance mode, and the corresponding frequency is called the resonance

frequency.

Figure 5.4 qualitatively demonstrates the basic principle of the conventional resonance

mode analysis method, where

ωp

(

p∈Z

) denotes

pω0

. Imagine that the physical current

108

5.3 LTP-Theory Based Harmonic Resonance Analysis

Modal 1

Modal 2

Modal 3

Modal 4

Nodal Current Injection

Modal Space

Nodal Voltage Response

λ1r11

λ1r21

λ1r31

λ1r41

Modal index

l11

l21

l31

l41

Figure 5.4: Basic principle of the conventional resonance mode analysis method

injection vector I(jω)has only one non-zero element at Bus 1, namely

I(jω) = 











the corresponding nodal voltage response can be obtained following two steps. First,

(

jω

)

projects I(jω)into the modal space, described by













⏞⏟⏟ ⏞

J(jω)

=





l11(jω)l12(jω)l13(jω)l14(jω)

l21(jω)l22(jω)l23(jω)l24(jω)

l31(jω)l32(jω)l33(jω)l34(jω)

l41(jω)l42(jω)l43(jω)l44(jω)







⏞⏟⏟ ⏞

L(jω)













⏞⏟⏟ ⏞

I(jω)

=





l11I1

l21I1

l31I1

l41I1







(5.24)

where

lmn

(

jω

)characterizes the significance of the current injection at the nth bus to excite

the mth modal current Jm.

109

5. Harmonic Resonance Analysis

Next, the physical nodal voltage response is given by the superposition of four modes

V(jω)=[r1(jω)r2(jω)r3(jω)r4(jω)]

⏞⏟⏟ ⏞

R(jω)







λ1(jω)

λ2(jω)

λ3(jω)

λ4(jω)







⏞⏟⏟ ⏞

Λ(jω)













⏞⏟⏟ ⏞

J(jω)

=∑︂4

m=1 λm(jω)rm(jω)Jm

(5.25)

where rm(jω)denotes the mth column of R(jω).

As shown in Figure 5.4, without loss of generality, it is assumed that

λ1

exhibits a resonance

mode at the resonance frequency

. Its magnitude is much larger than that of

λ2

λ3

and

λ4

Then, the nodal voltage V(jω)can be approximated by













⏞⏟⏟ ⏞

V(jω)

=∑︂4

m=1 λm(jω)rm(jω)Jm≈λ1(jω)





r11(jω)

r21(jω)

r31(jω)

r41(jω)







⏞⏟⏟ ⏞

r1(jω)

J1(5.26)

where

rnm

(

jω

), the nth element of

(

jω

), quantifies the observability of the mth modal

voltage at the nth Bus. The excitability and observability are combined into the participation

factor, which is the product of lmn(jω)and rnm(jω).

Recently, the influence of the grid-following converter is considered in [101, 102] by adding its

equivalent impedance in the diagonal entries of

(

jω

), yet the dynamic of the non-holomorphic

control units of the VSC (e.g., PLL) is neglected, and only balanced operation conditions are

considered.

5.3.2 LTP-Theory Based Generalized RMA method

Time-Periodic Impedance Matrix

To precisely investigate harmonic resonance characteristics of the system shown in Figure 5.3

considering also unbalanced conditions, the harmonic transfer matrices of the converter and

the network derived in Section 5.2 should be used to build the admittance and impedance

matrices, yielding

Z=Y−1=







Z1O,1O· · · Z1O,4O

.....

Z4O,1O· · · Z4O,4O

Z1O,1C· · · Z1O,4C

.....

Z4O,1C· · · Z4O,4C

Z1C,1O· · · Z1C,4O

.....

Z4C,1O· · · Z4C,4O

Z1C,1C· · · Z1C,4C

.....

Z4C,1C· · · Z4C,4C







(5.27)

110

5.3 LTP-Theory Based Harmonic Resonance Analysis

where the elementary impedance

ZmO,nO =







....

· · · Z−1,−1

mO,nO Z−1,0

mO,nO Z−1,1

mO,nO · · ·

· · · Z0,−1

mO,nO Z0,0

mO,nO Z0,1

mO,nO · · ·

· · · Z1,−1

mO,nO Z1,0

mO,nO Z1,1

mO,nO · · ·

....







(5.28)

quantifies the relation between the spectrum of the current injected at the nth bus

In=[︂· · · I−2

nI−1

nI0

nI1

nI2

n· · ·]︂T

and that of the nodal voltage at the mth bus

Vm=[︂· · · V−2

mV−1

mV0

mV1

mV2

m· · ·]︂T.

Specifically, Zp, q

mO,nO (jω)maps the current harmonic

nej(ω+qω0)t

to the voltage harmonic

mej(ω+pωn)t

m=Iq

n·Zp, q

mO,nO(jω).(5.29)

The physical meaning of other entries in Eq. (5.27) is defined in the same way. For the sake of

brevity, jω for the description of frequency dependence is omitted.

Now, assume the current injected at each bus only contains the harmonic component at

the frequency ω, namely

I1=[︂· · · 0 0 I0

10 0 · · ·]︂T

I2=[︂· · · 0 0 I0

20 0 · · ·]︂T

I3=[︂· · · 0 0 I0

30 0 · · ·]︂T

I4=[︂· · · 0 0 I0

40 0 · · ·]︂T.

(5.30)

It gives that the infinite-dimensional nodal current injection vector

I= [(I1)T(I2)T(I3)T(I4)T(I∗

1)T(I∗

2)T(I∗

3)T(I∗

4)T]T

has only eight non-zero terms, thus,

can be reduced to an eight-dimensional column vector

I=[︂I0

1I0

2I0

3I0

4(︂I0

1)︂∗(︂I0

2)︂∗(︂I0

3)︂∗(︂I0

4)︂∗]︂T.

111

5. Harmonic Resonance Analysis

Next, nodal voltage spectrum is determined by the central column of Eq. (5.28), yielding







V∗







⏞⏟⏟ ⏞







Z−1,0

1O,1OZ−1,0

1O,2OZ−1,0

1O,3OZ−1,0

1O,4OZ−1,0

1O,1CZ−1,0

1O,2CZ−1,0

1O,3CZ−1,0

1O,4C

Z0,0

1O,1OZ0,0

1O,2OZ0,0

1O,3OZ0,0

1O,4OZ0,0

1O,1CZ0,0

1O,2CZ0,0

1O,3CZ0,0

1O,4C

Z1,0

1O,1OZ1,0

1O,2OZ1,0

1O,3OZ1,0

1O,4OZ1,0

1O,1CZ1,0

1O,2CZ1,0

1O,3CZ1,0

1O,4C

Z−1,0

4C,1OZ−1,0

4C,2OZ−1,0

4C,3OZ−1,0

4C,4OZ−1,0

4C,1CZ−1,0

4C,2CZ−1,0

4C,3CZ−1,0

4C,4C

Z0,0

4C,1OZ0,0

4C,2OZ0,0

4C,3OZ0,0

4C,4OZ0,0

4C,1CZ0,0

4C,2CZ0,0

4C,3CZ0,0

4C,4C

Z1,0

4C,1OZ1,0

4C,2OZ1,0

4C,3OZ1,0

4C,4OZ1,0

4C,1CZ1,0

4C,2CZ1,0

4C,3CZ1,0

4C,4C













(︁I0

1)︁∗

(︁I0

2)︁∗

(︁I0

3)︁∗

(︁I0

4)︁∗







⏞⏟⏟ ⏞

(5.31)

The impedance matrix in Eq. (5.31) has eight columns and theoretically an infinite number of

rows. The eigenvalues of the matrix are not defined. However, if the frequency coupling effect

is described in the time domain, Eq. (5.29) becomes

mej(ω+pω0)t=Iq

nej(ω+qω0)t·Zp, q

mO,nO (jω)ej(p−q)ω0t.

By treating the entries in each column of the impedance matrix in Eq. (5.31) as Fourier series

coefficients, the time-domain nodal voltage can be derived as

v(t) = Z(jωt)i(t)

v(t)=[v1(t)v2(t)v3(t)v4(t)v∗

1(t)v∗

2(t)v∗

3(t)v∗

4(t)]T

i(t) = [︂I0

1ejωt I0

2ejωt I0

3ejωt I0

4ejωt (︂I0

1)︂∗e−jωt (︂I0

2)︂∗e−jωt(︂I0

3)︂∗e−jωt (︂I0

4)︂∗e−jωt]︂T

Z(jωt) = [︄ZOO (jωt)ZOC (−jωt)

ZCO (jωt)ZCC (−jωt)]︄8×8

(5.32)

where

ZOO (jωt) = 





Z1O,1O(jωt)Z1O,2O(jωt)Z1O,3O(jωt)Z1O,4O(jωt)

Z2O,1O(jωt)Z2O,2O(jωt)Z2O,3O(jωt)Z2O,4O(jωt)

Z3O,1O(jωt)Z3O,2O(jωt)Z3O,3O(jωt)Z3O,4O(jωt)

Z4O,1O(jωt)Z4O,2O(jωt)Z4O,3O(jωt)Z4O,4O(jωt)







ZOC (−jωt) = 





Z1O,1C(−jωt)Z1O,2C(−jωt)Z1O,3C(−jωt)Z1O,4C(−jωt)

Z2O,1C(−jωt)Z2O,2C(−jωt)Z2O,3C(−jωt)Z2O,4C(−jωt)

Z3O,1C(−jωt)Z3O,2C(−jωt)Z3O,3C(−jωt)Z3O,4C(−jωt)

Z4O,1C(−jωt)Z4O,2C(−jωt)Z4O,3C(−jωt)Z4O,4C(−jωt)







are time-periodic matrices with

ZmO,nO (jωt) = ∑︂∞

p=−∞ Zp, 0

mO,nO (jω)ejpω0t

ZmO,nC (−jωt) = ∑︂∞

p=−∞ Zp, 0

mO,nC (−jω)ejpω0t.

