Small-Signal Modeling and Stabil i ty Analysis of

Grid-Converter Interactions

2 0 1 9 - 0 6 - 0 3 , I E E E P E D G 2 0 1 9 , X i ’ a n , C h i n a

X i o n g f e i Wa n g , P r o f e s s o r, D e p a r t m e n t o f E n e r g y Te c h n o l o g yR e s e a r c h P r o g r a m L e a d e r f o r E l e c t r o n i c P o w e r G r i d ( w w w. e g r i d . e t . a a u . d k )

Instructor BioXiongfei Wang is a Professor and Research Program Leader on Electronic Power Grid (eGrid)with the Department of Energy Technology, Aalborg University, Denmark. His research interestsinclude modeling and control of grid-interactive converters, stability and power quality of power-electronic-based power systems, harmonic analysis and mitigation.

In 2016, he was selected into Aalborg University Strategic Talent Management Program, whichaims at developing next-generation research leaders for Aalborg University. He received six PrizePaper Awards in IEEE Transactions and conferences, the 2017 Outstanding Reviewer Award forthe IEEE Transactions on Power Electronics, the 2018 IEEE PELS Richard M. Bass OutstandingYoung Power Electronics Engineer Award, and the 2019 IEEE PELS Sustainable Energy SystemsTechnical Achievement Award. He serves as an Associate Editor for the IEEE Transactions onPower Electronics, the IEEE Transactions on Industry Applications, and the IEEE Journal ofEmerging and Selected Topics in Power Electronics.

CONTENT

INTRODUCTION

SMALL-SIGNAL MODELING

IMPEDANCE-BASED STABILITY ANALYSIS

PROSPECTS AND CHALLENGES

INTRODUCTION

Grid-converter interactions

Grid-forming/-following converters

Grid-Converter InteractionsLess physical properties, more control dependency

Larg

e W

ind

Gen

erat

orLa

rge

Win

d G

ener

ator

Turbine P & ω ControlTurbine P & ω Control

Grid Q &V ControlDC-Link V ControlGrid Q &V ControlDC-Link V Control

Grid SynchronizationGrid Synchronization

Current Control Current Control

IGBT Switching IGBT Switching

Trad

ition

al G

ener

ator

Trad

ition

al G

ener

ator

Prime MoverPrime Mover

Excitation ControlExcitation Control

0.1Hz

1Hz

100Hz

0.1Hz

10Hz

500Hz

2500Hz

Traditional Generator Output Voltage

Wind Generator Output Voltage

Harmonic Spectrum

Modulator

DC-Bus Voltage Control

Vector Current Control

Vdc

Vdc

Phase-Locked Loop (PLL)

iac

iac

θg

Vac

AC-Bus Voltage Control

id iq

Vacd

Vacd

Vacd

dqαβ

* *

*

* *

Grid-Converter InteractionsNegative damping induced by converter controllers

Lg

Cf VgGcl iac

Zg

*iac

YclVdcVg

Grid

ZgLf Lg

Cf

- Re{Ycl}>0: stable, yet under-damped- Re{Ycl}=0: resonant, zero damping - Re{Ycl}<0: unstable, negative damping

Grid-Converter Interactions Mapping from control loops to instability phenomena

Lg

Cf VgGcl iac

Zg

*iac

YclVdcVg

Grid

ZgLf Lg

Cf

f1: Grid fundamental frequency, fs: Switching frequency

Near-synchronous oscillations

Sub-synchronous oscillations

Harmonic oscillations

f1

2f1

fs/2

fs

Sideband (fs) oscillations

Sideband (f1) oscillationsModulator

DC-Bus Voltage Control

Vector Current Control

Vdc

Vdc

Phase-Locked Loop (PLL)

iac

iac

θg

Vac

AC-Bus Voltage Control

id iq

Vacd

Vacd

Vacd

dqαβ

* *

*

* *

GRID

Short-Circuit Ratio (SCR) and inertia

AC interconnect: low SCR, low-frequency

parallel/series LC resonances

DC interconnect: control interactions with

HVDC converter station

Wind power plant

Grid-Converter Interactions Weak grid with multiple resonance frequencies

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

350

400

450

500

|Z| [

]

Frequencu [Hz]

10 Major sections5 Major sections3 Major sections

©TenneT

VSC-HVDC + Offshore Wind

Filter resonance in the offshore HVDC converter station

Electrification of railways

Locomotives is out of control because of abnormal harmonics

MMC-HVDC Transmission

Transformer resonance with current control of MMC

©CSG

Real-Life ChallengesDamping: positive (harmonics), negative (instabilities), zero (resonances)

0 0.01 0.02 0.03 0.04 0.05 0.06-400

-300

-200

-100

0

100

200

300

400Måling i Grenå kl. 13:06, 28 kHz sampling

Time [sec]

Cur

rent

[A]

IL1

IL2IL3

0 0.01 0.02 0.03 0.04 0.05 0.06-30

-20

-10

0

10

20

30Måling på Platformen kl. 13:06, 28 kHz sampling

Time [sec]

Cur

rent

[A]

IL1IL2IL3

Current

PoCPCC

PCC

Source: C. F. Jensen, Energinet.dk, “Harmonic assessment in modern transmission network,” Harmony Symposium, Aalborg, August 2015.

Real-Life ChallengesHarmonic current generated from offshore wind power plant

Grid-Forming/-Following Converters Synchronization control is the key: from voltage-based to power-based

vg

Lg

vp

PLLθp

Current ControlPWM

i

u

vc

Lf

vdc

idqref

PLL vg

Xg

vpθp

i

idqrefejθp

Grid-following control

Voltage-based synchronization

vg

Lg

vp

Power Sync. Control

Voltage and Current ControlPWM

i

u

vc

Lf

vdc

Vpref

vp i

Grid-forming control

Power Sync. vg

Xg

vpθp

i

vmrefejθp

i

Power-based synchronization

Grid-forming convertersState of the art of power-based synchronization control

Load-frequency controlDroop controlVirtual synchronous machine (VISMA)

Power synchronization control Synchronverter Synchronous machine

matching control

Virtual synchronous generator (VSYNC)

1986199320072008

Virtual oscillator control

2016201620112011

PCC

vgvpvc Zgi j

cv Ve P

V Q

Pmax 1 p.u. @ SCR=1

SMALL-SIGNAL MODELING

Review of small-signal modeling methods

Dynamic representations

OutputFilter

Input Filter

DC Source

Power Grid

Modulator

Controller

d(t)

General diagram of grid-connected converters

Sources of nonlinearity and time-variance - Hybrid: Both continuous filter dynamics and discrete

switching events

- Nonlinear: Feedback control - dependence of the

duty cycle, i.e. d(t), on the input variables

- Time-variant: Switching modulation process and

time-periodic operating trajectory

Dynamic Properties of Grid-Connected Converters Hybrid, nonlinear and time-variant systems

Historical Review of Small-Signal ModelsHarmonic analysis and controller design

1970

1985

1997

2000

2003 2014Persson [7] - thyristor HVDCFrequency response analysis Describing Function with single sinusoidal inputs

For control design

Sakui and Fujita [8] - thyristorrectifier, Switching Function model w/o firing angle variation considered

