2015 KPS autumn meeting

Heetae Kim1, Sang Hoon Lee2, Petter Holme1,* 1 Department of Energy Science, Sungkyunkwan Univ., Suwon, South Korea 2 School of Physics, Korea Institute for Advanced Study, Seoul, South Korea

KPS 2015 Fall conference21–23 Oct. 2015, Gyeongju, South Korea

Building Blocks of Synchronization Stability Transition on Power Grid

Heetae Kim1, Sang Hoon Lee2, Petter Holme1,* 1 Department of Energy Science, Sungkyunkwan Univ., Suwon, South Korea 2 School of Physics, Korea Institute for Advanced Study, Seoul, South Korea

KPS 2015 Fall conference21–23 Oct. 2015, Gyeongju, South Korea

Building Blocks of Synchronization Stability Transition on Power Grid

Producer ✓Net power generation > 0 ✓e.g. Power plants

Consumer ✓Net power generation < 0 ✓e.g. Substations

Producers

Consumers

Nodes ✓Power plants and substations ✓Current flow and phase synchronization

Edges ✓Transmission lines ✓Direction: Undirected (bidirectional) ✓Weight: Transmission strength

Network structure

Attributes

Power grid components

Power grid components

Interactions ✓Rotational motions of rotors are synchronized with

the rated frequency (60Hz). ✓e.g. generators, transformers. ✓A perturbation on a rotor can be absorbed by the

synchronous interaction of the network. ✓Accordingly, rotors can recover the synchrony.

Phase synchronization

d 2θidt 2

= Pi −α idθidt

+ Kij sin(θ j −θi )j∑

i j

Rated frequency Ω (= 2π × 50Hz)

∅i(t)=Ωt+θi(t)

Not in synchrony

Synchronized (Phase-locked)

d 2θidt 2

= dθidt

= 0

Power plant (P>0)

Power plant (P>0)

Consumer (P<0)

3Ω

Ω Ω

2Ω

Synchronization dynamics

Kuramoto-type model

https://youtu.be/tiKH48EMgKE

!!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 net power input

the dissipation (damping) constant

the coupling (transmission) strength

i’s frequency deviation from Ωr

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

θi

Aij Pi α K ωi

Synchronization dynamics

Phase synchronization ✓The dynamics of a generator at node i is affected

by its neighbours.

Kuramoto-type model

P. J. Menck, J. Heitzig, N. Marwan, and J. Kurths, Nat Phys 9, 89 (2013).

Basin stability∈[0,1]

=

https://youtu.be/dFjf_d69HtY

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

✓How much a node can recover synchrony against a large perturbation from a phase space

Basin stability

K

K

Bas

in s

tabi

lity

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

Synchronization stability transition

H. Kim, S. H. Lee, P. Holme, New J Phys. (in press) arXiv:1504.05717.

✓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

1

0

Community consistency

1

0

∆K/∆Kmax

Previously…

H. Kim, S. H. Lee, P. Holme, New J Phys. (in press) arXiv:1504.05717.

0

1

0 20 40

Producer Consumer

Bas

in s

tabil

ity

K

Node A, DNode B, C

A B C D

Diverse transition shapes

✓The basin stability transition curves vary in a network.

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

1

0 20 40

Bas

in s

tabil

ity

K

ProducerConsumer

2 / 4-nodes network motifs

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

Transition pattern analysis

The same attributes

The same structure

The same transition

X O X

X O X

X X O(?)

X O O

2 / 4-nodes network motifs

Difficult to find a rule, which is always valid. ✓Not only a factor divides the transition pattern. ✓Synchronization undergoes non-linear dynamics.

Finding building rules

0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pro

bab

ilit

y

Basin stability

0

1

0 0.2 0.4

6-nodes network motifs


…


≒630 days

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

3-points classification

3-points classification - result

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 classification - result

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

Conclusions

Basin stability transition is important✓Basin stability measures synchronisation stability. ✓The basin stability does not monotonically increases

as a function of K.

✓Network motifs with 2, 4, 6-nodes with two attributes. ✓Some patterns are found. ✓further analysis is under investigation.

✓The functional form of the basin stability transition has patterns. ✓It could provide information about function or meso-scale

characteristics of the power grid.

The transition shape has diversity

Building block investigation on basic network motifs

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 question?

Prof. Petter Holme Heetae Kim Eun Lee Minjin LeeDr. Sang Hoon Lee

[email protected]

Thank you for listening!

National Research Foundation in Korea

0

1

0 20 40 60 80

(a)

Bas

in s

tab

ilit

y

K

N1

(b)

N2-1 N2-2

(f)

N12-1 N12-2 N12-3 N12-4 N12-5

+ 6 nodes

N12-6

(e)

N8-1 N8-2 N8-3

+ 4 nodes

N8-4

(d)

N6-1 N6-2

+ 3 nodes

N6-3

(c)

N4-1

+ 2 nodes

N4-2


0

1

0 20 40 60 80

(a)

Bas

in s

tab

ilit

y

K

N1

(b)

N2-1 N2-2

(f)

N12-1 N12-2 N12-3 N12-4 N12-5

+ 6 nodes

N12-6

(e)

N8-1 N8-2 N8-3

+ 4 nodes

N8-4

(d)

N6-1 N6-2

+ 3 nodes

N6-3

(c)

N4-1

+ 2 nodes

N4-2

Symmetric structure

✓Example 1

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


✓Example 2


Node 1

0

1

0

Bas

inst

abil

ity

K

Node 7

Node 2

Node 8


Node 1

0

1

0

Bas

inst

abil

ity

K

Node 7

Node 2

Node 8


✓Example 3

Graph A Graph Bis isomorphic to

f(A)=7, f(B)=4, f(C)=3, f(D)=6, f(E)=5, f(F)=2, f(G)=1.http://math.stackexchange.com/questions/393416/are-these-2-graphs-isomorphic

5 6

4

1 2

7

3

ED

B

GF

A

C

Origin ✓Iso- : “equal” ✓Morphosis: “to form”

Meaning ✓Formally, an isomorphism is bijective morphism. ✓Informally, an isomorphism is a map that preserves sets

and relations among elements.

Isomorphism screening

Network ensemble generation

Isomorphic motifs

For ensembles of small networks ✓2-nodes network: 1 motif out of 2 ✓4-nodes network: 11 motifs out of 228

24 24 24

124824 24

6126 24

2