Netsci 2016

Understanding the Transition Pattern of Synchronization Stability in Power Grids

May 30– June 3, 2016 Seoul, South Korea

Heetae Kim , Petter Holme Department of Energy Science, Sungkyunkwan University, South Korea

Sang Hoon Lee Korea Institute for Advanced Study, South Korea

AMERICA REVEALED - Power black-outs, PBS

http://www.nydailynews.com/news/national/northeast-blackout-2003-ten-year-anniversary-article-1.1426561



1 Department of Energy Science, Sungkyunkwan University, South Korea

2 Korea Institute for Advanced Study, South Korea

Sang Hoon Lee 2 Petter Holme 1Heetae Kim 1


• Power-grid generation models

• Static network ‣ Vulnerability against local failure or terrorist attack.

‣ The structural analyses of real power grids(North America, Italy, and so on)

• Dynamic network ‣ The dynamic phenomena such as cascading propagation.

‣ Synchronization stability


Synchronization in power grids

https://youtu.be/RT1ySBc-Blshttps://youtu.be/tiKH48EMgKE

Phase locked (synchronized)

i j

Power plant (P>0)

Power plant (P>0)

Consumer (P<0)

✓Phase of alternating current (AC) of motors at load or rotors in power generators

Synchronization in power grids

i j

Power plant (P>0)

Power plant (P>0)

Consumer (P<0)

Not in synchrony

Synchronized (Phase-locked)

✓Phase of alternating current (AC) of motors at load or rotors in power generators

Sync, done? Kuramoto-type model

!!θi = !ωi = Pi −αωi −K Aij sin(θi −θ j )∑

the phase at node i (measured in a reference frame that co-rotates with the

grid’s rated frequency Ωr)

adjacency matrix

the power input (P>0) or output (P<0)

the dissipation constant

the coupling constant

i’s frequency deviation from Ωr

θi

Aij Pi α K ωi

G. Filatrella, A. H. Nielsen, and N. F. Pedersen, Eur. Phys. J. B 61, 485 (2008).

• The dynamics of the generator at node i

θi θ j

Sync, done? Kuramoto-type model

the phase angle of voltage at node i (measured in a reference frame that co-

rotates with the grid’s rated frequency Ωr)

adjacency matrix

the power input (P>0) or output (P<0)

the dissipation constant

the coupling constant

i’s frequency deviation from Ωr

θi

Aij Pi α K ωi

θi θ j

G. Filatrella, A. H. Nielsen, and N. F. Pedersen, Eur. Phys. J. B 61, 485 (2008).

K

!!θi = !ωi = Pi −αωi −K Aij sin(θi −θ j )∑• The dynamics of the generator at node i

Sync, stable? Basin stability

Basin stability∈[0,1] =

✓How much a node can recover synchrony

P. J. Menck, J. Heitzig, N. Marwan, and J. Kurths, Nat Phys 9, 89 (2013).https://youtu.be/dFjf_d69HtY

Synchronization

Desynchronization

Basin stability: application example

P. J. Menck, J. Heitzig, J. Kurths, and H. Joachim Schellnhuber, Nat Comms 5, 3969 (2014).

<Northern European power grid>

Sync stability, change!

Otherwise, it will converge to a different solution of (1)! (2): anon-synchronous limit cycle characterized by

onsðtÞ $Paþ aK

Pcos

Pa

t! "

ð3Þ

(provided |P|/a2c1, |P|2/a2cK, see Supplementary Note 1).Other, possibly serial, faults may push the generator from the

synchronous state to perturbed states anywhere in state space. If,for instance, the turning on of a major load or a large fluctuationin renewable generation temporarily drove P below zero, thegenerator’s state would deviate into the lower half of state space(see illustrative trajectory 2 in Fig. 1a). Clearly, the synchronousbasin should be as large as possible. We therefore quantify howstable the synchronous state is against general large perturbationsin terms of basin stability S, a measure of the basin’s volume20.