For harmonic analysis

Mattavelli, Verghese, Stankovic[11] - thyristor FACTS devicesDynamic Phasor with time-variant Fourier coefficients

For control design

Wang, Harnefors, Blaabjerg [17] -Unified Impedance Model from dq-frame to αβ-frame, 2nd-orderHarmonic Transfer Matrix

For control design

Cespedes and Sun [16] - stability effect of PLL on PWM converterHarmonic Balance, Multi-Input Describing Functions

For control design

Rico, Madrigal, Acha [13] -STATCOM with phase angle control, Extended Harmonic Domain (EHD)

For harmonic analysis

Mollerstedt [12] - locomotive inverter, Harmonic State-Space (HSS) modelling, Harmonic Transfer Matrix

For harmonic stability analysis

1989Larson, Baker, McIver [10] –thyristor HVDC, numerical simulations derived Harmonic Cross-Coupling Matrix

For harmonic/control analysis

2007Harnefors [14] - PWM converterDQ-frame linearized model with the phase variationWen, Boroyevich, et, al [15], 2016

For control design

1986Ngo [9] - PWM converterState-Space Averaging with Park transformation DQ-frame linearized model

For control design

2016

Linearized (dq) modelAveraged (dq) modelAveraged (abc) model Switching function model

Three-phase balanced converters

p

n

j

Neglect switching parasitic parametersNonlinear, time-variant , continuous

1( ) ( )s

t

j jt Ts

d t s t dtT

Small-ripple approximationHigh pulse ratios (fs/f1 > 20) Nonlinear, time-periodic dabc(t)

q

ω1

α

β

Vα

Vβ V

dVdVq

ω1t

Nonlinear, time-invariant ddq(t)

A

1

1⋯ 1

⋮ ⋯ ⋮

1⋯

Jacobian matrix (Taylor series)Linear, time-invariant

State-Space Averaging with Park TransformationAveraged dq-frame model based on single real space vectors

Linearized (dq) modelGeneralized averaging over the fundamental frequency (±ω1) Switching function model

Three-phase unbalanced/single-phase converters

p

n

j

Neglect switching parasitic parametersNonlinear, time-variant , continuous

11( ) ( )

t jk tj jt T

d t s t e dtT

Small-ripple approximationUnbalanced three-phase systems (k = ±1)

A

1

1⋯ 1

⋮ ⋯ ⋮

1⋯

Jacobian matrix (Taylor series)Linear, time-invariant

Nonlinear, time-invariant, but multiple ddq(t)

0 f1-f1 f

Vabc

Va(t) Vb(t) Vc(t)

Generalized Averaging (Dynamic Phasor) Averaged dq-frame model based on multiple complex space vectors

Harmonic State-Space Averaged αβ-frame model based on multiple real space vectors

Nonlinear time-variant system

( ) ( ), ( ),

( ) ( ), ( ),

x t f x t u t t

y t g x t u t t

Linear time-periodic (LTP) systems

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

x t A t x t B t u ty t C t x t D t u t

Linearization about a time-periodically

varying trajectory

Fourier series expansion for A(t), B(t), C(t), D(t);

Exponentially modulated periodic (EMP) input u(t)

1 1( ) , ( )

jk t jk tstk k

k k

A t A e u t e U e

Harmonic state-space (HSS) model

( )

X A N X BUY CX DUs

Harmonic transfer function:

1( ) ( ) G C I A N B Ds s

1 1[ 0 ] N Tdiag j j

Three-phase unbalanced/single-phase converters

Switching Model

DQ-Frame Averaged Model(Balanced Three-Phase)

Harmonic Linearization(Harmonic Balance)DQ-Frame LTI Model

Alpha-Beta-Frame Model(Balanced Three-Phase)

- Nonlinear- Time-variant - Both continuous and discrete dynamics

AC-DC Converters (e.g. VSCs)

- Nonlinear - Time-variant- Continuous

- Nonlinear - Time-invariant- Continuous

- Linear - Time-invariant- Continuous

- Linear - Time-invariant- Continuous - Harmonic Transfer Function

Dynamic Phasor Model (Unbalanced Three-Phase)

Dynamic Phasor Model(Switching Frequency)

Harmonic State-Space Model (Unbalanced Three-Phase)

Harmonic State-Space Model (Extended to Switching Frequency)

Linear Time-Periodic Model (Harmonic State-Space)

Harmonic Domain Model1( ) ( )

t

t Tx t x d

T

1( ) ( )t jk

t Tx t x e d

T

State-Space Averaging

Nonlinear Time-Periodic Model (Multiple Sinusoids)

Generalized Averaging

f < fs/2 fs fs/2 f < fs

- Nonlinear - Time-invariant- Continuous

Comparisons of Small-Signal Modeling Methods Linearization around time-invariant/-periodic operating point/trajectory

- Dynamic phasor model is a generalization of the dq-frame averaged model, which extracts the time-invariant operating point in the

frequency domain

- Harmonic state-space model is a generalization of the stationary- (αβ-) frame model, which linearizes the system on time-periodic

operating trajectories in the time domain

Comparisons of Small-Signal Modeling Methods Model adequacy for analyzing grid-converter interactions

Model Adequacy DQ-frame(averaged) Model

Dynamic Phasor Model

Harmonic State-Space Model

Sideband (f1) oscillations + + +

Harmonic oscillations + + +

Sideband (fs) oscillations − + +

Low pulse-ratio (fs/f1) − + +

Unbalanced three-phase systems − + +

L

L

LN

n

p

ia

ib

ic

vaN

vbN

vcN

van

vbn

vcn

vdcis

rdc

Cdc

rL

rL

rL

idc

Voltage-Source Converter (VSC) with a non-ideal dc-link

1

1

00

d d dN ddc

q q qN q

i d v iLdL vi d v iLdt

32

ddcs d q

q

idvC i d d

idt

vdcis Cdc

idc

–ddid –dqiq

dqvdc

ddvdc

iq

id L

L

vdN

vqN

ω1Liq

ω1Lid

23

23

Averaged (dq-frame) model for three-phase balanced system

p

n

van

vbn

vcn

vdc Cdc N

vaN

vbN

vcN

L ia

ib

icis

idc

Switching function model of VSC

Multiple-Input Multiple-Output (MIMO) Representation Averaged dq-frame model of converter power stage

Simplified Model with Symmetrical DynamicsLinear time-invariant (LTI) approximation with nearly ‘ideal’ dc-link

vdcis Cdc

idc

–ddid –dqiq

dqvdc

ddvdc

iq

id L

L

vdN

vqN

ω1Liq

ω1Lid

23

23

Averaged (dq-frame) model for three-phase balanced system

dqVdc

ddVdc

iq

id L

L

vdN

vqN

ω1Liq

ω1Lid

LTI model for vector current control

Vd d dNdc

q q qN

dL L i d vdti d vdL L

dt

_ ( )L dq

Ls LZ s

L Ls

Single-Input Single-Output (SISO) Representation

L1: SISO complex transfer functions based on complex vectors equal to symmetrical transfer matrices based on real vectors

- Symmetrical transfer function matrix of L-filter can be represented by SISO complex transfer function:

