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
That looks good! Let’s write a manuscript.
It’s on today’s arXiv 1504.05717
Thank you for your attention!