Specifically, we define basin stability as

SðBÞ ¼Z

wBðy;oÞ rðy;oÞ dy do: ð4Þ

Here

wBðy;oÞ ¼1 ifðy;oÞ 2 B0 otherwise

#ð5Þ

is the indicator function of the synchronous state’s basin B and r isa density with

Rr(y, o) dydo¼ 1 that reflects to which states in

state space the system may be pushed by large perturbations. Thenumber SA[0,1] expresses the likelihood that the system returns tothe synchronous state after having been hit by a large perturbationoccuring randomly according to the probability density r. S¼ 0when synchrony is unstable, and S¼ 1 when it is globally stable.We estimate basin stability by means of a numerical Monte-Carloprocedure20–22: draw T random initial states according to r,simulate the associated trajectories, and count the number U oftimes the system converges to the synchronous state. Then SEU/T. We use T¼ 500 throughout this paper, which yields20 astandard error of e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS 1! Sð Þ

p=ffiffiffiffiTp

o0:023.Intuition suggests that the synchronous state should become

more stable when the transmission line’s transfer capacity Kincreases. This is indeed what we find: the expanding green areain Fig. 1a–c and the characteristic in Fig. 1d show that basinstability S, starting from S¼ 0 for Ko|P|, improves substantiallyas K goes up, until finally synchrony becomes the only stable state(S¼ 1). Here we have chosen a uniform distribution restricted to

a large box in state space, namely

rðy;oÞ ¼ 1= jQj ifðy;oÞ 2 Q0 otherwise

#;whereQ ¼ ½0; 2p()½! 100; 100(:

ð6ÞThis choice allows to clearly distinguish the three important casesin which B covers (i) significantly less than half of statespace (Fig. 1a); (ii) about half of state space (Fig. 1b); and(iii) all of state space (Fig. 1c). We keep using this choice of r inthe following.

Multinode model. The one-node model of equations (1 and 2) isof course a strong simplification: there will be some interplaybetween multiple nodes after one of them has been hit by a largeperturbation, and whether or not the grid will return to syn-chrony depends on the affected node’s properties, particularly itsposition within the grid topology. Hence, we now turn to anN-node version of the model that captures in a coarse-grainedway12,13,17 the decisive electromechanical interactions takingplace in the transmission grid after a large perturbation (seeMethods). It reads

_yi ¼ oi ð7Þ

_oi ¼ ! aioiþPi!XN

j¼1

Kij sin ðyi! yjÞ ð8Þ

where yi and oi denote phase and frequency of the generator atnode i, and ai and Pi are its damping constant and net powerinput. We refer to nodes with Pi40 as net generators and tonodes with Pio0 as net consumers. The matrix {Kij} reflects thewiring topology, with Kij¼Kji40 if nodes i and j are connected,and Kij¼ 0 otherwise.

Power grids do possess stable non-synchronousstates5,12,13,17,18. We assume that there is also a stablesynchronous state with constant phases ys

i and frequenciesoi¼ 0, and with basin of attraction B. How stable is this stateagainst large local perturbations that affect a single node? Andhow does this depend on the network topology? Before turning toa case study of the Northern European power grid, we addressthese questions statistically by studying an ensemble of 1,000randomly generated power grids with N¼ 100 nodes and E¼ 135transmission lines. These numbers yield the average degree/dS¼ 2.7, a value typical of power transmission grids23. Tofocus on the topology, we simplify generator and transmission

1

0.75

0.25

0.50

0

K

S

0 10 20 30 40 50 60 70

15

0

–15

15

0

–15

0 0

!

!

15

0

–15

!