112

5.3 LTP-Theory Based Harmonic Resonance Analysis

Generalized Resonance Mode Analysis

If the harmonic current at

is only injected into Bus 1, the nodal voltage response is obtained

from Eq. (5.32)







v1(t)

v2(t)

v3(t)

v4(t)







=ZOO (jωt)





1ejωt







+ZOC (−jωt)





(︁I0

1)︁∗e−jωt







.(5.33)

Taking ZOO(jωt)as an example, according to Eq. (4.25), it has the eigen decomposition

ZOO (jωt) = (︂ROO (jωt)ΛOO (jω) + R

˙OO (jωt))︂LOO (jωt)

LOO (jωt) = R−1

OO (jωt)

(5.34)

where the LTP eigenvalue/mode matrix

ΛOO(jω) = 





λOO,1(jω)

λOO,2(jω)

λOO,3(jω)

λOO,4(jω)







is a constant diagonal matrix. The corresponding LTP eigenvector matrix and its inverse,

namely

ROO(jωt) = [︂rOO,1(jωt)rOO,2(jωt)rOO,3(jωt)rOO,4(jωt)]︂

=





r1O,1O(jωt)r1O,2O(jωt)r1O,3O(jωt)r1O,4O(jωt)

r2O,1O(jωt)r2O,2O(jωt)r2O,3O(jωt)r2O,4O(jωt)

r3O,1O(jωt)r3O,2O(jωt)r3O,3O(jωt)r3O,4O(jωt)

r4O,1O(jωt)r4O,2O(jωt)r4O,3O(jωt)r4O,4O(jωt)







and

LOO(jωt) = 





lOO,1(jωt)

lOO,2(jωt)

lOO,3(jωt)

lOO,4(jωt)







=





l1O,1O(jωt)l1O,2O(jωt)l1O,3O(jωt)l1O,4O(jωt)

l2O,1O(jωt)l2O,2O(jωt)l2O,3O(jωt)l2O,4O(jωt)

l3O,1O(jωt)l3O,2O(jωt)l3O,3O(jωt)l3O,4O(jωt)

l4O,1O(jωt)l4O,2O(jωt)l4O,3O(jωt)l4O,4O(jωt)







are time periodic.

As shown in Figure 5.5,

lmO,1O(jωt)

∑︁plp

mO,1Oejpω0t

, the element in the mth row and

1st column of

LOO

(

jωt

), projects

1ejωt

into the modal currents

(

)containing multiple

113

5. Harmonic Resonance Analysis

Nodal Current Injection

Modal Space

Nodal Voltage Response

l1O,1O(t)

l2O,1O(t)

l3O,1O(t)

l4O,1O(t)

λOO,1 +jpω0 r1O,1O

pejpω0t

λOO,1+jpω0 r3O,1O

pejpω0t

Modal 2

Modal 3

Modal 4

Modal 1

Modal index

Figure 5.5: Basic principle of the LTP theory-based generalized resonance mode analysis method

harmonics, given by

J(jωt) = 





J1(t)

J2(t)

J3(t)

J4(t)







=





∑︁pJp

1ej(ω+pω0)t

∑︁pJp

2ej(ω+pω0)t

∑︁pJp

3ej(ω+pω0)t

∑︁pJp

4ej(ω+pω0)t







=I0

1ejωt 





∑︁plp

1O,1Oejpω0t

∑︁plp

2O,1Oejpω0t

∑︁plp

3O,1Oejpω0t

∑︁plp

4O,1Oejpω0t







.(5.35)

For modal currents at ω(also applies to other frequencies), namely

J(jωt) = [︂J0

1ejωt J0

2ejωt J0

3ejωt J0

4ejωt ]︂T

the corresponding nodal voltage response is given by the superposition of four LTP modes







v1(t)

v2(t)

v3(t)

v4(t)







=(︂ROO (jωt)ΛOO (jω) + R

˙OO (jωt))︂J(jωt)

=∑︂4

m=1 (λOO,m (jω)rOO,m (jωt) + r˙OO,m (jωt)) J0

mejωt

(5.36)

114

5.3 LTP-Theory Based Harmonic Resonance Analysis

where

rOO,m

(

jωt

)denotes the mth column of

ROO

(

jωt

). Inserting the Fourier series expansion

of rOO,m(jωt), Eq. (5.36) becomes







v1(t)

v2(t)

v3(t)

v4(t)







=





1ejωt ∑︁p(λOO,1(jω) + jpω0)rp

OO,1ejpω0t

+J0

2ejωt ∑︁p(λOO,2(jω) + jpω0)rp

OO,2ejpω0t

+J0

3ejωt ∑︁p(λOO,3(jω) + jpω0)rp

OO,3ejpω0t

+J0

4ejωt ∑︁p(λOO,4(jω) + jpω0)rp

OO,4ejpω0t







(5.37)

where

OO,m

is the pth order Fourier coefficient of

rOO,m

(

jωt

). Comparing Eq. (5.35) and

(5.37) with Eq. (5.24) and (5.25), it is seen that

lmO,nO

(

jωt

)and

rnO,mO

(

jωt

)characterize the

excitability and observability of the mth LTP mode at the nth bus, where

rnO,mO

(

jωt

)is the

nth element of rOO,m(jωt).

The generalization of the resonance mode definition for the LTP system is not straight-

forward, since each LTP mode is associated with a group of multipliers, for instance,

{λOO,m (jω) + jpω0, p ∈Z}

is related to the mth LTP mode. Inspired by following two

facts:

1. the definition of LTP resonance mode should be compatible with LTI systems.

2. rp

OO,m decreases to zero as papproaches infinity.

the mth LTP modal impedance is defined as

λm(jω)∆

= max 



rp

OO,m1

r0

OO,m1

· |λOO,m (jω) +jpω0|, p ∈Z





.(5.38)

The LTP resonance mode is accordingly defined as the peak of LTP modal impedances. For the

test system, it is found

λm(jω) = |λOO,m (jω)|

. In Figure 5.5, it is assumed that

λ1

exhibits a

peak value (i.e., an LTP resonance mode) at ω, and it is much larger than λ2,λ3and λ4.

Similarly, the other impedance matrix

ZOC

(

−jωt

)in Eq. (5.33) has also the eigen

decomposition with LTP eigenvalue and eigenvector matrices, i.e.,

ΛOC (jω)

ROC (−jωt)

and

LOC (−jωt)

. It can be expected that the coupling of original and conjugate coordinates caused

by the non-holomorphic control or unbalanced physical components may introduce new LTP

resonance modes, indicated by the LTP resonance mode of ZOC(−jωt).

Sensitivity Analysis

To detect the affected area and the most involved buses of a particular resonance mode, the

participation factor analysis can be used. According to previous analysis,

lmO,nO

(

jωt

)and

rnO,mO

(

jωt

)characterize the excitability and observability of mode

ZOO

(

jωt

)at Bus

Similarly,

lmO,nC

(

−jωt

)and

rnO,mC

(

−jωt

)characterize the excitability and observability of

the mth mode of

ZOC

(

−jωt

)at the nth bus. Then, participation factors of the mth mode of

115

5. Harmonic Resonance Analysis

ZOO(jωt)and ZOC(−jωt)at the nth bus are defined as

PFnO,mO =



lmO,nO (jωt)·rnO,mO (jωt)



1

∑︁4

b=1 



lmO,bO (jωt)·rbO,mO (jωt)



1

PFnO,mC =



lmO,nC (jωt)·rnO,mC (jωt)



1

∑︁4

b=1 



lmO,bC (jωt)·rbO,mC (jωt)



1

.(5.39)

Generally, the larger the participation factor of a bus is, the greater the risk of resonance.

Harmonic resonance results from the energy exchange between capacitive and inductive

elements. Thus, all parameter changes that influence the system admittance or impedance will

affect the system resonance characteristics. Sensitivity analysis is a useful tool to determine the

impact of a network component or a converter control parameter on the resonance frequency

and the resonance mode. In this paper, a local sensitivity analysis method, one-at-a-time

(OAT) [117], is used, since it is easy to implement and requires relatively low computational

effort. In the OAT method, the individual system parameter

is varied in the vicinity of

its nominal value

, and the corresponding sensitivity is defined as the ratio of the relative

variation of the output yto the percentage change of xi, given by

Si= (Xi/y)·(∂y/∂xi)(5.40)

where

stands for the resonance frequency and the resonance mode, while

is the network

parameter and the converter control parameter.

5.4 Case Study

In this section, the proposed generalized resonance mode analysis method is tested and validated

using the three-converter system shown in Figure 5.3. The converter and network parameters

are listed in Table 5.2 and Table 5.3. The model is implemented in MATLAB/Simulink and

simulated with the real-time simulator OP5600. Parallel harmonic resonance studies are

carried out based on the generalized resonance mode analysis method. The accuracy of the

analytical admittance/impedance model and the effectiveness of the generalized resonance

mode analysis method are validated by frequency sweep simulation results.

Table 5.2: Parameters of the Converters

Symbol Description VSC 1 VSC 2 VSC 3

L / R Converter-side filter inductance 5.6 mH /0.1 Ω 5.6 mH /0.1 Ω 7 mH /0.1 Ω

CFilter capacitor 16 µF 16 µF 30 µF

BW Bandwidth of the DSRF-PLL 25 Hz

τTime constant of the current controller 1 ms

SrPower reference 2 kW 2 kW 3 kW

Lg/ RgGrid-side filter inductor 1.5 mH /1 mΩ 1 mH /1 mΩ 1.5 mH /1 mΩ

The network is first assumed to be balanced. The results of the generalized resonance mode

analysis are shown in Figure 5.6. No critical resonance modes are introduced by

ZOC

(

−jωt

)

under this test scenario, thus, they are omitted. In the conventional analysis, the resonance

116

5.4 Case Study

Table 5.3: Parameter of the Network

Symbol Description Value

y1Line impedance between Bus 1 and the stiff grid 1 mH + 1 mΩ

y12 Line impedance between Bus 1 and Bus 2

0.5 mH + 1 mΩ

y13 Line impedance between Bus 1 and Bus 3

y14 Line impedance between Bus 1 and Bus 4

frequency and modal impedance of positive and negative sequence components are the same.

However, it is noted in Figure 5.6 that the negative sequence component has smaller resonance

frequencies and larger modal impedances, which is caused by the non-holomorphic control

units of the grid-following converters. This effect becomes more obvious as the line impedance

Lline1

increases from

1 mH

2 mH

, where the resonance frequency approaches the bandwidth

of the PLL. Another observation is that an additional resonance peak at

96 Hz

is introduced by

the larger grid impedance. This resonance mode is strongly linked to system stability, which

has been intensively studied in Chapter 4. The focus of this chapter is the system steady-state

resonance behavior in a higher frequency range.