( )sL L

L sL

L_dqZ s _ 1( ) ( )L dqZ s s j L

sL

ω1L

sL

–ω1L

id

iq vq

vd

(s+jω1)Lid + jiq vd + jvq

Complex transfer functions ↔ symmetrical transfer matrices

SISO Representation in Stationary FrameFrequency translation of complex transfer functions from dq- to αβ-frame

L2: Park transformations for symmetrical transfer matrices equal to frequency shifts for complex transfer functions

- Representing Park transformations by complex exponential functions (Euler’s formula)

1 1

1 1

cos( ) sin( )sin( ) cos( )

d

q

i it tt ti i

1 1cos( ) sin( ) ( )d qi ji t j t i ji 1

1

( )( )

j td q

j td q

i ji e i jii ji e i ji

- Frequency shift of complex transfer functions

1( )( )d q d qv jv L s j i ji 1 11( ) ( ) ( )j t j tde v jv L j e i ji

dt

( )v jv Ls i ji

_ 1( )L dqZ s j _ ( )LZ s _ 1_ ( ) ( )L dq LZ s Z s j

SISO Model of Vector Current ControlConstant dc-link voltage and no phase (θg) variation

DQ-frame PI current control with digital modulator

Plant Modulator PI

*iq

*id

dq

id

iqPI

αβ iα

iβ dq

αβ Vdc

1Umα

Umβθg θg

z-1 ZOHdα

dβ Vdc

Vvscα

Vvscβ L-Filter

Vacα Vacβ

iα

iβ

SISO model of dq-frame PI current control SISO model of αβ-frame PR current control

PI*idq

idq

Gd(s) L(s+jω1)+rL

1Vac_dq

idqPR

*iαβ

iαβ

Gd(s) Ls+rL

1Vac_αβ

iαβ

u(t)

s(t)

Time Delay of Digital Modulator 1.5 sampling period: ZOH + one sampling period

Vg

GridZg

Vac

PLLθg

Current Control PWM

i

C

Um

Vvsc

Lf

Vdc

Vector current control with PLL included

DQ-frame PLL

Vα cos

Vβ ω1

ωe θg

sin

-sin cos

Vd

Vq PI 1s

q

ω1

α

β

Vd

VdVq = 0 ω1tqV

dV

V

Basic principle

Dynamic Effect of Phase-Locked Loop (PLL)Including phase (θg) variation into current control

Small-Signal Modeling of PLLPerturbations on the input voltage vector and the output phase

Vα cos

Vβ ω1

ωe θg

sin

-sin cos

Vd

Vq PI 1s

DQ-frame PLL

q

ω1

α

β

Vd

VdVq = 0 ω1tΔVd

ΔVqδ

d cqc

Vdc

Vqc

10 0 0

1

0,

, g

j tdq d dq

jcg

dq

dq

V V j V V e

t V V e

V

Small-Signal Modeling of PLLAsymmetrical dynamics between d-axis and q-axis

Vα cos

Vβ ω1

ωe θg

sin

-sin cos

Vd

Vq PI 1s

DQ-frame PLL

ΔVqGPI(s) 1

sδ

Vd0

Vq Δωc

Second-order model of PLL

10 0 0

1

, 0dj t

dq dq d

g

qV V e V

t

V V j

1 1 )0

0

(

1 ,

gj j t j tcdq d

q d

q dq

cq

j

V V e V eV

V V V

e

e j

0

( )( ) , ( )

( )PI

PLL q PLLPI d

G sH s V H s

s G s V

iGd(s)PI

Vac

Vvsc

idq L-Filter

PLL

αβ dq

αβ dq

θg

θg

*idq

Ls1Udq

c

cU

DQ-frame current control with PLL dynamics

PLL Bridges Voltage Disturbance Current Negative damping introduced on the q-q axis

Equivalent block diagram with PLL dynamics

idq

idq

*idq Vvsc_dqUdq

Vac_dq

iPLL_dq

UPLL_dq

L-Filter

GPI(s) Gd(s)

GPLL(s)

YPLL(s)

Yp(s)Udq

c

c

Asymmetrical [YPLL(s)]

Vac_d iPLL_d

Vac_q

0

0

HPLL (s) iq0

HPLL (s) –id0iPLL_q

θe

θe

1 1( )0 0 0

0

_ 0 0

1 (1 ) )

( )

, (1

j t jcdq dq dq

j cdq d

t jdq dq d

q

PLL dq PLL q

q

dq

dq dq

i i i

e j i j i j

i H s

i i e e i e

i

i

ji j Vi

iGd(s)PI

Vac

Vvsc

idq L-Filter

PLL

αβ dq

αβ dq

θg

θg

*idq

Ls1Udq

c

cU

DQ-frame current control with PLL dynamics

MIMO Representation of PLL EffectAsymmetrical transfer function matrices in the dq-frame

Equivalent block diagram with PLL dynamics

idq

idq

*idq Vvsc_dqUdq

Vac_dq

iPLL_dq

UPLL_dq

L-Filter

GPI(s) Gd(s)

GPLL(s)

YPLL(s)

Yp(s)Udq

c

c

Vac_d UPLL_d

Vac_q

0

0

HPLL (s) –Uq0

HPLL (s) Ud0UPLL_q

θe

θe

Asymmetrical [YPLL(s)]

Vac_d iPLL_d

Vac_q

0

0

HPLL (s) iq0

HPLL (s) –id0iPLL_q

θe

θe

Asymmetrical [GPLL(s)]

Complex-valued equivalent of asymmetrical transfer matrices (dq-frame)MIMO Representation of PLL Effect

gdd (s)ud

uq yq

yd

gdq (s)

gqd (s)

gqq (s)

Asymmetrical transfer matrix for real vector(dq-frame, ω)

G+(s)udq

udq ydq

ydq

G–(s)

G–(s)

G+(s)

Equivalent transfer matrix for complex vector (dq-frame, ω → ω, –ω)

,dq d q dq d qu u ju y y jy ( ) ( ) ( ) ( )( )

2 2( ) ( ) ( ) ( )

( )2 2

dd qq qd dq

dd qq qd dq

g s g s g s g sG s j

g s g s g s g sG s j

( ) ( ) ( ) ( )( )

2 2dd qq qd dqg s g s g s g s

G s j

Complex-valued equivalent of asymmetrical transfer matrices (αβ-frame)Generalized MIMO Representation

G+(s)udq

udq ydq

ydq

G–(s)

G–(s)

G+(s)

Equivalent transfer matrix for complex vector(dq-frame, ω → ω, –ω)

G+(s–jω1)uαβ

yαβ

yαβ

G–(s–jω1)

G–(s–jω1)

G+(s–jω1) e-j2ω1tej2ω1tuαβ

Equivalent transfer matrix for complex vector(αβ-frame, ω → ω, 2ω1–ω)

( ) ( )dq dq dqy G s u G s u 1 1 1( ) ( )j t j t j te y G s e u G s e u

12

1 1( ) ( ) j ty G s j u G s j e u

( ) ( )dq dq dqy G s u G s u 1 1 1( ) ( )j t j t j te y G s e u G s e u

1 12 2

1 1( ) ( )j t j te y G s j u G s j e u

Physical insights revealed by the αβ-frame transfer matricesGeneralized MIMO Representation

