Wokaun Assessment of the proposals: without donors Marco ... · From coupled oscillators to AC power systems (simpliﬁed) swing equation model of interconnected synchronous generators:

Complex oscillator networks

and applications in power systems

ETH-ITS lecture series “Collective dynamics, control and imaging”

Florian Dorfler

Acknowledgements

Dominic Groß Marcello Colombino

!

! ! ! ! ! ! ! ! ! !!!!!!!!!!!!!!!!Contact:!!!!!!!!!!!!!!!!!!!!!Dr.!Christian!Schaffner!ESC!!!!!!!!!!!!!!!!!Tel:!+41!44!632!72!55!

! ! ! ! ! ! !!!!!!!!!!!!!!!!E<mail:[email protected]!

!

Seed Projects Energy Fund – Workshop Agenda Monday, 24 August, 2015, 12:45 – 19:00

ETH Zurich, Room CLA J1, Tannenstrasse 3, CH-8092 Zurich

Attendees Energy Fund Donors (only until 18:15) Evaluation Committee ETH Zürich P.I.’s (only until 17:20)

Programme

Start Finish Item Lead Proposal Number

12:45 12:50 Welcome & Introductions Christian Schaffner

12:50 13:10 Exact Boundary Conditions for modelling surface topography in finite-difference methods PI: Johan Robertsson 1

13:10 13:30 Navigating the renewable energy transition: modeling power, heat, climate impacts, and the feasibility of future energy visions

PI: Anthony Patt 2

13:30 13:50 Vulnerabilities of the Water-Energy Nexus PI: Giovanni Sansavini 3

13:50 14:10 Development of highly active and selective methanol synthesis catalysts to provide a versatile storage concept for surplus renewable energy.

PI: Christophe Copéret 4

14:10 14:25 Refreshment Break 1

14:25 14:45 Computationally tailored redox materials for efficient solar fuel production from CO2 and H2O PI: Aldo Steinfeld 5

14:45 15:05 Novel control approaches for low-inertia power grids PI: Florian Dörfler 6

15:05 15:25 High-efficiency thin-film CIGS photovoltaic/thermal hybrid system for building integration PI: Arno Schlüter 7

15:25 15:45 Behavioural and technical optimisation for building energy efficiency and flexibility PI: John Lygeros 8


16:00 16:20 Smart Monitoring of Dams for Sustainable Hydropower Energy Infrastructure PI: Eleni Chatzi 9

16:20 16:40 A novel theoretical and numerical framework to predict and control aero-acoustic instabilities in energy systems PI: Nicolas Noiray 10

16:40 17:00 Enabling Deployment while Leaving Room for Experimentation: Toward a Framework for Renewable Energy Financing in Emerging Economies

PI: Tobias Schmidt 11

17:00 17:20 Storage of renewable energy: methanol via hydrothermal processing of wet biomass

PI: Alexander Wokaun 12

17:20 17:30 Break

17:30 18:15 Assessment of the proposals: with donors Marco Mazzotti

18:15 19:00 Assessment of the proposals: without donors Marco Mazzotti

!

! ! ! ! ! ! ! ! ! !!!!!!!!!!!!!!!!Contact:!!!!!!!!!!!!!!!!!!!!!Dr.!Christian!Schaffner!ESC!!!!!!!!!!!!!!!!!Tel:!+41!44!632!72!55!

! ! ! ! ! ! !!!!!!!!!!!!!!!!E<mail:[email protected]!

!

Seed Projects Energy Fund – Workshop Agenda Monday, 24 August, 2015, 12:45 – 19:00

ETH Zurich, Room CLA J1, Tannenstrasse 3, CH-8092 Zurich

Attendees Energy Fund Donors (only until 18:15) Evaluation Committee ETH Zürich P.I.’s (only until 17:20)

Programme

Start Finish Item Lead Proposal Number

12:45 12:50 Welcome & Introductions Christian Schaffner

12:50 13:10 Exact Boundary Conditions for modelling surface topography in finite-difference methods PI: Johan Robertsson 1

13:10 13:30 Navigating the renewable energy transition: modeling power, heat, climate impacts, and the feasibility of future energy visions

PI: Anthony Patt 2

13:30 13:50 Vulnerabilities of the Water-Energy Nexus PI: Giovanni Sansavini 3

13:50 14:10 Development of highly active and selective methanol synthesis catalysts to provide a versatile storage concept for surplus renewable energy.

