An oral presentation slides at 2015 KPS autumn meeting in 21–23 Oct. 2015 Authors are Heetae Kim, Sang Hoon Lee, and Petter Holme.

1. 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 conference
21–23 Oct. 2015, Gyeongju, South Korea
Building Blocks of
Synchronization Stability Transition
on Power Grid

2. 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 conference
21–23 Oct. 2015, Gyeongju, South Korea
Building Blocks of
Synchronization Stability Transition
on Power Grid

3. 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

4. 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

5. d2
θi
dt2
= Pi −αi
dθi
dt
+ Kij sin(θj −θi )
j
∑
i j
Rated frequency Ω
(= 2π × 50Hz)
∅i(t)=Ωt+θi(t)
Not in synchrony
Synchronized
(Phase-locked)
d2
θi
dt2
=
dθi
dt
= 0
Power plant
(P>0)
Power plant
(P>0)
Consumer
(P<0)
3Ω
Ω Ω
2Ω
Synchronization dynamics
Kuramoto-type model

6. https://youtu.be/tiKH48EMgKE
!!θi = !ωi = Pi −αωi − K Aij sin(θi −θj )∑
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

7. 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

8. K
K
Basinstability
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

9. 1
0
Community
consistency
1
0
∆K/∆Kmax
Previously…
H. Kim, S. H. Lee, P. Holme, New J Phys. (in press) arXiv:1504.05717.

10. 0
1
0 20 40
Producer Consumer
Basinstability
K
Node A, D
Node B, C
A B C D
Diverse transition shapes
✓The basin stability transition curves vary in a network.
e5n1-1e4n2-1e3n2-2e3n2-1e2n1-0e1n2-0e1n2-1
e1n1-1
e1n2-1
e1n2-0
e2n1-0
e3n2-1
e3n2-1
e3n2-1
e3n2-1
Basinstability
K
e1n1-1

11. 0
1
0 20 40
Basinstability
K
Producer
Consumer
2 / 4-nodes network motifs
For ensembles of small networks
✓2-nodes network: 1 motif
✓4-nodes network: 11 motifs
Transition pattern analysis

12. 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

13. 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
Probability
Basin stability
0
1
0 0.2 0.4

14. 6-nodes network motifs

15. 6-nodes network motifs

16. …
6-nodes network motifs
≒630 days

17. 0
1
0 50 100 150
Basinstability
K
BS of 2-nodes networks
Producer
Consumer
0
1
0 20 40
Basinstability
K
BS of 4-nodes networks
Producer
Consumer
0
1
0 50 100 150
Basinstability
K
BS of 2-nodes networks
Producer
Consumer
0
1
0 20 40
Basinstability
K
BS of 4-nodes networks
Producer
Consumer
0
1
Basin
stability
Node 1
Node 2 Node 3
Node 4
0
1
0 10 20 30 40
Basin
stability
K
Node 1
Node 2 Node 3
Node 4
1
n
ity
Node 2
0
1
0 10 20 30 40
Basin
stability
Node 1
Node 2
3-points classification

18. 3-points classification - result
0
1
0 50 100 150
Basinstability
K
BS of 2-nodes networks
Producer
Consumer
0
1
0 20 40
Basinstability
K
BS of 4-nodes networks
Producer
Consumer

19. 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 10
The number of triangles
including the node

20. 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
e5n1-1e4n2-1e3n2-2e3n2-1e2n1-0e1n2-0e1n2-1
e1n1-1
e1n2-1
e1n2-0
e2n1-0
e3n2-1
e3n2-1
e3n2-1
e3n2-1
Basinstability
K
e1n1-1
0
0.5
1
0 20 40 60 80
Basinstability
K
Node a1
How and why…?

21. Acknowledgement
Any question?
Prof. Petter Holme Heetae Kim Eun Lee Minjin LeeDr. Sang Hoon Lee
kimheetae@gmail.com
Thank you for listening!
National Research
Foundation in Korea

22. 0
1
0 20 40 60 80
(a)
Basinstability
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
Synchronization stability transition
0
1
0 20 40 60 80
(a)
Basinstability
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

23. 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
Basin
stability
K
Node 6 Node 5 Node 14 Node 13
Node 15
Synchronization stability transition
✓Example 2

24. Node 3 Node 4 Node 5 Node 6
Node 1
0
1
0
Basin
stability
K
Node 7
Node 2
Node 8
Node 3 Node 4 Node 5 Node 6
Node 1
0
1
0
Basin
stability
K
Node 7
Node 2
Node 8
Synchronization stability transition
✓Example 3

25. 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

26. 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