Vdc Va

PLLθg

Current Control PWM

ia

Um

ic

ib

Vb

Vc

L

L

L

Three-phase three-wire converter

0 f1-f3 f

ia, ib, ic

f7-f5 f3 f9

Output current response

2ω1–ω

ω

0 f1-f1 f

Va, Vb, Vc

-f7 f7f5-f5

Input voltage perturbation

ω

Non-zero-sequence triplen harmonics in the three-phase three-wire converters with asymmetrical

control dynamics in the dq-frame

DQ-transformation

( ) ( )( ) ( )

d dd dq d

q qd qq q

V Z s Z s iV Z s Z s i

DQ-frame impedance model(real vector)

Zdd (s)id

iq Vq

Vd

Zdq (s)

Zqd (s)

Zqq (s)

Complex equivalence

DQ-frame impedance model(complex vector)

( ) ( )

( ) ( )dq dq

dq dq

V iZ s Z sV iZ s Z s

Z+(s)idq

idq Vdq

Vdq

Z–(s)

Z–(s)

Z+(s)

Frequency translation

Stationary (αβ)-frame impedance model(complex vector)

Z+(s–jω1)iαβ

Vαβ

Vαβ

Z–(s–jω1)

Z–(s–jω1)

Z+(s–jω1) e-j2ω1tej2ω1tiαβ

1 1

1 1

2 21 1

( ) ( )

( ) ( )j t j t

V iZ s j Z s j

Z s j Z s je V e i

Mathematical equivalence between different transfer matricesGeneralized MIMO Representation

IMPEDANCE-BASED STABILITY ANALYSIS

Basic principle: minor feedback loop and Nyquist criterion

Stability effects of different control loops

Yc(s)ic(s) Vs(s)

Zs(s)

Vo(s)

io(s)

Equivalent circuit of grid converter

Im

Re(-1, j0)

Gain Margin

PhaseMargin

Yc(s)Zs(s)

io

Yc(s)Vs

ic

Concept of minor feedback loop

Stable Zs(s)!

T (s) = Yc(s)Zs(s)

SISO Impedance-Based Stability Analysis Minor-feedback loop gain and Nyquist criterion

[Yc(s)]ic(s) Vs(s)

[Zs(s)]

Vo(s)

io(s)

Equivalent circuit of grid converter

[Yc(s)][Zs(s)]

io

[Yc(s)]Vs

ic

MIMO minor feedback loop

MIMO Impedance-Based Stability AnalysisReturn-ratio matrix and generalized Nyquist criterion

- Generalized Nyquist stability criterion

det I- ( ) ( ) 0c sY s Z s

- Nyquist diagrams of λ1 and λ2

Current Control for Grid-Following Converters Time delay destabilizes the stability of current control with L-filter

Current control w/o time delay - always stable! Phase response within 180°

Current

Voltage

Measured waveforms for current control

Current control w/ time delay - adding phase lagPhase response out of 180°

PR*iαβ

iαβ

Gd(s) Ls+rL

1Vac_αβ

iαβ

PRiαβ

iαβ

Ls+rL

1Vac_αβ

iαβ

Current Control for Grid-Following Converters Time delay introduces a frequency-dependent negative resistance

Vdc Vg

GridZg

Vac

PLLθg

Current Control PWM

i2

L2

i1

C

Um

Vvsc

Current control with LCL-filter (i1)

L1s1 1 1i1 ic

i2

Vac

Gd(s)PRVcVvsc

LCL-Filter

i1

i1Cs L2s

*

Single-loop converter current control

1 1 1 11

1 ,p o c d pL

Y Y T G G YZ

- L-filter plant and open-loop gain

1

1

1

11

1,1

o d

dc d

Y

YG

YG

- Closed-loop gain and output admittance

111 1

1 1

,1 1

ocl cl

YTG YT T

11 1 cos( ) sin( )dj T

d d dp p

Y e T j Tk k

Vc Vac

i2L2

i1

ic

CY1o

Y1d

G1cli1* Y1cl

Impedance model

Current Control for Grid-Following Converters Impedance-based stability analysis with LCL-filter only

* 1 21 1 1

1 2 1 2 2

11 1

c L C accl

c L C c L C L

Y Y Vi G i

Y Y Y Y Z

-40

-20

0

20

40

Mag

nitu

de (d

B)

10 100 1000 5000-360

-270

-180

-90

0

Phas

e (d

eg)

Frequency (Hz)

Stable w/ L-Filter

Open-loop gain T1

-100

-50

0

50

100

Mag

nitu

de (d

B)

10 100 1000 5000-180

-90

0

90

180

Phas

e (d

eg)

Frequency (Hz)

Y1c

YL2C

PM<0

fs/6

Unstable w/ LCL-Filter

Impedance ratio Y1c/YL2C

Vc Vac

i2L2

i1

icCfY1cG1cli1

YL2C

*

d sT T=1.5

Current Control for Grid-Following Converters Reducing time delay for robust stability of current control

Current

Voltage

1.5( ) 6, 2]sT sd s sG s e ( ) 6, 2]sT s

d s sG s e

Ts: computation delay; 0.5Ts: PWM delay 0.5Ts: computation delay; 0.5Ts: PWM delayInterrupt shift with 0.5Ts

Voltage Control for Grid-Forming ConvertersDual-loop voltage and current control scheme

Dual-loop voltage Control

Voltage controller (Gv):

Current controller (Gi):

≈ =

L1

CfVdc

GiPWMiref

v

Gv

i1

vref

L1

Cf

PWM

vi1 L2i2u

Controller

Loador

Grid

Small-signal model of power stage

Block diagram of LC-filteri piG = K

2 21+

rvv pv

K sG = Ks ω

Voltage Control for Grid-Forming ConvertersSmall-signal modeling of dual-loop voltage control

LC-filter plant

1

1 f1

L

olL C

ZZZ Y 1 f

11

uv

L C

GZ Y

1 f

11

ii

L C

GZ Y

f

1 f1

C

uiL C

YGZ Y

iT s =T1( )Current loop gain:

2( )1v

i

TT s =+T

Voltage loop gain:uv d i vT s = G G G G2( )

Small-signal model of dual-loop voltage control

Output impedance:

ol 1

o1 2

1( )

1uv d i iiZ T + G G G G

Z s =+T T

Voltage Control for Grid-Forming ConvertersStability of inner current loop, fLC < fs/6

-50

0

50

100

101 102 103 104-720

-540

-360

-180

0

180

Bode Diagram

Frequency (Hz)fs/6 fs/2

( )i ui d piT s = G G K

GM20dB

pis

100

6

ui

KωG j

Critcal Kpi for stable current loop:

GMs 20dB10

6

i

ωT j

f f2

1 f 1 f1 1

C

uiL C

Y sCGZ Y s LC

Phase lag: 90° → −90°

cos sin d d dG ωT j ωT

Additional phase lag of 90° at ωs/6

Phase crossingover −180° at ωs/6

Lager Kpi → smaller GM

Voltage Control for Grid-Forming ConvertersStability of outer voltage loop, fLC < fs/6

-60

-40

-20

0

20

40

101 102 103 104-540

-360

-180

0

180

Bode Diagram

Frequency (Hz)

21 f f1 d pis LC sC G K

v i d uvv

i

G G G GT s =

+T( )

1

Larger Kpi → more damping to the LC resonance

1 f f1 1

d piv i d uv

v vi L C C d pi

G KG GG GT GT Z Y Y G K

21 f f1 d pis LC sC G K

2 21+

rv rvv pv pv

K s KG = K Kss ω

Kpv, Krv → GM

20dB10v cT jω

ωc

fs/6 fs/2

Low bandwidth

Voltage Control for Grid-Forming ConvertersUnstable inner current loop leads to unstable voltage control, fLC < fs/6

-50

0

50

100

101 102 103 104-540

-360

-180

0

180

TiTv

Bode Diagram

Frequency (Hz)

- Current loop, GM<0, unstable- Voltage loop, 2 RHP poles → phase

leading → no crossing over ±180° within the bandwidth

Entire system unstable !

fs/6 fs/2

Voltage Control for Grid-Forming ConvertersUnstable inner current loop, fs/6 < fLC < fs/2

-50

0

50

100

101 102 103 104-540

-360

-180

0

180

360

Bode Diagram

Frequency (Hz)

( )i ui d piT s = G G K

Current loop and consequently entire system unstable !

Phase crossing over −180° at fLC

pi 0K

dG

uiG

ωc

fs/6 fs/2

Phase crossing over 0° at fs/6

Phase crossing over −360° at fs/2

Voltage Control for Grid-Forming ConvertersImpedance-based stability analysis

-40

-20

0

20

40

60

101 102 103 104

-90

-45

0

45

90

ZoZL2

Bode Diagram

Frequency (Hz)

L1

Cf

PWM

vi1 L2i2u

Controller

Grid L2i2

GridPCC

ref1v

v

Tv

TZo

Non-passive

>180°

Equivalent block diagram

idq

*idqGPI(s)

UdqGd(s)

Vvsc_dq

L-Filter

Yp(s)idq

Yp(s)

YPLL(s)

iPLL_dq

Vac_dqYp(s)GPLL(s) Gd(s)

Vac_dq

cidq

idq

*idq Vvsc_dqUdq

Vac_dq

iPLL_dq

UPLL_dq

L-Filter

GPI(s) Gd(s)

GPLL(s)

YPLL(s)

Yp(s)c

c Udq

Structural characterization of PLL effect

Instability Effect of PLL for Grid-Following ConvertersImpedance model of current control with PLL dynamics

Current control

Instability Effect of PLL for Grid-Following ConvertersTwo paralleled admittances added by the PLL

Equivalent block diagram

idq

*idqGPI(s)

UdqGd(s)

Vvsc_dq

L-Filter

Yp(s)idq

Yp(s)

YPLL(s)

iPLL_dq

Vac_dqYp(s)GPLL(s) Gd(s)

Vac_dq

c

Current control

idq

Gcl(s)

Ycl(s)

*idq

Vac_dq

GPLL(s) Gd(s) Ycl(s)

Gcl(s)YPLL(s)

Block diagram of impedance model

Impedance model

Gcl(s) *idq Ycl(s) YPLL,1(s) YPLL,2(s) Vac_dq

idq 1

( ) I ( ) ( ) ( ) ( ) ( ) ( )cl p d PI p d PIG s Y s G s G s Y s G s G s

1( ) I ( ) ( ) ( ) ( )cl p d PI pY s Y s G s G s Y s

,1( ) ( ) ( ) ( )PLL PLL d clY s G s G s Y s

,2 ( ) ( ) ( )PLL PLL clY s Y s G s

,1 ,, 2( ) ( ( ) ( )) PLLcl t c Pl LLY s s Y s Y sY

- Closed-loop output admittance matrix

- Grid impedance matrix

1

1( ) g g

gg g

sL LZ s

L sL

,20

,0

( ) (0 ( )

)(

)(0 )

PLL qcl t

PPLL

dLL

LP

L

Y s Y sH s i

Y sH s i

- Low-frequency approximation Ycl,t(s) with unity current loop gain - Generalized Nyquist stability criterion

,det I - ( ) ( ) 0g cl tZ s Y s

- Nyquist diagrams of λ1 and λ2

Negative resistance

Instability Effect of PLL for Grid-Following ConvertersLow-frequency negative resistance on the q-q axis

Instability Effect of PLL for Grid-Following Converters DQ-frame impedance matrices - PLL bandwidth 330 Hz, SCR = 7

Nyquist diagrams of eigenvalue transfer functions in the dq-frame

Stationary-frame impedance matrices - PLL bandwidth 330 Hz, SCR = 7

Nyquist diagrams of eigenvalue transfer functions in the αβ-frame

Instability Effect of PLL for Grid-Following Converters

Simulations - PLL bandwidth 330 Hz, 175 Hz, 20 Hz, SCR = 7

PLLBW = 330 Hz

PLLBW = 175 Hz

PLLBW = 20 Hz

Instability Effect of PLL for Grid-Following Converters

Experiments - PLL bandwidth 175 Hz, 20 Hz, SCR = 7

PLLBW = 175 Hz PLLBW = 20 Hz

Current

Voltage

Current

Voltage

Instability Effect of PLL for Grid-Following Converters

Instability Effect of PLL for Grid-Following ConvertersThree paralleled grid-following inverters with different SCRs

SCR = 8.4 SCR = 4.2

#1 current

Grid voltage

#2 current

#3 current

Instability Effect of DC-Link Voltage Control DC-link Voltage Control (DVC) for grid-following converters (rectifier mode)

Vdcref

V VdcVg

LIc

Cdc

+

DVC

CC

PWM

Lg

θg

+

+

Idc

PLL

Cg

- Unified power factor operation (Iq,ref =0)- Vg, Ic, V are space vectors of three-phase voltages and currents - PLL dynamic is not modeled - The coupling between dc-link and ac-side dynamics - Explicit analytical model of DVC loop

Closed-loop model with four input and three output variables

ˆdD

c-acG-c dcG

Plant

Control

ˆqD

d̂I

q̂I

d̂crefV

q̂refI

d̂V

q̂V

++

++

+-

p-dcG

p-acG

opY

+-

+-

d̂cV

- Gc-ac: current controller in matrix form; Gc-dc: dc-link voltage controller - Gp-ac: control plant of CC loop; Gp-dc: control plant of DVC loop;- Yop: open-loop output admittance;

Instability Effect of DC-Link Voltage Control

Plant of dc-link voltage loop, Gp-dc

V̂

p-acG TV1

dc dc dcsC V I

opY

d̂cV

Gac-dc

+ ++ +

‐

T

d qD D DQDT

d qV V VT

d qI I cIˆ

DQD+

TcΙ

ˆcΙ

sL TcΙ

- The dynamic coupling between dc-link and ac-side, Gac-dc is derived from the instantaneous power balance, i.e.