PI: Christophe Copéret 4


14:25 14:45 Computationally tailored redox materials for efficient solar fuel production from CO2 and H2O PI: Aldo Steinfeld 5

14:45 15:05 Novel control approaches for low-inertia power grids PI: Florian Dörfler 6

15:05 15:25 High-efficiency thin-film CIGS photovoltaic/thermal hybrid system for building integration PI: Arno Schlüter 7

15:25 15:45 Behavioural and technical optimisation for building energy efficiency and flexibility PI: John Lygeros 8


16:00 16:20 Smart Monitoring of Dams for Sustainable Hydropower Energy Infrastructure PI: Eleni Chatzi 9

16:20 16:40 A novel theoretical and numerical framework to predict and control aero-acoustic instabilities in energy systems PI: Nicolas Noiray 10

16:40 17:00 Enabling Deployment while Leaving Room for Experimentation: Toward a Framework for Renewable Energy Financing in Emerging Economies

PI: Tobias Schmidt 11

17:00 17:20 Storage of renewable energy: methanol via hydrothermal processing of wet biomass

PI: Alexander Wokaun 12

17:20 17:30 Break

17:30 18:15 Assessment of the proposals: with donors Marco Mazzotti

18:15 19:00 Assessment of the proposals: without donors Marco Mazzotti

2 / 28

A brief history of sync

Christiaan Huygens (1629 – 1695)

physicist & mathematician

engineer & horologist

observed “an odd kind of sympathy ”[Letter to Royal Society of London, 1665]

Recent reviews, experiments, & analysis[M. Bennet et al. ’02, M. Kapitaniak et al. ’12]

3 / 28

A field was born

sync in mathematical biology [A. Winfree ’80, S.H. Strogatz ’03, . . . ]

sync in physics and chemistry [Y. Kuramoto ’83, M. Mezard et al. ’87. . . ]

sync in neural networks [F.C. Hoppensteadt and E.M. Izhikevich ’00, . . . ]

sync in complex networks [C.W. Wu ’07, S. Bocaletti ’08, . . . ]

. . . and numerous technological applications (reviewed later)

Physics Reports 469 (2008) 93–153

Contents lists available at ScienceDirect

Physics Reports

journal homepage: www.elsevier.com/locate/physrep

Synchronization in complex networksAlex Arenas a,b, Albert Díaz-Guilera c,b, Jurgen Kurths d,e, Yamir Morenob,f,⇤, Changsong Zhoug

a Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spainb Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50009, Spainc Departament de Física Fonamental, Universitat de Barcelona, 08028 Barcelona, Spaind Institute of Physics, Humboldt University, D-12489 Berlin, Newtonstrasse 15, Germanye Potsdam Institute for Climate Impact Research, 14412 Potsdam, PF 601203, Germanyf Department of Theoretical Physics, University of Zaragoza, Zaragoza 50009, Spaing Department of Physics and Centre for Nonlinear Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

a r t i c l e i n f o

Article history:Accepted 17 September 2008Available online 24 September 2008editor: I. Procaccia

PACS:05.45.Xt89.75.Fb89.75.Hc

Keywords:SynchronizationComplex networks

a b s t r a c t

Synchronization processes in populations of locally interacting elements are the focus ofintense research in physical, biological, chemical, technological and social systems. Themany efforts devoted to understanding synchronization phenomena in natural systemsnow take advantage of the recent theory of complex networks. In this review,we report theadvances in the comprehension of synchronization phenomena when oscillating elementsare constrained to interact in a complex network topology. We also take an overview ofthe new emergent features coming out from the interplay between the structure and thefunction of the underlying patterns of connections. Extensive numerical work as well asanalytical approaches to the problem are presented. Finally, we review several applicationsof synchronization in complex networks to different disciplines: biological systems andneuroscience, engineering and computer science, and economy and social sciences.

© 2008 Elsevier B.V. All rights reserved.

