Heetae Kim, Sang Hoon Lee, Petter Holme Department of Energy Science, SKKU

23 April 2015, Daejeon, KPS spring meeting 2015

Community Consistency Determines the Stability Transition Window of Power-grid Nodes

Consistent violet community

Consistent orange community

Syncstable

Syncstable

unstable

Basin stability Synchronization on power grid

Community Consistency Determines the Stability Transition Window of Power-grid Nodes

Heetae Kim, Sang Hoon Lee, Petter Holme Department of Energy Science, SKKU

23 April 2015, Daejeon, KPS spring meeting 2015

Synchronization between power-grid nodes

https://youtu.be/GRk_qJxaxh8 https://youtu.be/RT1ySBc-Bls

https://youtu.be/tiKH48EMgKE

Sync and unsync Phase deviation

Rotational motion generates alternating phase electric current

Synchronizable (when connected on the power grid)

⤷

Synchronization on a network

!!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 constant

the coupling constant

i’s frequency deviation from Ωr

P =ViVj

Xij

sin(θ j −θi )

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).

Power transferred from j to i

The dynamics of the generator at node i

Basin stability

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 of nodes

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

<Northern European power grid>

Abrupt transition of basin stability

0

50

100

150

0 1

Num

ber o

f nod

es

Basin stabilityat K=1.2710

0 1Basin stabilityat K=1.2715

0 1Basin stabilityat K=1.2720

0 1Basin stabilityat K=1.2725

Num

ber o

f nod

es

0

1

Bas

in st

abili

ty

It is necessary to understand the entire transition

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

1 2

It is useful to understand the entire transition

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

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

(a)B

asin

stab

ility

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

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 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

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 is community consistency of node i. φij is the fraction of the case that node i and j are assigned to the same community for series of community detections.

N is 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

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

from community membership matrix

Result

1

0

Community consistency

1

0

∆K/∆Kmax

Conclusions

• Lessons learned • Basin stability transition window enables us to understand

power-grid synchronization in a comprehensive way. • Community consistency measures how a node closely belong to

communities. • Low community consistency → wide stability transition window.

• Further researches • Functional centrality measure of power-grid nodes. • Synchronization transition dynamics in various conditions.

Acknowledgement

We just have done!

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

National Research Foundation in Korea

kimheetae@gmail.com

That looks good! Let’s write a manuscript.

It’s on today’s arXiv 1504.05717

Thank you for your attention!