–" –"" "# – #s

(#, !)t1

(#s, 0) = (#, !)t0

Stable limit cycle

Trajectory 1

Trajectory 2

# – #s

0–" "

Figure 1 | Basin stability of the generator in the one-node model. (a–c) State space of the model (1)! (2), with a¼0.1, P¼ 1 and (a) K¼8,(b) K¼ 24, (c) K¼ 65 (see Methods). The solid black circle marks the desired synchronous state (ys,0), and the solid red line shows the undesirednon-synchronous limit cycle attractor. The basin of attraction of (ys, 0) is coloured green, and that of the limit cycle is coloured white. In (a), the greydashed line indicates the fault-on (K¼0) trajectory 1 and its end point (y, o)t1

. The dash-dotted line indicates the fault-on (K¼ 8, P¼ !6) trajectory 2.(d) Basin stability S of the synchronous state versus the transmission capacity K.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4969 ARTICLE

NATURE COMMUNICATIONS | 5:3969 | DOI: 10.1038/ncomms4969 | www.nature.com/naturecommunications 3

& 2014 Macmillan Publishers Limited. All rights reserved.

Otherwise, it will converge to a different solution of (1)! (2): anon-synchronous limit cycle characterized by

onsðtÞ $Paþ aK

Pcos

Pa

t! "

ð3Þ

(provided |P|/a2c1, |P|2/a2cK, see Supplementary Note 1).Other, possibly serial, faults may push the generator from the

synchronous state to perturbed states anywhere in state space. If,for instance, the turning on of a major load or a large fluctuationin renewable generation temporarily drove P below zero, thegenerator’s state would deviate into the lower half of state space(see illustrative trajectory 2 in Fig. 1a). Clearly, the synchronousbasin should be as large as possible. We therefore quantify howstable the synchronous state is against general large perturbationsin terms of basin stability S, a measure of the basin’s volume20.

Specifically, we define basin stability as

SðBÞ ¼Z

wBðy;oÞ rðy;oÞ dy do: ð4Þ

Here

wBðy;oÞ ¼1 ifðy;oÞ 2 B0 otherwise

#ð5Þ

is the indicator function of the synchronous state’s basin B and r isa density with

Rr(y, o) dydo¼ 1 that reflects to which states in

state space the system may be pushed by large perturbations. Thenumber SA[0,1] expresses the likelihood that the system returns tothe synchronous state after having been hit by a large perturbationoccuring randomly according to the probability density r. S¼ 0when synchrony is unstable, and S¼ 1 when it is globally stable.We estimate basin stability by means of a numerical Monte-Carloprocedure20–22: draw T random initial states according to r,simulate the associated trajectories, and count the number U oftimes the system converges to the synchronous state. Then SEU/T. We use T¼ 500 throughout this paper, which yields20 astandard error of e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS 1! Sð Þ

p=ffiffiffiffiTp

o0:023.Intuition suggests that the synchronous state should become

more stable when the transmission line’s transfer capacity Kincreases. This is indeed what we find: the expanding green areain Fig. 1a–c and the characteristic in Fig. 1d show that basinstability S, starting from S¼ 0 for Ko|P|, improves substantiallyas K goes up, until finally synchrony becomes the only stable state(S¼ 1). Here we have chosen a uniform distribution restricted to

a large box in state space, namely

rðy;oÞ ¼ 1= jQj ifðy;oÞ 2 Q0 otherwise

#;whereQ ¼ ½0; 2p()½! 100; 100(:

ð6ÞThis choice allows to clearly distinguish the three important casesin which B covers (i) significantly less than half of statespace (Fig. 1a); (ii) about half of state space (Fig. 1b); and(iii) all of state space (Fig. 1c). We keep using this choice of r inthe following.