Contents

1. Introduction............................................................................................................................................................................................. 942. Complex networks in a nutshell ............................................................................................................................................................ 953. Coupled phase oscillator models on complex networks ...................................................................................................................... 96

3.1. Phase oscillators.......................................................................................................................................................................... 973.1.1. The Kuramoto model ................................................................................................................................................... 973.1.2. Kuramoto model on complex networks..................................................................................................................... 983.1.3. Onset of synchronization in complex networks ........................................................................................................ 993.1.4. Path towards synchronization in complex networks................................................................................................ 1033.1.5. Kuramoto model on structured or modular networks.............................................................................................. 1053.1.6. Synchronization by pacemakers................................................................................................................................. 107

3.2. Pulse-coupled models ................................................................................................................................................................ 1083.3. Coupled maps.............................................................................................................................................................................. 109

4. Stability of the synchronized state in complex networks .................................................................................................................... 1124.1. Master stability function formalism.......................................................................................................................................... 112

4.1.1. Linear stability and master stability function ............................................................................................................ 113

⇤ Corresponding author at: Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50009, Spain. Tel.: +34976562212x223; fax: +34 976562215.

E-mail address: [email protected] (Y. Moreno).

0370-1573/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physrep.2008.09.002

4 / 28

Coupled phase oscillators

9 various models of oscillators & interactions

canonical coupled phase oscillator model:

[A. Winfree ’67, Y. Kuramoto ’75]

✓i

= !i

�X

n

j=1a

ij

sin(✓i

�✓j

)

In oscillators with phase ✓

i

2 S1

I non-identical natural frequencies !i

2 R1

I elastic coupling with strength a

ij

= a

ji

I undirected & connected graph G = (V, E , A)!1

!3!2

a12

a13

a23

5 / 28

Phenomenology & challenges in synchronization

Transition to synchronization is a trade-o↵ : coupling vs. heterogeneity

strong coupling & homogeneous weak coupling & heterogeneous

“Surprisingly enough, this seemingly obvious fact seems di�cult to prove.”

(Y. Kuramoto’s conclusion after proposing the model)

Two open central questions:

(still after 50 years of work)

quantify “coupling” vs. “heterogeneity”

basin of attraction for synchronization6 / 28

. . . starting from here a theory talk would

summarize many e↵orts leading to partial

(or negative) results and conjectures . . .

Outline

Introduction

From Coupled Oscillators to Inverter-Based Power Systems

Coupled Phase Oscillators and Inverter Droop Control

Consensus-Inspired Approach to Synchronization

Conclusions

From coupled oscillators to AC power systems

(simplified) swing equation model of

interconnected synchronous generators:

M

i

✓i

+D

i

✓i

= p

?i

�X

j

y

ij

sin(✓i

�✓j

)

where M

i

, D

i

, p

?i

are inertia, damping,

power injection set-point of generator i

weak coupling & heterogeneous strong coupling & homogeneous7 / 28

From coupled oscillators to AC power systems

(simplified) swing equation model of

interconnected synchronous generators:

M

i

✓i

+D

i

✓i

= p

?i

�X

j

y

ij

sin(✓i

�✓j

)

where M

i

, D

i

, p

?i

are inertia, damping,

power injection set-point of generator i

weak coupling & heterogeneousblackout India July 30/31 2012

7 / 28

Renewable/distributed/inverter-based generation on the rise

synchronous generator new workhorse scaling

new primary sources location & distributed implementation

focus today on inverter-based generation

8 / 28

Modeling: signal space in three-phase AC power systems

three-phase AC

"x

a

(t)x

b

(t)x

c

(t)

#=

"x

a

(t + T )x

b

(t + T )x

c

(t + T )

#

periodic with 0 average

1T

RT

0 x

i

(t)dt = 0

2. PRELIMINARIES IN CONTROL THEORY AND POWER SYSTEMS

-2� -� 0 � 2�

�1

0

1

�

xabc

(a) Symmetric three-phase AC signal with

constant amplitude

-2� -� 0 � 2�

�1

0

1

�

xabc

(b) Symmetric three-phase AC signal with

time-varying amplitude

-2� -� 0 � 2�

�1

0

1

�

xabc

(c) Asymmetric three-phase AC signal with

phases not shifted equally by 2�3

-2� -� 0 � 2�

�1

0

1

�

xabc

(d) Asymmetric three-phase AC signal re-

sulting of an asymmetric superposition of a

symmetric signal with signals oscillating at

higher frequencies

Figure 2.1: Symmetric and asymmetric AC three-phase signals. The lines correspond to

xa ’—’, xb ’- -’, xc ’· · · ’.