ˆˆ ˆˆ ˆ ˆ ˆ ˆqd dc

d d q q d d q q d q dc dc dc dc

dIdI dVV I V I V I V I LI LI C V V I

dt dt dt

0

d

d d

dcdc dc

dc dc

VsLI LI

IC V sC V

ac-dcG

RHP zero

Gp-dc

Instability Effect of DC-Link Voltage Control

Closed-loop input admittance for the rectifier mode

V̂

c-acGc-dcG delG p-acG TV1

dc dc dcsC V I

opY

d̂cVˆ 0dcrefV

Ycl1 Ycl2

+

‐

+

‐

+ ++ +

‐

Gac-dc

+ˆ

DQDˆcrefI

sL TcΙ

ˆcI

TcΙ

- Gdel: time delay effect of PWM in matrix form; E: unity gain matrix;

cl cl1 cl2Y Y + Y

GCL-ac

Instability Effect of DC-Link Voltage Control

Closed-loop input admittance – Ycl1

V̂

c-acG delG p-acG

opY

d̂cV

Ycl1

‐

++

c-dcG

‐

ac-dcG

ˆcrefI ˆ

cI

GCL-ac

1ˆˆ

cl1 Y1 p-ac del c ac opc

VY I G G G G YIY1 p-ac del c ac c-dc ac-dcG G G G G G

- Ycl1: closed-loop output admittance with Yop

Instability Effect of DC-Link Voltage Control

Closed-loop input admittance – Ycl2

V̂

c-dcG

TV1

dc dc dcsC V Id̂cV

Ycl2

++ +

‐

Gac-dc

ˆcrefI

ˆcI

TcΙ

sL TcΙ

CL-acG

- Ycl2: closed-loop output admittance with the steady-state operating point IT=[Id, Iq]

1

CL-ac c ac del p-ac c ac del p-acG G G G I G G G

1( )dc dc dcsC V I

Y2o CL-ac c dcG G G 1ˆ

1ˆ sL

T T T

cl2 Y2o c Y2o cc

VY G I V G II

Instability Effect of DC-Link Voltage Control

Impacts of DVC bandwidth on the closed-loop input admittance0

-50

-100

-150270

90

-90

-270

-10

-40

-70

0

-180

-360

1 10 100 1000 10000 1 10 100 1000 10000

Mag

nitu

de (d

B)Ph

ase (

degr

ee)

Mag

nitu

de (d

B)Ph

ase (

degr

ee)

Frequency (Hz)

Ydd Ydq

Yqd Yqq

280 Hz 400 Hz100 Hzno time delay 100 Hz

Instability Effect of DC-Link Voltage Control

Converter-grid interaction with Lg=5mH, Cg=20μF

-2000 -1500 -1000 -500 0 500 1000 1500 2000-100

-50

0

50

100

-2000 -1500 -1000 -500 0 500 1000 1500 2000-200

-100

0

100

200

Frequency (Hz)

1( )s

2 ( )s

-2000 -1500 -1000 -500 0 500 1000 1500 2000-100

-50

0

50

100

-2000 -1500 -1000 -500 0 500 1000 1500 2000-200

-100

0

100

200

Frequency (Hz)

-274 Hz 274Hz

1( )s

2 ( )s

Frequency responses of eigenvalue transfer functions

DLC Bandwidth 100Hz DLC Bandwidth 280Hz

Instability Effect of DC-Link Voltage Control

Converter-grid interaction with Lg=5mH, Cg=20μF

DVC Bandwidth 100Hz DVC Bandwidth 280Hz

Instability Effect of DC-Link Voltage Control

Ug.

Ig.

jXgIg.E

.

δ

Stability of Power Control for Grid-Forming ConvertersSmall-signal modeling of power-based synchronization control

ZgugPCC ige

Phasor diagram

Equivalent circuit

Lf ua

uc

ub

Zg=sLg+rgiaibic

PCC

+–θ 1s

P

– Q

Qset

ωn+

+

Power Calculation

idq+

En+ +Active Power Control

Reactive Power Control

E

edq

eabc abc / dqiabc

Voltage Controller

PWM

edq

dq/abc

edrefeqref

0

ea

ebec

Vdc

Pset1Js+Dp

1Kes+Dq

General control diagram of grid-forming converters0j

gU Ue jE Ee 1g g gZ j L jX

Ug.

Ig.

jXgIg.E

.

δ

Stability of Power Control for Grid-Forming ConvertersSmall-signal modeling of power-based synchronization control

ZgugPCC ige

Phasor diagram

Equivalent circuit

0jgU Ue jE Ee 1g g gZ j L jX

cossin

g g

E E UEUP QX X

0 0 0

0 0

0 0

0 0 0

s

ˆ

iˆ n

i

cos ˆ

2 coss n ˆ ˆ

ˆg

g

g

g

E UP

X

E UQ E

UE

X

EUX X

Linearization

Stability of Power Control for Grid-Forming ConvertersSmall-signal modeling of power-based synchronization control

ZgugPCC ige

Equivalent circuit

cossin

g g

E E UEUP QX X

0 0 0

0 0

0 0

0 0 0

s

ˆ

iˆ n

i

cos ˆ

2 coss n ˆ ˆ

ˆg

g

g

g

E UP

X

E UQ E

UE

X

EUX X

Only valid in low frequency range!

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

e−u g

(p.u

.)i g

(p.u

.)

Linearization

0 0 0 00

0 0 02 2

00 0 00

2 2

0 0 02 22 2

cos ˆˆ

cos ˆˆ

sin ˆ

sin ˆ

g

g

g

gg g

g

g g

gg

g

gg gg

X E E Up E

Xs

E

L X

X E E Uq E

X

sL E UE

L

XsL X

sL EU

X XsX sL

Stability of Power Control for Grid-Forming ConvertersSmall-signal modeling of power-based synchronization control

ZgugPCC ige

Equivalent circuit

1g gg gsL j L jsL X gdqZjd qe je Ee dqe

0jgd gqu ju Ue gdqu p jq *

dq gdqs e i

Synchronous resonance!

ˆˆ ˆ * *dq0 gdq dq0 gdq0s E i e I 0 0 0

ˆ( )0 0

ˆˆˆ (1 )j j jE e Ee E e j dqe ˆ ˆdq dq gdqi e Z

Stability of Power Control for Grid-Forming ConvertersOpen-loop gains of the power control loops with different parasitic Rg

Active power-angle (P-δ) loop Reactive power-voltage (Q-E) loop

Stability of Power Control for Grid-Forming ConvertersClosed-loop poles – reactive power loop is more robust than active power loop

Active power-angle (P-δ) loop Reactive power-voltage (Q-E) loop

Stability of Power Control for Grid-Forming ConvertersSimulation tests for Rg changed at the time instant of 2.0 s

Rg 0.08 Ω Rg 0.1 ΩRg 0.09 Ω

Power

Freq

× 1e4

3.15

3.20

3.25

3.30

3.35

3.40

3.45

1.0 1.5 2.0 2.5 3.0314.00

314.05

314.10

314.15

314.20

314.25

314.30

Power

Freq

× 1e4

3.15

3.20

3.25

3.30

3.35

3.40

3.45

1.0 1.5 2.0 2.5 3.0

314.00

314.05

314.10

314.15

314.20

314.25

314.30

Power

Freq

× 1e4

3.0

3.1

3.2

3.3

3.4

3.5

3.6

1.0 1.5 2.0 2.5 3.0

313.9

314.0

314.1

314.2

314.3

314.4

Stability of Power Control for Grid-Forming ConvertersDamping of synchronous resonance - active resistor

İ Lg

Vg0 Eδ Rg Zad

PQ Voltage Control

Dual-LoopPower ControlVPWMVdq *

Idq s+ωc

Kads

102

Frequency (Hz)

0

Phas

e (°)

Mag

nitu

de (d

B)

−100

50

−180

−270

−360

0

−150

100 101 103 104

−50

−90

Kad

102

Frequency (Hz)

0

Phas

e (°)

Mag

nitu

de (d

B)

−100

50

−180

−270

−360

0

−150

100 101 103 104

−50

−90

ωc

Active power-angle (P-δ)

loop

ωc = ω0/10 Kad = 0.05

Stability of Power Control for Grid-Forming Converters

İ Lg

Vg0 Eδ Rg Zad

PQ Voltage Control

Dual-LoopPower ControlVPWMVdq *

Idq s+ωc

Kads

Reactive power-voltage (Q-E)

loop

Rg = 0.01 Ω ωc = ω0/10

Damping of synchronous resonance - active resistor

Stability of Power Control for Grid-Forming ConvertersDamping of synchronous resonance - active resistor

Kad = 0.05, ωc = ω0/10

Kad = 0.05 & ωc = ω0/10 v.s. Rg = 0.05 Ω

Rg = 0.05 ΩKad = 0.05, ωc = ω0/10

PROSPECTS AND CHALLENGES

SISO modeling and control

Active stabilization techniques

Interoperability of multi-vendor converters

SISO Modeling and Control Symmetrical PLL for decoupled frequency response

DQ-frame PLL

Vα cos

Vβ ω1

ωe θg

sin

-sin cos

Vd

Vq PI 1s

Frequency-decoupled response to 80 Hz disturbanceVα

cos

Vβ ω1

ωe θq

sin

-sin cos

Vd

Vq PI 1s

PI

Vd0 1s 1

‒1

θdeθq

eθq

−j-j(θd +jθq)e

Symmetrical PLL[81]

Active Stabilization - Virtual Impedance ControlA multi-loop control for impedance shaping

Gvo(s):outer virtual impedance controller Gvi(s): inner virtual impedance controller Gc(s): current controller

- Modify the reference of modulator (duty cycle) with Gvi(s)

- Adjust the reference of controller reference with Gvo(s)

- Reconfigure the poles (oscillation modes) and zeros of the power system - No steady-state harmonic filtering, featuring low-power, high-frequency, high-bandwidth

Active Stabilization - Active DamperA power-electronic-based stabilizer for converter-based power systems

Interoperability of Multi-Vendor Power Electronic ConvertersStandard EMT modeling, dynamic specification, design-oriented analysis

New grid codes are demanded for self-

disciplined stabilization, i.e. grid-neutral

converters

Grid-forming converters for actively

regulating system voltage and frequency

Design-oriented stability analysis is critical

for utilizing the full controllability of power

electronics

Lack of unified models with physical

insights (virtual impedance/machine)

Power electronic based power systems

References [1] X. Wang and F. Blaabjerg, “Harmonic stability in power electronic based power systems: concept, modeling, and analysis,” IEEE Trans. Smart Grid, vol. 10, pp. 2858-2870, May 2019.

[2] J. D. Ainsworth, “Harmonic instability between controlled static converters and a.c. networks,” Proc. Inst. Elect. Eng., vol. 114, pp.949-957, Jul. 1967.

[3] A. Wood and J. Arrillaga, “Composite resonance: a circuit approach to the waveform distortion dynamics of an HVDC converter,” IEEE Trans. Power Del., vol. 10, no. 4, pp. 173-192, Oct. 1995.

[4] A. E. Hammad, “Analysis of second harmonic instability for the Chateauguay HVDC/SVC scheme,” IEEE Trans. Power Del., vol. 7, no. 1, Jan. 1992.

[5] I. Pendharkar, “Resonance stability in electrical railway systems – a dissipativity approach,” European Control Conference (ECC), 2013, pp. 4574-4579.

[6] J. Arrillaga, et al., Power system harmonic analysis, John Wiley & Sons, New York, Chapter 6, pp. 173-192

[7] E.V. Persson, “Calculation of transfer functions in grid controlled convertor systems,” IEE Proceedings, 117(5):989 to 997, May 1970.

[8] M. Sakui and H. Fujita, “Line harmonic current of three phase thyristor bridge rectifier with dc current ripple and overlap angle,” IEEJ vol. 105-B, no. 9, pp. 717 to 724, Sept. 1985.

[9] K. D. Ngo. “Low frequency characterization of PWM converters,” IEEE Trans. Power Electron., vol. PE-1, no. 4, pp. 223-230, Oct. 1986.

[10] E. V. Larson, D. H. Baker and J. C. McIver, “Low-order harmonic interaction on ac/dc systems,” IEEE Trans. Power Del., vol. 4, no. 1, pp. 493-501, Jan. 1989.

[11] P. Mattavelli, G. C. Verghese, A. M. Stankovic, “Phasor dynamics of thyristor-controlled series capacitor systems,” IEEE Trans. Power Syst., vol. 12, no. 3, pp. 1259-1267, Aug. 1997.

[12] E. Mollerstedt and B. Bernhardsson, “Out of control because of harmonics an analysis of the harmonic response of an inverter locomotive,” IEEE Control Systems Magazine, vol. 20, no. 4, pp.

70-81, Aug. 2000.

[13] J. J. Rico, M. Madrigal, and E. Acha, “Dynamic Harmonic Evolution Using the Extended Harmonic Domain,” IEEE Trans. Power Del., vol. 18, pp. 587–594, Apr. 2003.

[14] L. Harnefors, M. Bongiorno, and S. Lundberg, “Input-admittance calculation and shaping for controlled voltage-source converters,” IEEE Trans. Ind. Electron., vol. 54, no. 6, pp. 3323-3334,

Dec. 2007.

[15] B. Wen, D. Boroyevich, R. Burgos, P. Mattavelli, and Z. Shen, “Analysis of D-Q small-signal impedance of grid-tied inverters,” IEEE Trans. Power Electron., vol. 31, pp. 675-687, Jan. 2016.

[16] M. Cespedes and J. Sun, “Impedance modeling and analysis of grid-connected voltage-source converters,” IEEE Trans. Power Electron., vol. 29, no. 3, pp. 1254-1261, Mar. 2014.

[17] X. Wang, L. Harnefors, and F. Blaabjerg, “Unified impedance model of grid-connected voltage-source converters," IEEE Trans. Power Electron., vol. 33, no. 2, pp. 1775-1787, Feb. 2018.

[18] X. Yue, X. Wang, and F. Blaabjerg, “Review of small-signal modeling methods including frequency-coupling dynamics of power converters," IEEE Trans. Power Electron., vol. 34, pp. 3313-

3328, Apr. 2019.

References[19] M. Lu, X. Wang, P. C. Loh, F. Blaabjerg, and T. Dragicevic, “Graphical evaluation of time-delay compensation techniques for digitally controlled converters,” IEEE Trans. Power Electron., vol.

33, no. 3, pp. 2601-2614, Mar. 2018.

[20] S. Hiti, D. Boroyevich, et al. “Small-signal modeling and control of three-phase PWM converters,” IEEE IAS 1994, pp. 1143-1150.

[21] R. D. Middlebrook, “Predicting modulator phase lag in pwm converter feedback loop,” Powercon 1981, pp. 1-6.