Res. Mode 5

X:-1095

Y:39.49

Res. Mode 3

X:-917

Y:62.16

Res. Mode 1

X:-605

Y:65.86

Res. Mode 2

X:625

Y:46.06

Res. Mode 4

X:932

Y:51.88

Res. Mode 6

X:1107

Y:33.32

X:96

λOO,1

λOO,1 λOO,2

λOO,2 λOO,3

λOO,3 λOO,4

λOO,4

λOO,1 λOO,2 λOO,3 λOO,4

Res. Mode 5

X:-1095

Y:39.49

Res. Mode 3

X:-917

Y:62.16

Res. Mode 1

X:-605

Y:65.86

Res. Mode 2

X:625

Y:46.06

Res. Mode 4

X:932

Y:51.88

Res. Mode 6

X:1107

Y:33.32

X:96

λOO,1 λOO,2 λOO,3 λOO,4

Figure 5.6: Modal impedances of the test system for different grid impedances. Solid lines:

Lline1= 1 mH. Dashed lines: Lline1= 2 mH.

Application of the Generalized RMA Method

Following the definition given by Eq. (5.39), the participation factor calculation results are

listed in Table 5.4. It can be concluded that all buses are involved in the resonance mode 1 and

2, while the key participating buses for resonance mode 3 and 4 are Bus 1, 2 and 3. Moreover,

resonance mode 5 and 6 are mainly related to Bus 3. The participation factor analysis results

can be validated by the frequency sweep method: injecting a current disturbance at each bus

and checking whether the corresponding nodal voltage resonances appear.

In fact, the voltage response information is stored in the analytical impedance matrices

ZOO

(

jωt

)and

ZOC

(

−jωt

), as discussed in Section 5.3.2. Specifically, assume the current

disturbance is injected at Bus 2, Figure 5.7 shows the magnitude of the impedances in

ZOO

(

jωt

)

from Bus 2 to all the buses. Here, solid lines denote the analytical results, and the crosses denote

frequency sweep simulation results. For instance,

Z0,0

1O, 2O

(

jω

)maps the current disturbance

117

5. Harmonic Resonance Analysis

2ejωt

at Bus 2 to the voltage response

1ejωt

at Bus 1. Additionally,

ZOC

(

−jωt

)also contains

non-zero terms,

Z2,0

mO, 2C

(

−jω

), which maps (

)

∗e−jωt

at Bus 2 to

mej(−ω+2ω0)t

at Bus m.

The maximum magnitude of

Z2,0

mO, 2C

(

−jω

)is smaller than

3 Ω

, thus, they are ignored here. It

can be observed from Figure 5.7 that all four impedances exhibit peak values at the resonance

frequencies of resonance mode 1 and 2, which means that these resonances can be easily

excited and observed at Bus 2. Furthermore, among the four impedances,

Z0,0

4O, 2O

(

jω

)has the

largest values at - 605 and

625 Hz

, namely the corresponding observability at Bus 4 is stronger,

which is consistent with the result in Table 5.4 that Bus 4 has a larger participation factor for

resonance mode 1 and 2 (0.3825 and 0.3886). Similarly, it is seen that resonance mode 3 and 4

can also be easily excited at Bus 2, which indicate the resonance modes observability, share

the same order of participation factors. As for resonance mode 5 and 6, all impedances are

comparably small, which means it is difficult to excite and observe resonance mode 5 and 6 at

Bus 2. Nevertheless, the stronger participation of Bus 3 can still be confirmed by the fact that

Z0,0

3O, 2O(jω)is much larger than the other three impedances at -1095 and 1107 Hz.

Table 5.4: Participation Factor Analysis Results under Symmetric Condition

Resonance Frequency Participation Factor of each Bus

Mode (Hz) 1 2 3 4

1 -605 0.1938 0.2124 0.2119 0.3825

2 625 0.1938 0.2096 0.2096 0.3886

3 -917 0.1676 0.4719 0.3335 0.0281

4 932 0.1689 0.4670 0.3352 0.0306

5 -1095 0.0721 0.0319 0.8831 0.0251

6 1107 0.0735 0.0329 0.8853 0.0258

Mode 5

X:-1095

Mode 3

X:-917

Mode 1

X:-605

Mode 2

X:625

Mode 4

X:932

Mode 6

X:1107

Z3O,2O

0,0 jω

Z4O,2O

0,0 jω

Z1O,2O

0,0 jω

Z2O,2O

0,0 jω

Figure 5.7: Impedances from Bus 2 to all the buses. Solid lines: analytical results. Crosses:

frequency sweep results.

118

5.4 Case Study

Influence of the Grid Imbalance

In this subsection, the influence of the grid imbalance is investigated by considering the

following two cases:

•unbalanced grid voltage containing negative sequence component.

•unbalanced three-phase grid impedance.

First, the effect of the unbalanced grid voltage is investigated by sweeping the unbalance

ratio

V−/V+

, where

V−

and

V+

represent the magnitude of the negative and positive

sequence components of the grid voltage. Figure 5.8 shows the movement of the modal

impedance for different values of

. It is identified that the grid voltage imbalance has only a

small impact in the frequency range below

150 Hz

, while the resonance frequency and modal

impedance at higher frequencies are immune to the appearance of the negative sequence grid

voltage.

Mode 5

Mode 3

Mode 1

Mode 6

Mode 4

Mode 2

D: from 0 to 0.6D: from 0 to 0.6

λOO,1

λOO,1 λOO,2

λOO,2 λOO,3

λOO,3 λOO,4

λOO,4

λOO,1 λOO,2 λOO,3 λOO,4

Mode 5

Mode 3

Mode 1

Mode 6

Mode 4

Mode 2

D: from 0 to 0.6

λOO,1 λOO,2 λOO,3 λOO,4

Figure 5.8: Modal impedances for different unbalance ratios D

Then, the grid imbalance is introduced by modifying the impedance

Lline1

(

3 mH

1 mH

). Applying the generalized resonance mode analysis method, the modal

impedance of

ZOO

(

jωt

)and

ZOC

(

−jωt

)are shown in Figure 5.9. Compared to Figure 5.6, it

can be observed that, among the first six resonance modes, resonance mode 5 and 6 are almost

unaffected, while the resonance frequency and modal impedance of other modes decrease to

varying degrees. Additionally, an extra resonance mode (resonance mode 7) at

535 Hz

appears.

Meanwhile, one modal impedance of

ZOC

(

−jωt

)becomes critical and exhibits peak values

556 and

±908 Hz

, i.e., resonance mode 8, 9, 10 and 11. According to Eq. (5.32), the

resonance mode associated with

ZOC

(

−jωt

)indicates that the current injection

nejωt

the nth bus will excite voltage resonances

{︂Vp

mej(−ω+pω0)t, p ∈Z}︂

at the mth bus, and the

relation is quantified by the entries of ZOC(−jωt), for instance, Vp

m=Zp,0

mO,nC (−jω)(︁I0

n)︁∗.

The participation factor calculation results are listed in Table 5.5. As for the newly

introduced resonance modes, it is identified that all the buses are involved in resonance mode 7,

8 and 9, while Bus 4 is excluded from resonance mode 10 and 11. Similarly, the impedance

matrices

ZOO

(

jωt

)and

ZOC

(

−jωt

)are used to validate the participation factor analysis

results, as shown in Figure 5.10. The comparison between the impedances from Bus 2 to all

119

5. Harmonic Resonance Analysis

Res. Mode 5

X:-1093

Y:38.32

Res. Mode 3

X:-898

Y:56.08

Res. Mode 1

X:-545

Y:53.71

Res. Mode 2

X:624

Y:40.69

Res. Mode 4

X:928

Y:46.8

Res. Mode 6

X:1105

Y:32.38

Res. Mode7

X:535

Y:33.89

Res. Mode 9

X:556

Y:33.03 Res. Mode 11

X:908

Y:23.17

Res. Mode 8

X:-556

Y:33.03

Res. Mode 10

X:-908

Y:23.17

λOO,1

λOO,1 λOO,2

λOO,2 λOO,3

λOO,3 λOO,4

λOO,4 λOC,1

λOC,1

λOO,1 λOO,2 λOO,3 λOO,4 λOC,1

Res. Mode 5

X:-1093

Y:38.32

Res. Mode 3

X:-898

Y:56.08

Res. Mode 1

X:-545

Y:53.71

Res. Mode 2

X:624

Y:40.69

Res. Mode 4

X:928

Y:46.8

Res. Mode 6

X:1105

Y:32.38

Res. Mode7

X:535

Y:33.89

Res. Mode 9

X:556

Y:33.03 Res. Mode 11

X:908

Y:23.17

Res. Mode 8

X:-556

Y:33.03

Res. Mode 10

X:-908

Y:23.17

λOO,1 λOO,2 λOO,3 λOO,4 λOC,1

Figure 5.9: Modal impedances under unbalanced grid impedance condition

Table 5.5: Participation Factor Analysis Results under unbalanced Condition

Resonance Frequency Participation Factor of each Bus

Mode (Hz) 1 2 3 4

1 -545 0.2175 0.2293 0.2294 0.3249

2 624 0.2011 0.2102 0.2102 0.3806

3 -898 0.1829 0.4491 0.3442 0.0249

4 928 0.1776 0.4571 0.3362 0.0307

5 -1093 0.0759 0.0381 0.8737 0.0263

6 1105 0.0769 0.0393 0.8767 0.027

7 535 0.223 0.2328 0.2333 0.3121

8 -556 0.222 0.2275 0.2277 0.3243

9 556 0.222 0.2275 0.2277 0.3243

10 -908 0.1869 0.4452 0.344 0.0273

11 908 0.1869 0.4452 0.344 0.0273

the buses at the resonance frequency of each resonance mode confirms the bus participation

information given in Table 5.5. Again, the agreement between analytical results (solid lines)

and frequency sweep simulation results (crosses) in Figure 5.10 confirms the accuracy and

effectiveness of the proposed methodology. To give an intuitive illustration of the frequency

coupling effect, one of the frequency sweep simulation results is shown in Figure 5.11:

1 A

(

0.1 pu

) negative-sequence current disturbance at

550 Hz

(

−11 pu

) is injected at Bus 2, the

corresponding steady-state voltage responses at each bus are shown in Figure 5.11. As expected,

the nodal voltage at each bus contains harmonic components at

650,

550 and

450 Hz.

The impedances obtained from the bus voltages and the injected current agree well with the

analytical results.