30

balanced (nearly true)

= A(t)

"sin(�(t))

sin(�(t) � 2⇡3 )

sin(�(t) + 2⇡3 )

#

so that

x

a

(t)+x

b

(t)+x

c

(t)=0


-2� -� 0 � 2�

�1

0

1

�

xabc


constant amplitude

-2� -� 0 � 2�

�1

0

1

�

xabc



-2� -� 0 � 2�

�1

0

1

�

xabc



-2� -� 0 � 2�

�1

0

1

�

xabc




higher frequencies


xa ’—’, xb ’- -’, xc ’· · · ’.

30

synchronous (desired)

=A

"sin(�0 + !0t)

sin(�0 + !0t � 2⇡3 )

sin(�0 + !0t + 2⇡3 )

#

const. freq & amp

) const. in rot. frame


-2� -� 0 � 2�

�1

0

1

�

xabc


constant amplitude

-2� -� 0 � 2�

�1

0

1

�

xabc



-2� -� 0 � 2�

�1

0

1

�

xabc



-2� -� 0 � 2�

�1

0

1

�

xabc




higher frequencies


xa ’—’, xb ’- -’, xc ’· · · ’.

30

assumption : signals are balanced ) 2d-coordinates x(t) = [x↵(t) x�(t)]

(equivalent representation: complex-valued polar/phasor coordinates)9 / 28

Modeling: the inverter

v

dc,k

L

k

i

k

R

k

C

k

v

k

G

k

i

o,k

1

2

v

dc,k

m

k

Network

terminal signals : voltage v

k

2 R2 and output current i

o,k 2 R2

controllable signal : switching modulation signal m

k

) common abstraction:

direct control of current i

k

:

C

k

d

dt

v

k

= �G

k

v

k

� i

o,k + i

k

⇠ controllable voltage source

R

k

L

i

k

C

k

v

k

G

k

i

o

Network

v

k

, i

o,k

i

k

i = i

?

(v

k

, i

o,k

)

10 / 28

Modeling: the network equations

branch dynamics : each branch is series of resistance r

ij

& inductance ìj

time-scale separation : all network signals are assumed to be in

synchronous steady-state x(t) = !0J x(t) where J =⇥

0 �11 0

⇤⇠

p�1

admittance matrix Y = YT with admittances y

ij

= (rij

+ !0ìj

J)�1

1

2

3

4

5 Y =

2

64

.... . .

... . .. ...

�y

k1I2 ···P

n

j=1 y

kj

I2 ··· �y

kn

I2

... . .. ...

. . ....

3

75

⇠ generalized Laplacian matrix

) balance equations:i

o

= Yv

with terminal currents/voltages (io

, v)

11 / 28

Objectives for decentralized control design

We aim to stabilize a target trajectory (v(t), io

(t)) satisfying the following:

1 frequency stability at synchronous frequency !0 :

d

dt

v

k

(t) = !0J v

k

(t)

⇠ synchronization to desired harmonic waveform

2 voltage regulation to desired voltage magnitudes v

?k

:

kv

k

(t)k = v

?k

⇠ stabilization of desired amplitudes (assume wlog that v

?k

= v

?)