Multinode model. The one-node model of equations (1 and 2) isof course a strong simplification: there will be some interplaybetween multiple nodes after one of them has been hit by a largeperturbation, and whether or not the grid will return to syn-chrony depends on the affected node’s properties, particularly itsposition within the grid topology. Hence, we now turn to anN-node version of the model that captures in a coarse-grainedway12,13,17 the decisive electromechanical interactions takingplace in the transmission grid after a large perturbation (seeMethods). It reads

_yi ¼ oi ð7Þ

_oi ¼ ! aioiþPi!XN

j¼1

Kij sin ðyi! yjÞ ð8Þ

where yi and oi denote phase and frequency of the generator atnode i, and ai and Pi are its damping constant and net powerinput. We refer to nodes with Pi40 as net generators and tonodes with Pio0 as net consumers. The matrix {Kij} reflects thewiring topology, with Kij¼Kji40 if nodes i and j are connected,and Kij¼ 0 otherwise.

Power grids do possess stable non-synchronousstates5,12,13,17,18. We assume that there is also a stablesynchronous state with constant phases ys

i and frequenciesoi¼ 0, and with basin of attraction B. How stable is this stateagainst large local perturbations that affect a single node? Andhow does this depend on the network topology? Before turning toa case study of the Northern European power grid, we addressthese questions statistically by studying an ensemble of 1,000randomly generated power grids with N¼ 100 nodes and E¼ 135transmission lines. These numbers yield the average degree/dS¼ 2.7, a value typical of power transmission grids23. Tofocus on the topology, we simplify generator and transmission

1

0.75

0.25

0.50

0

K

S

0 10 20 30 40 50 60 70

15

0

–15

15

0

–15

0 0

!

!

15

0

–15

!

–" –"" "# – #s

(#, !)t1

(#s, 0) = (#, !)t0

Stable limit cycle

Trajectory 1

Trajectory 2

# – #s

0–" "

Figure 1 | Basin stability of the generator in the one-node model. (a–c) State space of the model (1)! (2), with a¼0.1, P¼ 1 and (a) K¼8,(b) K¼ 24, (c) K¼ 65 (see Methods). The solid black circle marks the desired synchronous state (ys,0), and the solid red line shows the undesirednon-synchronous limit cycle attractor. The basin of attraction of (ys, 0) is coloured green, and that of the limit cycle is coloured white. In (a), the greydashed line indicates the fault-on (K¼0) trajectory 1 and its end point (y, o)t1

. The dash-dotted line indicates the fault-on (K¼ 8, P¼ !6) trajectory 2.(d) Basin stability S of the synchronous state versus the transmission capacity K.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4969 ARTICLE

NATURE COMMUNICATIONS | 5:3969 | DOI: 10.1038/ncomms4969 | www.nature.com/naturecommunications 3

& 2014 Macmillan Publishers Limited. All rights reserved.

K=8 K=24 K=65

P. J. Menck, J. Heitzig, J. Kurths, and H. Joachim Schellnhuber, Nat Comms 5, 3969 (2014).

✓Synchronization stability changes according to the transmission strength

Sync stability, change, abruptly!

0

50

100

150

0 1

Num

ber o

f nod

es

Basin stabilityat K=1.2710

0 1Basin stabilityat K=1.2715



Num

ber o

f nod

es

0

1

Bas

in st

abili

ty

✓It is necessary to understand the entire transition

H. Kim, S. H. Lee, P. Holme, New J. Phys. 17, 113005 (2015).

Various transition shapes

0

1

0 20 40

Producer Consumer

Bas

in s

tabil

ity

K

Node A, DNode B, C

A B C D

✓The basin stability transition form varies in a network.

0

0.5

1

0 10 20

(a) (b)

Bas

in s

tab

ilit

y

K

Node 2Node 5

1 2

3 4 5 6

7 8

0

0.5

1

0 10 20

(a) (b)

Bas

in s

tab

ilit

y

K

Node 2Node 5

1 2

3 4 5 6

7 8

0

1

0 20 40

Producer Consumer

Bas

in s

tabil

ity

K

Node A, DNode B, C

A B C D

Basin stability transition window

K

K

Bas

in st

abili

ty

Coupling strength

1

2

1

2

Basin stability transition window

Basin stability at K0

K0 K1

Basin stability at K1

Node 1

Node 2

Klow Khigh

✓The shape of basin stability transition curves are diverse for each node. ✓Both the position of attributes and the network structure affect the shape.