120

5.4 Case Study

Z3O,2O

-2,0 jω

Z4O,2O

-2,0 jω

Z1O,2O

-2,0 jω

Z2O,2O

-2,0 jω

Z3O,2O

-2,0 jω

Z4O,2O

-2,0 jω

Z1O,2O

-2,0 jω

Z2O,2O

-2,0 jω

Z3O,2O

0,0 jω

Z4O,2O

0,0 jω

Z1O,2O

0,0 jω

Z2O,2O

0,0 jω

Z3O,2O

0,0 jω

Z4O,2O

0,0 jω

Z1O,2O

0,0 jω

Z2O,2O

0,0 jω

Z3O,2O

2,0 jω

Z4O,2O

2,0 jω

Z1O,2O

2,0 jω

Z2O,2O

2,0 jω

Z3O,2O

2,0 jω

Z4O,2O

2,0 jω

Z1O,2O

2,0 jω

Z2O,2O

2,0 jω

Z3O,2C

2,0 jω

Z4O,2C

2,0 jω

Z1O,2C

2,0 jω

Z2O,2C

2,0 jω

Z3O,2C

2,0 jω

Z4O,2C

2,0 jω

Z1O,2C

2,0 jω

Z2O,2C

2,0 jω

Z3O,2C

0,0 jω

Z4O,2C

0,0 jω

Z1O,2C

0,0 jω

Z2O,2C

0,0 jω

Z3O,2C

0,0 jω

Z4O,2C

0,0 jω

Z1O,2C

0,0 jω

Z2O,2C

0,0 jω

Z3O,2C

-2,0 jω

Z4O,2C

-2,0 jω

Z1O,2C

-2,0 jω

Z2O,2C

-2,0 jω

Z3O,2C

-2,0 jω

Z4O,2C

-2,0 jω

Z1O,2C

-2,0 jω

Z2O,2C

-2,0 jω

Z3O,2O

-2,0 jω

Z4O,2O

-2,0 jω

Z1O,2O

-2,0 jω

Z2O,2O

-2,0 jω

Z3O,2O

0,0 jω

Z4O,2O

0,0 jω

Z1O,2O

0,0 jω

Z2O,2O

0,0 jω

Z3O,2O

2,0 jω

Z4O,2O

2,0 jω

Z1O,2O

2,0 jω

Z2O,2O

2,0 jω

Z3O,2C

2,0 jω

Z4O,2C

2,0 jω

Z1O,2C

2,0 jω

Z2O,2C

2,0 jω

Z3O,2C

0,0 jω

Z4O,2C

0,0 jω

Z1O,2C

0,0 jω

Z2O,2C

0,0 jω

Z3O,2C

-2,0 jω

Z4O,2C

-2,0 jω

Z1O,2C

-2,0 jω

Z2O,2C

-2,0 jω

Figure 5.10: Impedances from Bus 2 to all the buses under the unbalanced grid impedance

condition. Solid lines: analytical results. Crosses: frequency sweep results.

β-componentβ-componentα-componentα-component β-componentα-component β-componentα-component

-13-11-9 -1 1 9 1113-13-11-9 -1 1 9 1113-13-11-9 -1 1 9 1113 -13-11-9 -1 1 9 1113-13-11-9 -1 1 9 1113-13-11-9 -1 1 9 1113 -13-11-9 -1 1 9 1113-13-11-9 -1 1 9 1113-13-11-9 -1 1 9 1113 -13-11-9 -1 1 9 1113-13-11-9 -1 1 9 1113-13-11-9 -1 1 9 1113

SimulationSimulation AnalyticalAnalyticalSimulation Analytical

-13-11-9 -1 1 9 1113 -13-11-9 -1 1 9 1113 -13-11-9 -1 1 9 1113 -13-11-9 -1 1 9 1113

Simulation Analytical

β-componentα-component

-13-11-9 -1 1 9 1113 -13-11-9 -1 1 9 1113 -13-11-9 -1 1 9 1113 -13-11-9 -1 1 9 1113

Simulation Analytical

Figure 5.11: Voltage responses at each bus to a 1 A 550 Hz negative-sequence current disturbance

at Bus 2. Top: Steady-state time domain waveforms. Bottom: Comparison between the frequency

spectrums of the simulation and analytical results.

Sensitivity Analysis

The above presented results indicate that the generalized resonance mode analysis method is a

useful tool to identify the resonance frequencies of a converter-dominated grid, considering

121

5. Harmonic Resonance Analysis

balanced and unbalanced operation conditions. The proposed participation factor analysis

can further characterize the excitability and observability of the individual resonance mode.

In this section, the OAT method is used to assess the impact of system parameters on each

resonance mode and the corresponding resonance frequency. The sensitivity indices of following

parameters are evaluated:

1. Physical Components: line impedances Lline i and LCL filter of each VSC L1i, Ci, L2i;

Control Parameters: Time constant of the converter’s current controller

τi

and the

designed bandwidth (BWi) of the DSRF-PLL.

The index istands for the ith line or converter.

Taking the unbalanced line impedance

Lline1

{La= 3 mH, Lb=Lc= 1 mH}

as an

example, the value of

is swept in the range of [90%

110%] of

3 mH

, and the movement

of the modal impedances is shown in Figure 5.12. Then, the impact of

on each resonance

is quantified by the resonance frequency and the resonance mode sensitivity indices,

and

Sλ

, as shown in Figure 5.13. All resonance frequency indices

indicate negative values,

which means the resonance modes move towards lower frequencies as

increases. Another

observation is that the resonance modes resulting from

ZOC

(

−jωt

), that is, resonance mode 8,

9, 10 and 11, are highly sensitive to

, and enlarge as the grid imbalance becomes more

severe. Meanwhile, large

can damp resonance mode 1, 3 and 4 to some extent, since their

sensitivity indices Sλare negative.

Res. Mode 5

Res. Mode 3 Res. Mode 1 Res. Mode 2

Res. Mode 4

Res. Mode 6

Res. Mode7

Res. Mode 9

Res. Mode 11

Res. Mode 8

Res. Mode 10

Δ|λ|

Δf

Δ|λ|

Δf

λOO,1

λOO,2

λOO,3

λOO,4

λOC,1

Res. Mode 5

Res. Mode 3 Res. Mode 1 Res. Mode 2

Res. Mode 4

Res. Mode 6

Res. Mode7

Res. Mode 9

Res. Mode 11

Res. Mode 8

Res. Mode 10

Δ|λ|

Δf

λOO,1

λOO,2

λOO,3

λOO,4

λOC,1

Figure 5.12: Modal impedances for different La

The sensitivity indices of other parameters are calculated following the same procedure

and demonstrated by the heatmaps in Figure 5.14 and Figure 5.15. The following observations

can be obtained from Figure 5.14 and Figure 5.15:

Harmonic resonances are mainly affected by a few parameters, while most parameters

have little or negligible impact on the resonance frequencies and the resonance modes.

The resonance frequency sensitivity indices agree quite well with the PF analysis results,

122

5.4 Case Study

Sλ

Figure 5.13: Sensitivity indices of Lato all the resonance modes

i.e., the network components related to the bus with the largest PF will have larger

For instance, Bus 4 is most involved in resonance mode 7, 8 and 9 as given in Table 5.5,

the parameters of converter 3, e.g.,

L13

and

τ3

, indicate comparably larger resonance

frequency sensitivity indices.

All physical components have negative sensitivity indices

, while the sensitivity

indices

and

Sλ

of the control parameters are positive. It predicts that the resonance

frequencies will decrease when the values of capacitor and inductor are increased; both

the resonance frequencies and resonance modes become larger as the grid-following

converter’s current controller becomes slower. It is noteworthy that some physical

components have bidirectional resonance mode sensitivities. For instance, the resonance

mode sensitivities of

at resonance mode 5 and 10 are positive and negative, respectively.

It implies that these two resonance modes cannot be simultaneously damped by merely

increasing/decreasing

. This property should be carefully considered for designing

resonance mitigation strategies.

|Sf|<10-4

Figure 5.14: Resonance frequency sensitivity analysis results.

Harmonic resonance only occurs when the frequencies generated by the harmonic sources

are close to the resonance frequencies with large resonance modes. Therefore, based on the

sensitivity indices, the resonance mitigation scheme can be developed by optimizing the system

parameters to decrease the resonance modes and shift the resonance frequencies away from

the frequency contents of the harmonic sources.

123

5. Harmonic Resonance Analysis

Sλ

|Sλ |<0.1

Figure 5.15: Resonance mode sensitivity analysis results.

5.5 Summary

This chapter presents a systematic methodology for the harmonic resonance analysis of

converter dominated power systems. Unlike the classical RMA for the legacy power system,

the admittance/impedance matrix is represented by a time-periodic matrix obtained from

the reformulation of the HTMs to fully capture frequency coupling effects. The proposed

generalized RMA can accurately evaluate the effects of the non-holomorphic control units

and the unbalanced operation conditions on the resonance characteristics. The results of case

studies indicate that the DSRF-PLL of the grid-following converter can cause the difference of

harmonic resonances for positive and negative sequences, and the grid impedance imbalance

will introduce extra resonance modes. In addition, sensitivity analysis is used to assess the

impact of individual parameters on the resonance frequency and the resonance mode, which

provides useful information for resonance mitigation. Potential applications of the proposed

method include: harmonic resonance analysis of large-scale modern power systems with diverse

converters under different operation conditions, investigation of the impact of other nonlinear

control units, e.g., PWM, on the resonance behavior, and optimized design of passive and

active damping strategies.

124

Sensitivity-Based Stability Improvement

In previous chapters, the eigenvalue sensitivity analysis and participation analysis have been

proved to be useful tools for the identification of the most influential parameters and states

affecting the small-signal stability of converter-dominated power systems. In this chapter,

sensitivity and participation analysis are used as design-oriented tools to improve stability

margins and damping performance. Based on eigenvalue and damping ratio sensitivities, a

linear optimization problem is formulated and solved iteratively to achieve automatic tuning of

control parameters. Additionally, guided by the participation analysis, an auxiliary damping

loop design rule is proposed for the scenarios where the control parameters cannot be freely

alternated. Experimental tests have been carried out on both grid-following converters and

grid-forming converters to validate the effectiveness of the proposed methodology.

6.1 Sensitivity-Based Parameter Optimization

The LTI and LTP eigenvalue sensitivity analysis results provide linear approximation of the

movement of eigenvalues as physical and control parameters are modified. Take the Type I

GFL converter as an example, as the proportional gain

of the PI controller in the DSRF-PLL

is swept in the range of [

−

20%

20%] of the initially designed value, the 14 LTP eigenvalues

can be predicted with

λi=λi0+∂λi

∂kp∆kp, i = 1,...,14 (6.1)

where

λi0

is the initial value of the ith LTP eigenvalue, which are marked with black circles in

Figure 6.1. The crosses give their sensitivity-based predictions when

is changed. Meanwhile,

the system matrix is updated for each new value of

. Corresponding LTP eigenvalues

are recalculated, and the results are plotted as dots in Figure 6.1. The agreement of LTP

eigenvalues obtained from the prediction and recalculation confirms the effectiveness of the

proposed LTP eigenvalue sensitivity index

∂λi

∂kp

used in Eq. (6.1). Nevertheless, it should be

clarified that the eigenvalue sensitivity is a linear approximation around the initial control

parameters, and the accuracy cannot be guaranteed when the changes of parameter are too

large. The increasing deviations between actual LTP eigenvalues and their predictions can be

125

6. Sensitivity-Based Stability Improvement

observed in the enlarged plot of Figure 6.1. Such prediction error can be reduced by considering

also the second-order eigenvalue sensitivity, however, it brings nonlinearity and computational

complexity.