3 power injection set-points for active & reactive power {p

?k

, q?k

}:

v

>k

i

o,k = p

?k

, v

>k

Ji

o,k = q

?k

⇠ stabilization of desired angle set-points {✓?kj

}12 / 28

Overview of oscillator-based control strategies for inverters

⇡

classic droop control (brief review)

A desirable closed loop

d

dt

v

k

= !0J v

k

+ e✓(v) + ekvk(v)

0

bla

blav1

v2

!0J v1

!0J v2

K1v1 � R()io,1

K2v2 � R()io,2

�1(v1)

�2(v2)

Phase error

e✓,1(v) = v2 � R(✓?21)v1

e✓,2(v) = v1 � R(✓?12)v2

Magnitude error

ekvk,1(v) = (v? � kv1k)v1

ekvk,2(v) = (v? � kv2k)v2

22 / 1

consensus-inspired approach (today)

v!"

idc

m

iI

v

LI

⌧m

✓, !

vf

v

if

⌧e

is

L✓

C!"

Mrf

rs rsG!"

RI

synchronous generator emulation

[Zhong, Weiss, ’11, D’Arco, Suul ’13,

Bevrani, Sie, Miura ’14, Jouini, Arghir, FD ’16]

R CLg(v)v+

-PWM

dc,k

virtual van-der-Pol oscillator control

[Johnson, Dhople, Hamadeh, & Krein, ’13,

Sinha, Johnson, Ainsworth, FD, Dhople ’15]13 / 28

Outline

Introduction




Conclusions

Droop control of power inverters

key idea : replicate the generator

swing equation for the inverter:

M

i

✓i

+D

i

✓i

=p

?i

�X

j

y

ij

sin(✓i

�✓j

)

⇡

Standard implementation:

1 measure terminal currents

and voltages (v(t), io

(t))

2 process measurements:

active power p

k

= v

>k

i

o,k

reactive power q

k

= v

>k

Ji

o,k

3 proportional control of

terminal voltage waveform:

R

k

L

i

k

C

k

v

k

G

k

i

o

Network

v

k

, i

o,k

i

k

i = i

?

(v

k

, i

o,k

)

d

dt

✓k

kvkk

�= (v?

k

, p?k

, q?k

) � (kv

k

k, pk

, qk

)

14 / 28

Closer look at droop control for a lossless network

key idea : replicate the generator

swing equation for the inverter:

M

i

✓i

+D

i

✓i

=p

?i

�X

j

y

ij

sin(✓i

�✓j

)

⇡

Ip � ! droop mimicking a generator:

d

dt

✓k

= !0+k (p?k

� p

k

) = !0+kp

?k

�k ·X

j

y

kj

kv

j

k kv

k

k sin(✓k

� ✓j

)| {z }

generator / Kuramoto oscillator

I analogous q � kvk droop (many variations):

d

dt

kv

k

k = k (q?k

� q

k

) + k (v? � kv

k

k) · kv

k

k

15 / 28

Standard analysis of lossless droop/generators/oscillators

energy function: inductive energy + quadratic amplitude error

V (v) =X

j<k

y

jk

�kv

k

k2 � kv

k

kkv

j

k cos(✓k

� ✓j

)�

+ 12

Xk

(v? � kv

k

k)2

Ip � ! droop:

Iq � kvk droop:

d

dt

(✓k

� !0 t) = kp

?k

� k · @V

@✓

d

dt

kv

k

k = kq

?k

� k · kv

k

k · @V

@kv

k

k

) closed loop is weighted gradient flow if all p

?k

are identical and q

?k

= 0

standard analysis : if equilibria exists, if (p?k

, q?k

) are su�ciently small, if

lossless, if r2V (v?) is positive definite, . . . =) local asymptotic stability

analysis has many limitations & droop has similar practical limitations !

16 / 28

Outline

Introduction




Conclusions

Recall: objectives for decentralized control design

We aim to stabilize a target trajectory (v(t), io

(t)) satisfying the following:

1 frequency stability at synchronous frequency !0 :

d

dt

v

k

(t) = !0J v

k

(t)

⇠ synchronization to desired harmonic waveform

2 voltage regulation to desired voltage magnitudes v

?k

:

kv

k

(t)k = v

?

⇠ stabilization of desired amplitudes

3 power injection set-points for active & reactive power {p

?k

, q?k

}:

v

>k

i

o,k = p

?k

, v

>k

Ji

o,k = q

?k

⇠ stabilization of desired angle set-points {✓?kj

}17 / 28

Overview of design strategy

Step 1: construct target dynamics f

?(v(t)) for the terminal voltages so

that ddt

v(t) = f

?(v(t)) is stable and satisfies the three control objectives.