[22] J. Kwon, X. Wang, F. Blaabjerg, C. L. Bak, A. R. Wood, and N. Watson, “Linearized modeling methods of ac-dc converters for an accurate frequency response,” IEEE J. Emerg. Sel. Topics

Power Electron., vol. 5, no. 4, pp. 1526-1541, Dec. 2017.

[23] A. Wood, D. J. Hume, and C. M. Osauskas, “Linear analysis of waveform distortion for HVDC and FACTs devices,” in Proc. IEEE ICHQP 2000, pp. 1-7.

[24] S. R. Sanders, J. M. Noworolski, X. Liu, and G. C. Verghese, “Generalized averaging method for power conversion circuits,” IEEE Trans. Power Electron., vol. 6, no. 2, pp. 251-259, Apr. 1991.

[25] C. Buchhagen, M. Greve, A. Menze, and J. Jung, “Harmonic stability – practical experience of a TSO,” in Proc. Wind Integration Workshop, 2016, pp. 1-6.

[26] L. Harnefors, “Modeling of three-phase dynamic systems using complex transfer functions and transfer matrices,” IEEE Trans. Ind. Electron. vol. 54, no. 4, pp. 2239-2248, Aug. 2007.

[27] S. Chung, “A phase tracking system for three phase utility interface inverters,” IEEE Trans. Power Electron., vol. 15, no. 3, pp. 431-437, May 2000.

[28] D. Yang, X. Wang, and F. Blaabjerg, “Sideband-harmonic instability of paralleled inverters with asynchronous carriers,” IEEE Trans. Power Electron., vol. 33, no. 6, pp. 4571-4577, Jun. 2018.

[29] J. Kwon, X. Wang, F. Blaabjerg, C. L. Bak, A. R. Wood and N. R. Watson, “Harmonic instability analysis of single-phase grid connected converter using harmonic state space (HSS) modeling

method”, IEEE Trans. Ind. Appl.,, Vol. 52, No. 5, pp. 4188 – 4200. Sept./Oct. 2016.

[30] J. Holtz, “The representation of ac machine dynamics by complex signal flow graphs,” IEEE Trans. Ind. Electron., vol. 42, no. 3, pp. 263-271, Jun. 1995.

[31] L. Harnefors, X. Wang, A. G. Yepes, and F. Blaabjerg, “Passivity-based stability assessment of grid-connected VSCs – an overview,” IEEE J. Emerg. Sel. Topics Power Electron., vol. 4, no. 1,

pp. 116-125, Mar. 2016.

[32] X. Wang, F. Blaabjerg, and P. C. Loh, “Passivity-based stability analysis and damping injection for multiparalleled VSCs with LCL filters,” IEEE Trans. Power Electron., vol. 32, no. 11, pp. 8922-

8935, Nov. 2017.

[33] X. Wang, F. Blaabjerg, and W. Wu, “Modeling and analysis of harmonic stability in ac power-electronics-based power system,” IEEE Trans. Power Electron., vol. 29, no. 12, pp. 6421-6432,

Dec. 2014.

[34] N. M. Wereley, “Analysis and control of linear periodically time varying systems,” Ph.D. dissertation, Dept. of Aeronautics and Astronautics, MIT, 1991.

References[35] D. Maksimovic, A. M. Stankovic, V. J. Thottuvelil, and G. C. Verghese, "Modeling and simulation of power electronic converters," Proc. IEEE, vol. 89, pp. 898-912, Jun. 2001.

[36] M. K. Bakhshizadeh, X. Wang, F. Blaabjerg, J. Hjerrild, L. Kocewiak, C. L. Bak, and B. Hesselbaek, “Couplings in phase domain impedance moeling of grid-connected converters,” IEEE Trans.

Power Electron. vol. 31, no. 10, pp. 6792-6796, Oct. 2016.

[37] J. M. Undrill and T. E. Kostyniak, “Subsynchronous oscillations Part I – comprehensive stability analysis,” IEEE Trans. Power App. Syst., vol. PAS-95, no. 4, pp. 1446-1455, Jul./Aug. 1976.

[38] B. Ferreira, "Understanding the Challenges of Converter Networks and Systems: Better opportunities in the future," IEEE Power Electronics Mag., vol. 3, no. 2, pp. 46-49, Jun. 2016.

[39] B. Kroposki, et al., “Achieving a 100% renewable grid: operating electric power systems with extremely high levels of variable renewable energy,” IEEE Power & Energy Mag. vol. 15, pp. 61-73,

Mar./Apr. 2017.

[40] R. D. Middlebrook, “Input filter consideration in design and application of switching regulators,” in Proc. IEEE IAS 1976, 366-382.

[41] H. Zhang, L. Harnefors, X. Wang, H. Gong, J. Hasler, “Stability analysis of grid-connected voltage-source converters using SISO modeling,” IEEE Trans. Power Electron., vol. 34, pp. 8104-

8117, Aug., 2019.

[42] C. Zhang, X. Wang, and F. Blaabjerg, “The influence of phase-locked loop on the stability of single-phase grid-connected inverter,” in Proc. IEEE ECCE 2015, pp. 4734-4744.

[43] H. Zhang, X. Wang, L. Harnefors, J. Hasler, and H. Nee, “SISO transfer functions for stability analysis of grid-connected voltage-source converters,” IEEE Trans. Ind. Appl., vol. 55, pp. 2931-

2941, May-Jun., 2019.

[44] L. Harnefors, L. Zhang, and M. Bongiorno, “Frequency-domain passivity-based current controller design,” IET Power Electron., vol. 1, no. 4, pp. 455-465, Dec. 2008.

[45] X. Wang, Y. W. Li, F. Blaabjerg, and P. C. Loh, “Virtual-impedance-based control for voltage- and current-source converters,” IEEE Trans. Power Electron. vol. 30, pp. 7019-7037, Dec. 2015.

[46] D. Lu, X. Wang, and F. Blaabjerg, “Impedance-based analysis of dc-link voltage dynamics in voltage-source converters,” IEEE Trans. Power Electron. vol. 34, pp. 3973-3985, Apr. 2019.

[47] D. Yang, X. Wang, F. Liu, K. Xin, Y. Liu, and F. Blaabjerg，“Adaptive reactive power control of PV power plants for improved power transfer capability under ultra-weak grid conditions,“

IEEE Trans. Smart Grid, vol. 10, pp. 1269-1279, Mar. 2019.

[48] L. Zhang, L. Harnefors, and H.-P. Nee, “Power-synchronization control of grid-connected voltage-source converters,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 809–820, 2010.

[49] L. Zhang, H. Nee, and L. Harnefors, “Analysis of stability limitations of a VSC-HVDC link using power-synchronization control,” IEEE Trans. Power Syst., vol. 26, no. 3, pp. 1326–1337, 2011.

[50] D. Yang, X. Wang, and F. Blaabjerg, “Suppression of synchronous resonance for virtual synchronous generators,” in Proc. IET RPG, 2017.

[51] D. Yang, X. Wang, and F. Liu, K. Xin, Y. Liu, F. Blaabjerg, “A PLL-less vector current control of VSC-HVDC for ultra weak grid interconnection,” in Proc. IEEE ECCE, 2018.

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