Percentage change of kp (%)

20-20

Figure 6.1: Distribution of LTP eigenvalues of Typy I GFL converter system with changes of

. Black circles: initial values of the eigenvalues. Crosses: sensitivity-based predictions. Dots:

recalculated/true eigenvalues.

According to the modal analysis presented in Chapter 4, the control parameter should

be optimized so that all eigenvalues lie in the left-half of the complex plane to ensure the

system stability. At the same time, damping ratio of each mode should be as large as possible

to achieve smooth dynamic responses. As given by Eq. (4.45), each LTP eigenvalue/mode

contains multiple oscillation components and its damping performance is quantified by a

damping ratio vector. To give a comprehensive evaluation of the damping performance of one

LTP mode, the sum of damping ratios weighted by the first-order norm of the eigenvector is

defined

¯i=∑︂H

h=−Hrh

i1

|ri|1

ξh

i=∑︂H

h=−Hrh

i1

|ri|1

−σi

√︂σ2

i+ (ωi+hω0)2.(6.2)

Then, the first-order sensitivity of ξ

¯iwith respect to an arbitrary parameter φis given by

∂ξ

¯i

∂φ =∑︂H

h=−H

∂|rh

i|1

∂φ |ri|1−∂|ri|1

∂φ rh

i1

|ri|2

ξh

i+rh

i1

|ri|1

∂ξh

∂φ (6.3)

where ∂ξh

∂φ can be determined with the eigenvalue sensitivity

∂ξh

∂φ =1

σ2

i+ (ωi+hω0)2(︄−√︂σ2

i+ (ωi+hω0)2Re {︃∂λi

∂φ }︃+σi·

σiRe {︁∂λi

∂φ }︁+ (ωi+hω0) Im {︁∂λi

∂φ }︁

√︁σ2

i+ (ωi+hω0)2)︄.

Moreover,

∂|rh

i|1

∂φ

and

∂|ri|1

∂φ

in Eq. (6.2) are related to the LTP eigenvector sensitivities, which

are deduced as following:

The linear independent LTP eigenvectors and their shifted copies are the basis vectors,

therefore, the sensitivity of the ith LTP eigenvector

(

)with respect to an arbitrary parameter

φcan be generally written as the linear superposition of those basis vectors

∂ri(t)

∂φ =∑︁N

j=1 ∑︁H

h=−Hψh

ijejhω0trj(t), h = 0 if j=i(6.4)

126

6.1 Sensitivity-Based Parameter Optimization

where ψh

ij denotes a constant coefficient to be determined.

According to Eq. (4.56a), the shifted copies of LTP eigenvalue and eigenvector satisfy the

relation

A(t)(︂ejhω0tri(t))︂−∂

∂t (︂ejhω0tri(t))︂= (λi−jhω0)(︂ejhω0tri(t))︂(6.5)

Taking the partial derivative of Eq. (6.5) with respect to φ, yields

∂A(t)

∂φ (︂ejhω0tri(t))︂+A(t)(︃ejhω0t∂ri(t)

∂φ )︃−∂

∂t (︃ejhω0t∂ri(t)

∂φ )︃

=∂λi

∂φ (︂ejhω0tri(t))︂+ (λi−jhω0)(︃ejhω0t∂ri(t)

∂φ )︃(6.6)

Multiplying both sides of Eq. (6.6) with

(

)and inserting the relation given by Eq. (4.56b),

yields

lj(t)∂A(t)

∂φ (︂ejhω0tri(t))︂−∂

∂t (︃ejhω0tlj(t)∂ri(t)

∂φ )︃= (λi−jhω0−λj)(︃ejhω0tlj(t)∂r(t)i

∂φ )︃

(6.7)

The second term on the left side of Eq. (6.7) contains no DC components, therefore, the

constant coefficient ψ−h

ij is determined by

ψ−h

ij ={︂lj(t)∂A(t)

∂φ (︂ejhω0tri(t))︂}︂0

λi−jhω0−λj

(6.8)

Then, ∂|rh

i|1

∂φ and ∂|ri|1

∂φ can be obtained based on ∂ri(t)

∂φ .

With the eigenvalue and damping ratio sensitivity deduced above, the following linear

optimization problem can be formulated for the automatic tuning of control parameters

min

∆φj

σmax (6.9a)

subjected to

σi= Re {λi0}+∑︂M

j=1

Re {∂λi}

∂φj

∆φj, i = 1, . . . , N (6.9b)

σmax ≥σi(6.9c)

¯i=ξ

¯i0+∑︂M

j=1

∂ξ

¯i

∂φj

∆φj, i = 1, . . . , N (6.9d)

¯min 0 ≤ξ

¯i0(6.9e)

¯min ≤ξ

¯i(6.9f)

¯min ≥ξ

¯min 0 (6.9g)

|∆φj| ≤ 1%φj0(6.9h)

The objective function Eq. (6.9a) is to minimize the real part of the right-most LTP eigenvalues.

Decision variables

{∆φi, i = 1,· · · , M}

are the changes of control parameters with initial

values

{φi0, i = 1,· · · , M}

.Mstands for the total number of adjustable control parameters.

The initial values are determined following the classical design methods described in Chapter 3.

127

6. Sensitivity-Based Stability Improvement

The constraint Eq. (6.9b) gives the linear prediction of the real part of each LTP eigenvalue

after control parameters are updated. To guarantee the accuracy of the sensitivity-based

linear approximation, changes of parameters are limited by inequalities Eq. (6.9h), namely

permissible variance must be within

1% of the initial values.

¯i0

denotes the initial value

of the comprehensive damping ratio, and its prediction is given by the constraint Eq. (6.9d).

The inequality Eq. (6.9g) ensures that the minimum comprehensive damping ratio will not be

degraded by updating parameters. This linear optimization problem is solved by using the

commercial software GUROBI.

Start (k=0)

Initialize control parameters with conventional design rules

{φi0, i=1, , M}

Power flow calculation and small-signal model development

LTP modal analysis

 Calculate LTP eigenvalues and eigenvectors

 Calculate eigenvalue and damping ratio sensitivies

Formulate and solve the optimization problem Eq. (6.9)

 Determine changes of control parameters

{Δφi, i=1, , M}

 Determine sensitivity-based eigenvalue predictions

{λpred, i, i=1, , N}

For {φi=φi0+Δφi, i=1, , M}, rebuild the small-signal model

and recalculate LTP eigenvalues {λi, i=1, , N}

k=100?

Stop

Reduce the percentage

limits in Eq. (6.9h) by

half

max 1%

pred i i



















Yes

Update control parameters:

{φi0=φi0+Δφi, i=1, , M}

Update iteration step:

k=k+1

Figure 6.2: Sensitivity-based automatic control parameter optimization framework

Figure 6.2 shows the flowchart of the eigenvalue-sensitivity-based parameter optimization

algorithm. In the first step, control parameters are initialized with the conventional tuning

procedure. Power flow analysis is performed to determine the steady-state operation trajectory.

Then, the small-signal model is established, and the modal analysis is carried out to determine

LTP eigenvalues, eigenvectors and corresponding sensitivities. Next, the linear optimization

problem given by Eq. (6.9) is formulated and solved. The validity of the sensitivity-based

prediction is checked by recalculating eigenvalues with updated parameters. If deviations

between actual eigenvalues and their predictions are too large, the linear optimization problem

is modified by reducing limits of parameter changes. Specifically, the maximum percentage

change of control parameters in Eq. (6.9h) decreases from

1% to

5%. Otherwise, parameter

updates are accepted and the next iteration starts. The algorithm stops when a predefined

maximum number of iteration is reached, which is set to be 100.

The proposed design method is tested with the Type I GFL converter system. The

current control time constant is fixed to

0.5 ms

. Parameters of the DSRF-PLL are selected for

optimization, which are initialized to a desired bandwidth of

20 Hz

. The grid impedance is

adapted to the maximum value of the test bench (

17.4 mH

and

0.53 Ω

). Figure 6.3

128

6.1 Sensitivity-Based Parameter Optimization

shows the movement of eigenvalues for different steady-state operation trajectories. As the

active power reference

increases, the critical eigenvalues move towards the right half plane

with decreasing damping ratio, and eventually enter the unstable region.

Pr (W)

15001000

Figure 6.3: Movement of LTP eigenvalues of Typy I GFL converter system for different active

power reference Pr

To extend the stability margin, parameters of the DSRF-PLL are optimized with the

proposed design method. Another difference worth mentioning from the conventional tuning

procedure is that the cutoff frequencies of the four low-pass filters within the DSRF-PLL are

treated to be independent rather than sharing the same value, namely the decision variables

become

{∆φ1,∆φ2,∆φ3,∆φ4,∆φ5,∆φ6}={∆kp, , ∆ki,∆ωf1,∆ωf2,∆ωf3,∆ωf4}

To improve computational efficiency, only damping ratio of LTP eigenvalues with real parts

larger than -200 are examined. Figure 6.4 shows the movement of LTP eigenvalues at each

iteration. The critical eigenvalues migrate gradually towards the left half plane, which confirms

the effectiveness of the proposed methodology. Initial and optimized control parameters are

listed in Table 6.1. It is noted that asymmetric tuning the cutoff frequencies of the low-pass

filters is beneficial for the system stability.

Iteration Step

1001

Figure 6.4: Movement of LTP eigenvalues of Type I GFL converter system at each optimization

iteration. Crosses: sensitivity-based predictions. Dots: recalculated/true eigenvalues.

Experimental tests have been carried out on the single GFL converter system to verify the

effectiveness of the proposed control parameter tuning method. Figure 6.5 and Figure 6.6 show

the comparison of time-domain free and forced responses for different DSRF-PLL parameters.

The improvement of the stability margin and the damping performance is validated.