Step 2: achieve desired target dynamics ddt

v(t) = f

?(v(t))

via fully decentralized current controllers i

?k

(vk

, ik,o).

R

k

L

i

k

C

k

v

k

G

k

i

o

Network

v

k

, i

o,k

i

k

i = i

?

(v

k

, i

o,k

) X local measurements (vk

, ik,o)

X local set-points (v?k

, p?k

, q?k

)

E unknown: global set-pointsfor angles {✓?

kj

} and nonlocalmeasurements of (v

j

, io,j)

Step 3: implement control via switching modulation signal (not today).

18 / 28

Step 1: desirable closed-loop target dynamicsobjectives: frequency, phase, and voltage stability

d

dt

v = !0 J v

| {z }rotation at ! = !0

+ ⌘ · e✓(v)| {z }

phase error

+ ↵ · e kvk(v)| {z }

magnitude error

(i) synchronous rotation: ddt

v = !0 J v = !0

2

4. . . h

0 �11 0

i

. . .

3

5v

(ii) phase stabilization: ddt

v = e✓,k(v) =P

n

j=1 w

jk

(vj

� R(✓?jk

)vk

)

(iii) magnitude stabilization: ddt

v = e kvk,k(vk

) = (v? � kv

k

k)vk

gains: ↵k

, ⌘k

> 0 for each k and w

jk

� 0 for relative phases amongst {j , k}19 / 28

Illustration of desirable closed-loop dynamics

d

dt

v

k

= !0 J v

k

+ ⌘ · e✓(v) + ↵ · e kvk(v)

0

bla

blav1

v2

!0J v1

!0J v2

K1v1 � R()io,1

K2v2 � R()io,2

�1(v1)

�2(v2)

phase error:

e✓,1(v) = v2 � R(✓?21)v1

e✓,2(v) = v1 � R(✓?12)v2

magnitude error:

e kvk,1(v) = (v? � kv1k)v1

e kvk,2(v) = (v? � kv2k)v2

20 / 28

Step 2: decentralized implementation of target dynamics

open-loop system: controllable voltage sources + network coupling

C

k

d

dt

v

k

= �G

k

v

k

� i

o,k + i

k

, k 2 {1, . . . , n} inverter dynamics

i

o

= Yv network interconnection

target dynamics:d

dt

v = !0 J v


+ ⌘ · e✓(v)| {z }

phase error

+ ↵ · e kvk(v)| {z }

magnitude error

R

k

L

i

k

C

k

v

k

G

k

i

o

Network

v

k

, i

o,k

i

k

i = i

?

(v

k

, i

o,k

) X known: local measurements(v

k

, ik,o) & set-points (v?

k

, p?k

, q?k

)

E unknown: global set-points forrelative angles {✓?

kj

} and nonlocalmeasurements of (v

j

, io,j)

21 / 28

Decentralized implementation of phase error dynamics

e✓,k(v) =X

j

w

jk

(vj

�R(✓?jk

)vk

)

| {z }need to know v

j

and ✓?jk

=X

j

w

jk

(vj

�v

k

)

| {z }negative Laplacian:�Lv

+X

j

w

jk

(I �R(✓?jk

))vk

| {z }local feedback: K

k

(✓?)vk

insight I: non-local measurements from communication through physics:

assume that all lines are homogeneous r

ij

/ìj

= = const., then

R() i

o

| {z }local feedback

= R() Y| {z }

Laplacian matrix

v = Lv

|{z}Laplacian feedback

insight II: angle set-points & line-parameters from power flow equations:

p

?k

=v

?2P

j

R

jk

(1�cos(✓?jk

))�!0Ljk

sin(✓?jk

)

R

2jk

+!20L

2jk

q

?k

=v

?2P

j

!0Ljk

(1�cos(✓?jk

))+R

jk

sin(✓?jk

)

R

2jk

+!20L

2jk

9>>=

>>;) K

k

(✓?)