Various transition pattern

Network generation

<Transmission system dada>

Node (Poser plant)

Link (Transmission line)

Agua santa

PlacillaNode

(Substation)

CDEC-SIC Annual report (2014)

• 420 nodes ↳129 power plants

291 substations • 543 edges

Transition windows of Chilean power grid

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

(a)

Bas

in st

abili

ty

K

Node ANode BNode C

0 5 10 15

(a) (b)

10-3−10-2

10-2−10-1

10-1−100

100−101

101−102

<K range>

K 0 1

Proportion

0 20

∆K

Kmid

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

(a)

Bas

in st

abili

ty

K

Node 80Node 286Node 283

0 5 10 15

(a) (b)

10-3−10-2

10-2−10-1

10-1−100

100−101

101−102

<K range>

K 0 1

Proportion

0 20

∆K

Kmid

0

∆K max


✓Heterogeneous distribution of ∆K range

(a) (b)

(c) (d)6

2

4

A 0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16

Bas

in st

abili

ty

K

node Anode Bnode Cnode Dnode Enode Fnode G

B

C

D

E

FG

1

5

3 0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

Bas

in st

abili

ty

K

node 1node 2node 3node 4node 5node 6

(a) (b)

(c) (d)6

2

4

A 0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16

Bas

in st

abili

ty

K


B

C

D

E

FG

1

5

3 0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

Bas

in st

abili

ty

K


Community detection

Mucha P J and Porter M A GenLouvain http://netwiki.amath.unc.edu/GenLouvain/GenLouvain

✓Consistent vs inconsistent community membership Simulations

Community consistency

φi : community consistency of node i.φij : the fraction of the case that node i and j are assigned to the same community for series of community detections.N : the number of nodes.

�i =1

N�1

Pj 6=i(1� 2�ij)2

1

3

2 1

3

21

3

2 1

3

2

1 1 0.51 1 0.50.5 0.5 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

community membership matrix

trial #1 trial #2 trial #3 trial #4

Φi = 0 or 1: completely consistent0<Φi<1: inconsistent


φij : Fraction of being the same community together

Result 1

1

0

Community consistency

1

0

∆K/∆Kmax

Φ: community consistencyk: degreeC: clustering coefficientF: current flow betweenness centrality.

Figure 5.ΔK and community consistency. This is basically a scatter plot but the darker points representmore points, drawnwithtransparency.

Table 1.Pearson correlation coefficient r ofΔK versuscommunity consistency (Φ), degree (k), clustering coeffi-cient (C), and current flowbetweenness (F) centrality.

Φ k C F

r −0.581 0.033 −0.054 0.072p-value < 10−3 0.500 0.266 0.139

Figure 6.ΔK /ΔKmax (left panel) and community consistency (right panel) of theChilean power grid, withΔKmax is themaximumΔK value among the nodes. The insets show the area of interest with the nodeswith largeΔK and small community consistency.

7

New J. Phys. 17 (2015) 113005 HKim et al

Investigation on toy motifs

(a) (b)

(c) (d)6

2

4

A 0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16

Bas

in st

abili

ty

K


B

C

D

E

FG

1

5

3 0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

Bas

in st

abili

ty

K


Acknowledgment: Taro Takaguchi

Topology vs transition

Node 7 Node 4 Node 8 Node 9 Node 12

Node 16Node 2 Node 3 Node 10 Node 11

Node 17Node 1

Node 18

0

1

0 25

Bas

inst

abil

ity

K

Node 6 Node 5 Node 14 Node 13

Node 15



Node 17Node 1

Node 18

0

1

0 25

Bas

inst

abil

ity

K


Node 15Node 7 Node 4 Node 8 Node 9 Node 12


Node 17Node 1

Node 18

0

1

0 25

Bas

inst

abil

ity

K


Node 15



Node 17Node 1

Node 18

0

1

0 25

Bas

inst

abil

ity

K


Node 15

Various transition shapes

✓The basin stability transition curves vary in networks.

