community consistency determines the stability transition window of power-grid nodes
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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
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.
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