129

6. Sensitivity-Based Stability Improvement

Table 6.1: Control parameters of the DSRF-PLL for different iterations

Iteration kpkiωf1ωf2ωf3ωf4

0 (conventional design) 125.7 6580 301.6 301.6 301.6 301.6

50 171.1 9502 180.6 206.9 207.8 207.8

100 176.3 10290 142.5 212.9 155.3 215.8

Conventional Design

100 iterations

50 iterations

Figure 6.5: Measurements of the Type I GFL

2 Hz

disturbance applied to

2.0 s

for different DSRF-PLL parameters

Conventional Design

50 iterations

100 iterations

Figure 6.6: Measurements of the Type I GFL

changing from 1300 to

1400 W

2.0 s for different DSRF-PLL parameters

130

6.2 Sensitivity-Based Damping Loop Design

For a given control structure, it may happen that the stability margin cannot be further

expanded by adjusting the control parameters alone. Moreover, there exist also application

scenarios where the control parameters cannot be freely selected, for instance the droop

coefficients of GFM converters are defined by grid codes. For these cases, auxiliary damping

loops need to be developed. To this end, the participation factor analysis can provide insightful

guidance on the location selection of the damping loop.

Taking the single GFM converter system as an example, Figure 4.29 shows magnitudes

of participation factors between state variables and oscillation modes. The right-most low-

frequency modes

λ1,2

mainly contribute to dynamics of the power control loop, which suggests

that a local feedback loop can be added around the state variable

to improve the damping

performance, as shown in Figure 6.7.

Power

Calculation

GLPF(s)

1/s1/Jω0s

Swing equation

LPF

1/kQ

Outer Control Gaux(s)

ωθ

ω0

LPF

Aux. Damping Loop

Figure 6.7: Outer control diagram of the VSG converter with auxiliary damping loop

Based on experience in design of the power system stabilizer for conventional rotational

generators, the transfer function of the extra feedback loop can be generally expressed as

Gaux (s) = Kaux ·sTw

1 + sTw

⏞⏟⏟ ⏞

washout filter

·1 + sT1

1 + sT2

⏞⏟⏟ ⏞

phase compensation

(6.10)

where the phase compensation unit is used to guarantee that the newly introduced control loop

provides damping torque to the target oscillation. For the GFM converter a negative feedback

is adopted, namely the phase compensation is

180°

. The washout filter is a high-pass filter with

a time constant of

, which allows oscillations at the frequency of

λ1,2

to pass, while ensuring

that the system steady state is not affected by the use of the damping loop. The value of

commonly in the range of 1 to 20 seconds, here it is set to

5 s

. The gain

Kaux

determines the

amount of damping introduced by the auxiliary loop. Figure 6.8 plots the eigenvalue loci for

different values of

Kaux

. It is observed that the low-frequency mode becomes better damped as

Kaux

increases, and it can even turn into two real-valued eigenvalues. Moreover, the adoption

of the auxiliary damping loop brings a pure real eigenvalue near the origin which does not

threaten the system stability.

The proposed damping strategy is tested with the three-VSG laboratory prototype microgrid

shown in Figure 3.28. The inertia constant

is increased to

0.264 kg ·m2

. To excite dynamic

responses, an active power reference step of

600 W

is given to VSG 1. The output active

and reactive power responses of all three converters are calculated using voltage and current

131

6. Sensitivity-Based Stability Improvement

Kaux

15000

Figure 6.8: Movement of eigenvalues of the single GFM converter system for different values of

Kaux

measurements, and an apparent low-frequency oscillation between VSGs can be observed from

Figure 6.9. To validate the effectiveness of the proposed auxiliary damping loop, the same

active power reference step is repeated for different values of

Kaux

, experimental results are

plotted in Figure 6.10. Since dynamic responses of all three VSGs’ output active and reactive

power exhibit the same damping characteristic, only those of VSG 1 are plotted in Figure 6.10.

It is seen that, after enabling the auxiliary damping loop, the low-frequency oscillation can be

damped well, and the damping gets larger as Kaux increases.

VSG 1

VSG 2

VSG 3

Figure 6.9: Measurement of dynamic responses of VSGs output active power (left) and reactive

power (right) to 600 W active power reference step of VSG 1 at t= 2.0 s

Kaux=0

Kaux=300

Kaux=500

Kaux=700

Figure 6.10: Measurement of active and reactive power responses of VSG 1 for different Kaux

The sensitivity-based damping loop design principle is also tested with the GFL converters.

One conclusion drawn in Section 4.5.1 is that as the grid becomes weaker (namely SCR

decreases), LTP eigenvalues

λ3,4

in Figure 4.17 threaten the system stability. Those two

modes exhibit larger participation factor in dynamics of the PLLs, therefore, adding a negative

132

6.3 Summary

feedback loop around the state variable

in PLLs (see Figure 6.11) is an efficient way to move

λ3,4towards left.

Gaux(s)

vq+ ω

Figure 6.11: Control diagram of the SRF-PLL with auxiliary damping loop

In the test, the operational SCR is reduced to 1.8 by increasing both the power reference

and the grid line impedance. After the system reaches steady state, a

2 Hz

disturbance is given

to the state variable

. Free responses of

are shown in Figure 6.12 and Figure 6.13. It is seen

that the Type I GFL converter can be stabilized by the proposed damping loop. Additionally,

the damping performance of both types of converters can be improved by increasing the gain

Kaux. The experimental tests confirm the effectiveness of the proposed damping loop.

Kaux changes from 50 to 0

Kaux= 200 Kaux = 50

Figure 6.12: Measurement of free response of

of the Type I GFL converter for different values

of Kaux.2 Hz disturbance is applied to ηat t= 1.0 s and t= 2.0 s

aux

= 200 K

aux

= 50 K

aux

= 0

Figure 6.13: Measurement of free response of

of the Type II GFL converter for different values

of Kaux.2 Hz disturbance is applied to ηat t= 1.0 s,t= 2.0 s and t= 3.0 s

6.3 Summary

In this chapter, the sensitivity and participation analysis are used as design-oriented tools

to guide the optimization of control parameters and structures. By using the proposed

sensitivity-based optimization method, control parameters are iteratively modified to shift

133

6. Sensitivity-Based Stability Improvement

critical eigenvalues towards the left half plane without degrading the damping performance.

Though the same effect can be achieved by using a brute force methods (namely iterating over

all possible parameter combinations), the proposed method exhibits superior computational

efficiency because the sensitivity index indicates the fastest direction in which the critical

eigenvalues move towards the left. The application to the Type I GFL converter system

reveals that asymmetric tuning of d-channel and q-channel control parameters can improve

the system stability. For scenarios where the control parameters are predefined, extra damping

loops consisting of the proportional gain, high-pass filter and phase compensation unit can

be developed based on the participation analysis results. For both GFL converters and GFM

converters, negative feedback around the frequency state variable is an effective way to improve

the system stability. Theoretical analysis results have been confirmed with experimental

measurements.

134

Conclusion and Outlook

7.1 Conclusions

This thesis aimed at the modeling, stability and resonance analysis of modern converter-

dominated power systems. To this end, a modular, scalable and flexible numerical modeling

framework is developed on the MATLAB/Simulink platform. To gain detailed and insightful

understanding of the system dynamics, small-signal models are derived by using the component

connection method. Modal analysis is carried out to identify causes of abnormal resonances and

instability. Experimental tests on a laboratory multi-converter PHIL system validate the fidelity

of the simulation results. Cross-validation between stability and resonance evaluation results

obtained from numerical models, analytical models and hardware measurements confirms the

effectiveness of the proposed methodology. The major findings and conclusions are summarized

as follows.

•

The frequency coupling effect can be classified into the sequence coupling effect and

the frequency shift effect. Specifically, unbalanced three-phase branches will result in

couplings between voltages and currents at frequencies of

and

−ω

. Time periodic

operation trajectories caused by grid voltage harmonics will bring couplings at frequencies

and

{ω+nω0, n ∈Z}

. The influence of the asymmetric converter controller can

be regarded as the superposition of both, for instance, the PLL causes the coupling at

frequencies of ωand −ω+ 2ω0.

•

Similar to the LTI system, there exists a time-invariant modal space for the LTP system.

The system stability is determined by real parts of the system matrix in the modal

space. The eigenvalues of the LTP system are affected by the frequency coupling effect

to varying degrees. The major difference between LTP and LTI systems is that the

space transformation matrix of the LTP system, namely the LTP eigenvector matrix, is

time-periodic.

•

The eigenvalue-based stability analysis reveals that the GFL converter with DSOGI-PLL

and PR current controller has a larger stability margin. The DSRF-PLL with higher

135

7. Conclusion and Outlook

bandwidths can exhibit instability behavior when the input voltage becomes unbalanced.

The instability issue of the GFL converter under weak grid conditions can be overcome by

the GFM converter, while it is challenging to guarantee the stability of GFM converters

connected to strong grids. Moreover, the inner-loop control parameters of the GFM

converter can degrade the performance of the outer power loop.

•

The results of the generalized resonance mode analysis give that the integration of

GFL converters can cause a difference of harmonic resonances for positive and negative

sequences. The resonance frequencies and resonance modes are hardly affected by grid

voltage imbalances, while the grid impedance imbalance can introduce extra resonance

modes.

•

The LTP eigenvalue and damping ratio sensitivities can be used to optimize control

parameters. It is found that independently tuning the

d/α

-channel and

q/β

-channel

control parameters can extend the system stability margin. Guided by the participation

factor analysis, auxiliary damping loops can be added around the frequency state variable

of both GFL and GFM converters to improve the damping performance.

7.2 Future Work

There are still some open questions left unaddressed in this thesis, which are worth to be

extended and investigated:

•

The proposed power flow analysis does not cover the integration of induction machine

loads, constant power loads and GFM converters under unbalanced conditions. To

provide the fault-ride-through service, converters are commonly equipped with sequence-

decoupling structures and dual sequence current controllers, which should also be further

studied to evaluate their impact on the system stability.

•

In this thesis, the converters are assumed to be supplied with ideal DC voltage sources.

Average models of converters are used for the steady-state and small-signal analysis.

In practice, the DC side can be energy storage systems with DC-DC converters,

the interaction between DC and AC sides can bring new frequency coupling effects.

Moreover, the impact of other nonlinearities including PWM, saturation of controllers

and nonlinearity of passive components deserves further investigation.

•

The proposed power flow analysis gives whether there is a steady-state operation point

or trajectory. The eigenvalue-based small-signal analysis evaluates the system stability

near such equilibrium. However, the stability of converter-dominated power systems

against large disturbances, namely the transient stability, is still a challenging problem.

Dynamic modal analysis could be a promising solution.

•The establishment of the state-space model requires detailed modeling of the converter

systems, which could be critical for practical applications. Data-driven system

identification techniques should be further investigated to achieve the trade-off between

modeling accuracy and privacy protection. The sparse identification of nonlinear

dynamical systems (SINDy) technique [118] could be an interesting option.

136

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146

List of Figures

1.1 Electricity production in Germany by sources [3] . . . . . . . . . . . . . . . . . 1

1.2 Extended classification of power system stability [4] . . . . . . . . . . . . . . . 2

2.1

Qualitative illustration of the frequency coupling effect, current disturbance

∆

iin

at the frequency of

fin

can excite voltage response ∆

at multiple frequencies.

2.2 Small-signal models of the nonlinear time-periodic system. . . . . . . . . . . . . 8

2.3 Relation between eigenvalues and time-domain free responses . . . . . . . . . . 11

2.4

Principle of the impedance-based stability analysis. Top: impedance model.

Bottom: equivalent control system . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Nyquist diagram for single-input-single-output system . . . . . . . . . . . . . . 12

3.1

Waveform of a three-phase current or voltage. Blue: the complex-valued variable.

Red: the real part (αcomponent). Green: the imaginary part (βcomponent). . 18

3.2 Complex-domain description of three-phase variables . . . . . . . . . . . . . . . 18

3.3 Schematic and control of a single grid-following converter system . . . . . . . . 20

3.4 BasicschemeofSRF-PLL.............................. 21

3.5 Basic scheme of DSRF-PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.6 Basic scheme of DSOGI-PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.7

Block diagrams of PI current controller. Left: real domain implementation.

Right: complex domain representation. . . . . . . . . . . . . . . . . . . . . . . . 24

3.8

Block diagrams of PR current controller. Left: real domain implementation.

Right: complex domain representation. . . . . . . . . . . . . . . . . . . . . . . . 25

3.9

Signal flow graphs of the negative-sequence current. Top: the PI current

controller. Bottom: the PR current controller. . . . . . . . . . . . . . . . . . . . 26

3.10

Steady-state positive (left) and negative (right) sequence equivalent circuits of

the grid-following converter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.11 Schematic and control of a single VSG converter system . . . . . . . . . . . . . 27

3.12 Block diagram of the outer power control loop . . . . . . . . . . . . . . . . . . . 28

3.13

Block diagram of the inner cascaded voltage-current control loop with the virtual

impedance....................................... 29

3.14 Block diagram of the virtual impedance Zv(s)................... 29

3.15 Steady-state positive sequence equivalent circuit of the grid-forming converter . 31

3.16 Simplified small-signal model of the advanced PLLs. . . . . . . . . . . . . . . . 37

3.17 Simplified block diagram of the current control loop . . . . . . . . . . . . . . . 38

3.18 Simplified block diagram of the voltage control loop . . . . . . . . . . . . . . . 41

147

LIST OF FIGURES

3.19 Complete small-signal modeling of the converter-dominated power systems . . . 45

3.20

General framework for the modeling and stability analysis of converter-

dominatedpowersystems............................... 47

3.21 Configuration of the laboratory power-hardware-in-the-loop test setup. . . . . . 47

3.22

Free responses of the Type I GFL converter

2 Hz

frequency disturbance at

1.0 s 49

3.23

Forced responses of the Type I GFL converter to

100 W

power reference step at

t= 1.0 s ........................................ 49

3.24

Free responses of the Type II GFL converter to

2 Hz

frequency disturbance at

t= 1.0 s ........................................ 50

3.25

Forced responses of the Type II GFL converter to

100 W

power reference step

at t= 1.0 s ...................................... 50

3.26 Free responses of GFL converters to 2 Hz disturbances at t= 1.0 s ....... 51

3.27

Forced responses of the VSG converter to

100 W

power reference step at

1.0 s 52

3.28 Configuration of the multiple-converter test system for Case 4 . . . . . . . . . . 53

3.29 Configuration of the multiple-converter test system for Case 5 . . . . . . . . . . 53

3.30 Topology of the modified IEEE 13-bus system . . . . . . . . . . . . . . . . . . . 53

3.31

Forced responses of three VSG converters to

200 W

power reference step given

to VSG 1 at t= 1.0 s ................................. 54

3.32 Forced responses of converters and induction motor to 0.2 pu load torque jump 54

3.33 Steady-state bus voltages of the modified IEEE 13-Bus system . . . . . . . . . 56

3.34

Dynamic responses of the PLL estimated frequency to a 5% active power

reference step of VSC 1 (Solid lines – average model. Dashed lines – small signal

model) Left: PLL bandwidth is 30 Hz. Right: PLL bandwidth is 40 Hz ..... 56

3.35

Dynamic responses of the

-component of the grid-side inductor current to a 5%

active power reference step of VSC 1 (Solid lines – average model. Dashed lines

– small signal model) Left: PLL bandwidth is

30 Hz

. Right: PLL bandwidth is

40 Hz .......................................... 56

4.1 Physical meaning of the eigenvalue sensitivity . . . . . . . . . . . . . . . . . . . 63

4.2

Qualitative eigenvalue loci of

A − N

. Each element of the diagonal matrix

(a) corresponds to one vertical line in the Eigenvalue map (b). (c) Eigenvectors

of A−N associated with λ1+jkω0(left) and λi+j(k+ 1)ω0(right) . . . . . 67

4.3

Free response of

to an initial condition of

[︂1 0 ]︂T

0 s

. The dynamic

evolution of

shown in (c) is the superposition of two LTP modes, namely

λ1

in (a) and

λ2

in (b). Each LTP mode corresponds to five oscillation components.

4.4

Distribution of non-zero Fourier coefficients of the 14-dimensional time-periodic

matrix

(

). Each colored surface corresponds to a 14

14 Fourier series

coefficient of

(

), and their non-zero elements are denoted by dots. Extra

black dots result from the negative-sequence grid voltage. . . . . . . . . . . . . 72

4.5

Free response of PLL estimated frequency ∆

to initial condition ∆

η0

= 2

different starting time instances . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.6

Small-signal analysis for

20 Hz

. (a) LTP eigenvalue map. (b) Oscillation

components of each LTP mode. (c) PF of each LTP mode in state ∆η. . . . . . 74

148

LIST OF FIGURES

4.7

Small-signal analysis for

33 Hz

. (a) LTP eigenvalue map. (b) Oscillation

components of each LTP mode. (c) PF of each LTP mode in state ∆η. . . . . . 74

4.8

Convergence process of the eigenvalue sorting method. All eigenvalues of

A− N

for different truncation orders (top). Eigenvalues located in the fundamental

strip(bottom). .................................... 77

4.9

Eigenvalue map of

A − N

for different truncation orders

. The selected

eigenvalues in each step of the eigenvector sorting method are marked with circles.

4.10 Convergence process of the truncation order selection method . . . . . . . . . . 80

4.11

Maximum percentage deviation between

eQT0

and eigenvalues of

(

T0,

0). Left:

Balanced scenario. Right: Unbalanced scenario. . . . . . . . . . . . . . . . . . . 80

4.12

Maps of required truncation order for different parameter combinations. Left:

Balanced scenario. Right: Unbalanced scenario. . . . . . . . . . . . . . . . . . . 81

4.13

Distribution of diagonal elements of

eQT0

(blue circles) and eigenvalues of

Φ(T0,0) (orangecrosses). .............................. 82

4.14

LTP eigenvalue map of the Type I grid-following converter obtained by using

Algorithm 2 with different groupings of the state observation . . . . . . . . . . 84

4.15 LTP eigenvalue map of the grid-following converter . . . . . . . . . . . . . . . . 85

4.16 Magnitude of participation factors of the grid-following converter . . . . . . . . 86

4.17

Eigenvalue locus of the grid-following converter for different combinations of

SCRandBW..................................... 87

4.18

Eigenvalue locus of the grid-following converter for different combinations of

andBW........................................ 88

4.19

Eigenvalue locus for

30 Hz

and different

. Left: Type I grid-following

converter. Right: Type II grid-following converter . . . . . . . . . . . . . . . . . 89

4.20

Eigenvalue locus for different

. Left: Type I grid-following converter. Right:

Type II grid-following converter . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.21

Real part of the right-most eigenvalues of the Type I grid-following converter for

different parameter combinations. White areas are the unstable region. Dashed

black lines indicate stability boundaries determined based on conventional LTI

models. ........................................ 90

4.22

Real part of the right-most eigenvalues of the Type II grid-following converter for

different parameter combinations. White areas are the unstable region. Dashed

black lines indicate stability boundaries determined based on conventional LTI

models. ........................................ 90

4.23

Forced responses of the Type II GFL converter to a

−200 W

power reference

step at

1.0 s

for

20 Hz

and

0.5 ms

. Grey lines: measurements.

Solid lines:nonlinear average model. Dashed lines:small-signal model . . . . . . 91

4.24

Forced responses of the Type II GFL converter to a

−200 W

power reference

step at

1.0 s

for

20 Hz

and

3 ms

. Grey lines: measurements.

Solid lines:nonlinear average model. Dashed lines:small-signal model . . . . . . 91

4.25

Forced responses of the Type I GFL converter to a

−200 W

power reference

step at

1.0 s

for

20 Hz

and

0.5 ms

. Grey lines: measurements.

Solid lines:nonlinear average model. Dashed lines:small-signal model . . . . . . 92

149

LIST OF FIGURES

4.26 Measurements of the Type I grid-following converter for the increase of τ. . . 93

4.27 Measurements of the Type I grid-following converter for the increase of BW . . 93

4.28 LTP eigenvalue map of the grid-forming converter . . . . . . . . . . . . . . . . 93

4.29 Magnitude of participation factors of the grid-forming converter . . . . . . . . . 93

4.30 Eigenvalue locus of the GFM converter for different combinations of kPand J. 94

4.31 Eigenvalue locus of the GFM converter for different combinations of kQand J. 94

4.32

Eigenvalue locus of the GFM converter for different combinations of

SCR

and

Zv94

4.33 Eigenvalue locus of the GFM converter for different combinations of τand P M 95

4.34 Forced responses of the VSG converter for different inertia constants . . . . . . 95

4.35

Forced responses of the VSG converter for

0.264 kg ·m2

different active

powerdroopcoefficients ............................... 96

4.36

Forced responses of the VSG converter for

0.264 kg ·m2

and different

reactive power droop coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.37 Forced responses of the VSG converter for different virtual impedances . . . . . 97

4.38 Instability of the VSG converter caused by small virtual impedance . . . . . . . 98

4.39 Instability of the VSG converter caused by slow inner current control . . . . . . 98

5.1 Qualitative forced response of the LTP system . . . . . . . . . . . . . . . . . . 100

5.2

Equivalent admittances of the Type I grid-following converter for changes of

different control and physical parameters. Solid lines: analytical results. Crosses:

frequencysweepresults. ...............................105

5.3 Topology of a grid-connected multiple-converter system . . . . . . . . . . . . . 107

5.4 Basic principle of the conventional resonance mode analysis method . . . . . . 109

5.5

Basic principle of the LTP theory-based generalized resonance mode analysis

method ........................................114

5.6

Modal impedances of the test system for different grid impedances. Solid lines:

Lline1= 1 mH. Dashed lines: Lline1= 2 mH.....................117

5.7

Impedances from Bus 2 to all the buses. Solid lines: analytical results. Crosses:

frequencysweepresults. ...............................118

5.8 Modal impedances for different unbalance ratios D................119

5.9 Modal impedances under unbalanced grid impedance condition . . . . . . . . . 120

5.10

Impedances from Bus 2 to all the buses under the unbalanced grid impedance

condition. Solid lines: analytical results. Crosses: frequency sweep results. . . . 121

5.11

Voltage responses at each bus to a 1 A 550 Hz negative-sequence current

disturbance at Bus 2. Top: Steady-state time domain waveforms. Bottom:

Comparison between the frequency spectrums of the simulation and analytical

results..........................................121

5.12 Modal impedances for different La.........................122

5.13 Sensitivity indices of Lato all the resonance modes . . . . . . . . . . . . . . . . 123

5.14 Resonance frequency sensitivity analysis results. . . . . . . . . . . . . . . . . . . 123

5.15 Resonance mode sensitivity analysis results. . . . . . . . . . . . . . . . . . . . . 124

150

LIST OF FIGURES

6.1

Distribution of LTP eigenvalues of Typy I GFL converter system with changes

. Black circles: initial values of the eigenvalues. Crosses: sensitivity-based

predictions. Dots: recalculated/true eigenvalues. . . . . . . . . . . . . . . . . . 126

6.2 Sensitivity-based automatic control parameter optimization framework . . . . . 128

6.3

Movement of LTP eigenvalues of Typy I GFL converter system for different

active power reference Pr..............................129

6.4

Movement of LTP eigenvalues of Type I GFL converter system at each optimiza-

tion iteration. Crosses: sensitivity-based predictions. Dots: recalculated/true

eigenvalues. ......................................129

6.5

Measurements of the Type I GFL to

2 Hz

disturbance applied to

2.0 s

for different DSRF-PLL parameters . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.6

Measurements of the Type I GFL to

changing from 1300 to

1400 W

t= 2.0 s for different DSRF-PLL parameters . . . . . . . . . . . . . . . . . . . . 130

6.7 Outer control diagram of the VSG converter with auxiliary damping loop . . . 131

6.8

Movement of eigenvalues of the single GFM converter system for different values

of Kaux ........................................132

6.9

Measurement of dynamic responses of VSGs output active power (left) and

reactive power (right) to

600 W

active power reference step of VSG 1 at

2.0 s132

6.10

Measurement of active and reactive power responses of VSG 1 for different

Kaux132

6.11 Control diagram of the SRF-PLL with auxiliary damping loop . . . . . . . . . 133

6.12

Measurement of free response of

of the Type I GFL converter for different

values of Kaux.2 Hz disturbance is applied to ηat t= 1.0 s and t= 2.0 s . . . . 133

6.13

Measurement of free response of

of the Type II GFL converter for different

values of

Kaux

2 Hz

disturbance is applied to

1.0 s

2.0 s

and

3.0 s133

151

List of Tables

3.1 Physical Parameters of the Two-Level IGBT Converter System . . . . . . . . . 48

3.2 Power Flow Analysis Results for Single-Converter Cases . . . . . . . . . . . . . 48

3.3 Control Parameters of the single GFM Converter . . . . . . . . . . . . . . . . . 52

3.4 Steady-state Values of the Frequency and Filter Capacitor Voltages . . . . . . . 53

3.5 Branch Parameters of the Modified IEEE 13-Bus System . . . . . . . . . . . . . 55

3.6 Load Parameters of the Modified IEEE 13-Bus System . . . . . . . . . . . . . . 55

4.1 Parameters of the Type I Grid-Following Converter for Participation Analysis . 72

4.2

Parameters of the Type I Grid-Following Converter for the Study of the LTP

Eigenvalue Calculation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Parameters of Different Test Cases of the IEEE 13-bus System . . . . . . . . . 81

4.4 Coupling Effect Quantification of LTP Eigenvalue . . . . . . . . . . . . . . . . . 84

4.5 Impact of Parameter Changes on the System Stability . . . . . . . . . . . . . . 86

4.6 Base Case Parameters of the GFM Converter . . . . . . . . . . . . . . . . . . . 92

5.1 Parameters of the Single Type I Grid-Following Converter System . . . . . . . 105

5.2 Parameters of the Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.3 Parameter of the Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.4 Participation Factor Analysis Results under Symmetric Condition . . . . . . . 118

5.5 Participation Factor Analysis Results under unbalanced Condition . . . . . . . 120

6.1 Control parameters of the DSRF-PLL for different iterations . . . . . . . . . . 130

153

Wirtinger Calculus

Let z=x+jy, for xand yreal, denote a complex number and let

f(z) = F(x, y) = fRe (x, y)+1j·fIm (x, y)

be a general complex-valued function of the complex number

. In standard complex analysis

courses, f(z)is defined as differentiable/holomorphic when the limit

∂f

∂z = lim

∆z→0

f(z+ ∆z)−f(z)

∆z

exists for ∆zapproaches zero from any directions.

Setting

∆z= ∆x+j∆y

two possible paths for ∆

z→

0are considered. The first path goes in the horizontal direction

with ∆y= 0 and ∆x→0yielding

∂f

∂z = lim

∆x→0

F(x+ ∆x, y)−F(x, y)

∆x

= lim

∆x→0{︃fRe (x+ ∆x, y)−fRe (x, y)

∆x+jfIm (x+ ∆x, y)−fIm (x, y)

∆x}︃

=∂fRe (x, y)

∂x +j∂fIm (x, y)

∂x

.(A.1)

The second path goes in the vertical direction with ∆x= 0 and ∆y→0yielding

∂f

∂z = lim

∆y→0

F(x, y + ∆y)−F(x, y)

j∆y

= lim

∆y→0{︃fRe (x, y + ∆y)−fRe (x, y)

j∆y+jfIm (x, y + ∆y)−fIm (x, y)

j∆y}︃

=∂fRe (x, y)

j∂y +∂fIm (x, y)

∂y

.(A.2)

155

A. Wirtinger Calculus

When

(

)is differentiable/holomorphic, both expressions (Eq. A.1 and Eq. A.2) should be

the same, namely

∂fRe (x, y)

∂x =∂fIm (x, y)

∂y

∂fRe (x, y)

∂y =−∂fIm (x, y)

∂x

which is the Cauchy-Riemann condition. It is proved that if the partial derivatives of

fRe

and

fIm

with respect to

and

are continuous, the Cauchy-Riemann condition are sufficient for

f(z)being holomorphic.

As presented in Chapter 3, the complex conjugate operator, magnitude calculation function,

real and imaginary extraction operator do not satisfy the Cauchy-Riemann condition. The

complex partial derivative defined above must be extended. To this end, the total differential

of the bivariate function F(x, y)is first deduced

∆F=∂F (x, y)

∂x ∆x+∂F (x, y)

∂y ∆y

=∂fRe (x, y)

∂x ∆x+j∂fIm (x, y)

∂x ∆x+∂fRe (x, y)

∂y ∆y+j∂fIm (x, y)

∂y ∆y

Inserting the relation

∆x=1

2(∆z+ ∆z∗)

∆y=1

2j(∆z−∆z∗)

yields

∆F=1

2[︃∂

∂x (fRe (x, y) + jfIm (x, y)) −j∂

∂y (fRe (x, y) + jfIm (x, y))]︃∆z

2[︃∂

∂x (fRe (x, y) + jfIm (x, y)) + j∂

∂y (fRe (x, y) + jfIm (x, y))]︃∆z∗

2(︃∂

∂x −j∂

∂y )︃F(x, y) ∆z+1

2(︃∂

∂x +j∂

∂y )︃F(x, y) ∆z∗

The Wirtinger derivatives are introduced by defining

∂

∂z

∆

2(︃∂

∂x −j∂

∂y )︃

∂

∂z∗

∆

2(︃∂

∂x +j∂

∂y )︃(A.3)

then the differential of the complex function f(z)can be written as

∆f=∂f (z)

∂z ∆z+∂f (z)

∂z∗∆z∗.

It can be immediately derived from Eq. (A.3)

∂

∂z z∗=∂

∂z∗z= 0

which means that

z∗

can be regarded as a constant value when differentiating

(

)with respect

to z, namely calculating ∂f(z)

∂z . The same goes for ∂f(z)

∂z∗.

156

Small-Signal Model of Induction

machine

For the sake of consistency, stator and rotor currents of the induction machine instead of fluxes

are used as state variables for the modeling of induction machines. Transform rotor currents

into the stationary reference frame of the stator, dynamics of the induction machine can be

described by the state-space equation

vαβs =rsiαβs +Lsd

dt iαβs +Lmd

dt iαβr

vαβr =rriαβr +Lmd

dt iαβs +Lrd

dt ir−jωr(Lmiαβs +Lrir)(B.1)

where

and

are the stator inductance and resistance.

and

are the rotor inductance

and resistance.

denotes the linkage inductance.

ωr

is the electrical speed of the rotor.

vαβs

and

iαβs

are the stator voltage and current.

iαβr

is the rotor current. All rotor variables are

referred to the stator windings.

The electromagnetic torque is given by

Tel =3

2npLmIm {︂iαβs ·i∗

αβr}︂(B.2)

where

is the pole pair number. Let

denote the load torque, the swing dynamic of the

induction machine is given by

ω˙r=(Tel −Tload)np

JIM

(B.3)

where JIM is the inertia constant of the induction machine.

157

B. Small-Signal Model of Induction machine

By linearizing Eq. (B.1) - Eq. (B.3) around steady-state operation trajectories, time-periodic

small-signal model of the induction machine can be obtained

∆x˙IM (t) = AIM (t) ∆xIM (t) + BIM (t)[︄∆vαβs (t)

∆v∗

αβs (t)]︄

[︄∆iαβs (t)

∆i∗

αβs (t)]︄

⏞⏟⏟ ⏞

∆yIM (t)

=[︄10000

01000]︄

⏞ ⏟⏟ ⏞

CIM (t)

∆xIM (t)(B.4)

where the stator current is selected as output yIM . The state vector is given by

∆xIM (t) = [︂∆iαβs (t),∆i∗

αβs (t),∆iαβr (t),∆i∗

αβr (t), ωr]︂T

158