e5n1-1e4n2-1e3n2-2e3n2-1e2n1-0e1n2-0e1n2-1e1n1-1

e1n2-1

e1n2-0

e2n1-0

e3n2-1

e3n2-1

e3n2-1

e3n2-1B

asin

sta

bil

ity

K

e1n1-1

2 / 4-nodes network motifs

0

1

0 20 40

Bas

in s

tabil

ity

K

ProducerConsumer

For ensembles of small networks ✓2-nodes network: 1 motif ✓4-nodes network: 11 motifs

✓Transition pattern analysis

6-nodes network motifs



…

≒630 days≒584 motifs

3-points classification

0

1

0 50 100 150

Bas

in s

tab

ilit

y

K

BS of 2-nodes networks

Producer Consumer

0

1

0 20 40

Bas

in s

tab

ilit

y

K


ProducerConsumer

0

1

0 50 100 150

Bas

in s

tab

ilit

y

K


Producer Consumer

0

1

0 20 40

Bas

in s

tab

ilit

y

K


ProducerConsumer

0

1

0 10 20 30 40

Bas

inst

abil

ity

K

Node 1

Node 2 Node 3

Node 4

0

1

0 10 20 30 40

Bas

inst

abil

ity

K

Node 1

Node 2 Node 3

Node 4

0

1

0 10 20 30 40

Bas

inst

abil

ity

K

Node 1

Node 2 Node 3

Node 4

0

1

0 10 20 30 40

Bas

inst

abil

ity

K

Node 1

Node 2 Node 3

Node 4


0

1

0 50 100 150

Bas

in s

tab

ilit

y

K


Producer Consumer

0

1

0 20 40

Bas

in s

tab

ilit

y

K


ProducerConsumer


3-Points 3D diagram ✓Basin stability at only three K values are necessary (K= 7, 14, and 21). ✓Nodes with the large number of triangles have the specific patterns

6-nodes motifs classification

0 10The number of triangles

including the node


(b) (c)

(d) (e)

(a)

Result 2

0

1

0 10 20 30 40

Bas

inst

abil

ity

K

Node 1

Node 2 Node 3

Node 4

0

1

0 10 20 30 40

Bas

inst

abil

ity

K

Node 1

Node 2 Node 3

Node 40

1

0 10 20 30 40

Bas

inst

abil

ity

K

Node 1

Node 2 Node 3

Node 4

BS

K (BS>0.4)

Conclusion

Transition shape matters✓Basin stability measures synchronisation stability. ✓The transition pattern explains the synchronization

stability

✓Transition pattern of BS is affected by topology (structure of network and attributes of nodes). ✓Betweenness predicts the shape of patterns

✓Transition window is correlated with CC ✓Synchronization characteristics is correlated

with meso-scopic network characteristics

Transition window vs Community consistency

Transition patterns vs Betweenness

e5n1-1e4n2-1e3n2-2e3n2-1e2n1-0e1n2-0e1n2-1e1n1-1

e1n2-1

e1n2-0

e2n1-0

e3n2-1

e3n2-1

e3n2-1

e3n2-1

Bas

in s

tab

ilit

y

K

e1n1-1

0

0.5

1

0 20 40 60 80

Bas

in s

tab

ilit

y

K

Node a1

0

0.5

1

Node b1

0

0.5

1

0 50 100 150

(a) (b)

Node b2

How and why…?

Acknowledgement

Any postdoc position?

Petter Holme Heetae Kim Eun Lee Minjin LeeSang Hoon Lee

[email protected]

Now you can graduate!

National Research Foundation in Korea

H. Kim, S. H. Lee, P. Holme, New J. Phys. 17, 113005 (2015). H. Kim, S. H. Lee, P. Holme, arXiv:1602.01712 (2016)