| {z }global parameters

=R()

v

?2

p

?k

q

?k

�q

?k

p

?k

�

| {z }local parameters

22 / 28

Surprising (?) connections revealed in polar coordinates

closed loop:d

dt

v = !0 J v


+ ⌘ · (�L + K) v)| {z }

phase error

+ ↵ · diag(v? � kv

k

k) v

| {z }magnitude error

polar coordinates in lossless case r

ij

= 0 & near nominal voltage kv

k

k ⇡ 1:

d

dt

kv

k

k = ⌘

✓q

?k

v

?2� q

k

kv

k

k2

◆kv

k

k +↵

v

?(v? � kv

k

k)kv

k

k magnitude

⇡ ⌘ (q?k

� q

k

) kv

k

k + ↵(v? � kv

k

k)kv

k

k q � kvk droop

d

dt

✓k

= !0 + ⌘

✓p

?k

v

?2� p

k

kv

k

k2

◆phase

⇡ !0 + ⌘ (p?k

� p

k

) p � ! droop

= !0 + p

?k

+ ⌘ ·X

j

w

jk

sin(✓j

� ✓k

) Kuramoto oscillator23 / 28

Closed-loop stability analysis

closed loop:d

dt

v =⇠⇠⇠⇠XXXX!0 J v

| {z }in rotating frame

+ ⌘ · (�L + K) v) + ↵ · diag(v?�kv

k

k) v

stationary target sets in rotating coordinate frame:

S = {v 2 Rn | v

k

= R(✓?k1)v1 } set of correct relative angles

A = {v 2 Rn | kv

k

k = v

? } set of correct magnitudes

main result : T = S \A is almost globally asymptotically stable if the grid

parameters, angle set-points, and control gains satisfy a mild condition.

24 / 28

Discussion of stability condition

energy-like function centered at {✓?kj

} and with free-floating amplitudes:

V (v) =X

j<k

w

jk

�kv

k

k2 � kv

k

kkv

j

k cos(✓k

� ✓j

� ✓?kj

)�

= v

>Pv

) closed loop is gradient flow v = �rV (v) for ↵ = 0 and {✓?kj

} = 0

condition , energy function non-increasing despite ↵ > 0 & {✓?kj

} 6= 0:

⌘⇣(K � L)>

P + P(K � L)⌘

+ 2 ↵P � 0

remark : the assumption can always be met in a connected grid if

I slow/fast loops : amplitude gain ↵ < synchronizing gain ⌘

I not too heavy loading : su�ciently small relative angle set-points {✓?kj

}

) stability condition is very reasonable (always met) for practical scenarios

25 / 28

Almost global synchronizationto trajectory with prescribed frequency, voltage

amplitudes, & active/reactive power injections

S

Z

AT

exponential stability of S:

under our assumption, kvk2S is

a Lyapunov function for S.

stability of A relative to S:

provided v(0) 2 S \ {0}, theset A is asymptotically stable.

Z has measure zero:

the region of attraction of {0},termed Z, has measure zero.

continuity argument:

almost all trajectories (not inZ) approaching S must (bycontinuity) converge to S \ A.

26 / 28

Simulation example

v

dc,1

L1

i1

R1

C1 v1G1

i

o,k

1

2v

dc,k

m1

v

dc,2

L2

i2

R2

C2 v2G2

i

o,2

1

2v

dc,2m2

v

dc,3

L3

i3

R3

C3v3 G3

i

o,3

1

2v

dc,3m3

0 5 10 15

�1

0

1

Time

Vol

tage

s a

voltage time series

�1 0 1�1

�0.5

0

0.5

1

v�

v

↵

voltage phase space

27 / 28

Conclusions

Summary:

oscillator networks & power applications

droop control & Kuramoto oscillators

design of decentralized oscillator control

Outlook:

robustness & performance analysis

fair comparison of di↵erent approaches

experimental validation

Insights for coupled oscillators:

Kuramoto models rarely (almost never)admit almost global synchronization

it pays o↵ to work in R2 rather than S1�1 0 1

�1

�0.5

0

0.5

1

v�

v

↵

voltage phase space

28 / 28

Documents

Wokaun Assessment of the proposals: without donors Marco ... · From coupled oscillators to AC power systems (simpliﬁed) swing equation model of interconnected synchronous generators: