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Network Analysis of Power Grids:
Synchronization Stability and Sustainability
10 June 2016, N-Center(86102)
Department of Energy Science, Sungkyunkwan University
Heetae Kim
Thesis examination
/ 47
Thesis outline
2
Kim, H., Lee, S.H., Holme, P.

Community consistency determines the stability
transition window of power-grid nodes

New J. Phys. 17, 113005 (2015).
Kim, H., Holme, P.

Network Theory Integrated Life Cycle
Assessment for an Electric Power System
Sustainability 7, 10961–10975 (2015).
Kim, H., Lee, S.H., Holme, P. 

Building blocks of basin stability of power grids
(Phys. Rev. E accepted)
Synchronization
Community
characteristics
Sustainability
(Greenhouse gas)
Isomorphic motifs
Network theory
Power grids
• Title: Network analysis of Power Grids: Synchronization Stability and Sustainability
/ 47
The seven bridges of Königsberg
3
L. Euler. Commentarii Academiae Scientiarum Petropolitanae, 8:128–140, 1741.
Chapter 1
Introduction
1.1 Introduction to network theory
C
DA
B
fb
g
✓Understanding connection or interaction with nodes (vertices) and links (edges)
/ 47
Applications of network theory
4
volution
dominated by single links, whereas the co-authorship data have many
dense, highly connected neighbourhoods. Furthermore, the links in
the phone network correspond to instant communication events, cap-
turing a relationship as it happens. In contrast, the co-authorship data
a Co-authorship
c d
b Phone call
14
0.6
0.5
Vol 446|5 April 2007|doi:10.1038/nature05670
Palla, G., Barabási, A.-L., Vicsek, T. Nature. 446, 664–667 (2007).
that the lack of necessary complexes does not tend to cause cancers, in
contrast to the existence of unnecessary complexes in Pattern 1.
It is known that one form of cancer can affect many tissues, not
only the tissue from which it originated. The expression patterns of
cancer-associated complexes may indicate the cancer-tissue rela-
tions. One interesting way to verify the cancer-tissue relations from
an external source is to use the Web search engine40
. Our basic
assumption is that the more Web pages Google finds from the search
query with ‘[cancer name][tissue name] ’, the more probably the
tissue is related to the cancer. We measure cancer-tissue ‘‘Google
correlation’’ (‘Google page’ column in Table S6). For a specific cancer
A, most Google correlation values for ‘[cancer A][originated tissue of
cancer A]’ pair are ranked on the top among all the ‘[cancer A][tissue
name].’ More precisely, 14 of the 39 cancers have the largest number
of Google correlation value with their originated tissues. This result
validates our assumption. In addition, from Table S6, for each cancer,
we calculated the Pearson correlation coefficient between columns
‘Google pages’ and column ‘Patterns 1–4,’ as shown in Table S8.
The statistical significance test suggested that cancer-associated
complexes are expressed according to Patterns 1, 2 or 4. Thus, we
took the maximum values of Pearson correlation coefficient for
Patterns 1, 2, and 4, and show them in the last column of Table S8.
Most (about 3/4) of the Pearson correlation coefficients in the last
column are positive, suggesting a positive correlation between can-
cer-tissues relations from Google correlation and those from the
number of cancer-associated complexes with differential abundance
levels.
Bipartite complex-cancer relations and common complexes
associated with the same cluster of cancers. The previous subsec-
tion suggests that most cancer-associated complexes are Pattern 1
complexes in the originated normal tissues, i.e., over-expressed in the
cancer tissue but under-expressed in the originated normal tissue.
Thus we focus on these Pattern 1 complexes, and investigate the
bipartite network between cancers and Pattern 1 complexes in
cancer tissues. We constructed a bipartite network between cancers
and Pattern 1 complexes, in which a cancer node is connected to a
complex node if and only if this complex is a Pattern 1 complex of
this cancer. In the bipartite network, we measured the topological
similarity of the vertices according to the following Jaccard similarity
index:
J(u,v)~
NuNvj j
Nu|Nvj j
,
where Nu is the set of neighbors of node u. Then Ward’s clustering, a
hierarchically agglomerative clustering method, was used to cluster
the nodes in the network41
. The hierarchical clustering starts off with
each node being its own cluster and the distance between nodes u and
v is defined as d(u, v) 5 1 2 J(u, v). At each step, pair of clusters (u, v)
with the smallest distance d(u, v) is selected to be merged as a single
cluster and distance measures between clusters are updated as the
weighted sum of distances according to the Lance-Williams algori-
thm42
, and the process is repeated until all nodes have been combined
into one cluster, represented as a dendrogram with a hierarchical
structure. In our case, d(u, v) 5 2 is used as the threshold for cut-
ting the hierarchical tree to yield the clustering structure. Figure 5
shows that some cancers are clustered because of their common over-
expressed complexes, and also some complexes are clustered together.
We classify the 39 cancers under study into six categories accord-
ing to Medical Subject Headings (MeSH43
) annotation of their origi-
nated tissue categories: nerve tissue neoplasm, connective and soft
tissue neoplasm, head and neck neoplasm, urogenital tissue neo-
plasm, digestive system neoplasm, and respiratory tract neoplasm.
Biologically, cancers originated from same tissue should be corre-
lated to some extent. In Table 4, we list the cluster indexes of the
cancers in Figure 5 and their originated tissues. It can be seen that
cancers originated from the same tissue category are clustered
together. Figure 5 shows that cancers in the clusters 4, 5, 6 tend to
link with complexes in clusters 10, 20 and 18 respectively, suggesting
Figure 5 | Bipartite network of cancers and protein complexes of Pattern 1. Triangles (circles) represent cancers (complexes), respectively. The numbers
(and corresponding colors) on vertices show the clustering structure defined with the Jaccard similarity index (see the text).
www.nature.com/scientificreports
SCIENTIFIC REPORTS | 3 : 1583 | DOI: 10.1038/srep01583 6
Zhao, J., Lee, S.H., Huss, M., Holme, P. Sci. Rep. 3, (2013).Patric Hagmann, CHUV-UNIL, Lausanne, Switzerland
✓Brain network
✓Co-authors ✓Phone call
✓Protein network
Wang, Pu et al., Sci. Rep. 2 (2012).
✓Urban road network
/ 47
Network analysis of power girds
5
1. Simpson-Porco, J.W., Doerfler, F., Bullo, F.Nat Comms. 7, (2016).
2. Rikvold, P.A., Hamad, I.A., Israels, B., Poroseva, S.V.:Proceed. on CSP2011. 34, 119–123 (2012).
4. Rohden, M., Sorge, A., Witthaut, D., Timme, M.Chaos. 24, 013123 (2014).
5. Xu, Y., Gurfinkel, A.J., Rikvold, P.A., Physica A 401, 130–140 (2014).
3. Rohden, M., Sorge, A., Timme, M., Witthaut, D., Phys. Rev. Lett. 109, 064101 (2012).
demands in the
e point of voltage
overly optimistic.
ysis to the coupled
mentary Note 5.
ility condition (7)
goal of increasing
rid in Fig. 5a is
nt capacitors have
to support voltage
black in Fig. 5b)
Node 8 is under
er factor of 0.82,
ough 9 have been
le all voltages are
calculate using the
etwork is actually
so apparent by
r flow equations
o the open-circuit
e into account the
ncing the greatest
equipment being
e of supplying an
rid. Our goal is to
stability margins.
power electronic
consumption; in
tal control action.
tability metric (7)
mediately observes
mation on where
example, suppose
s seven and nine,
this example that
ol action at node
ing stress at node
plied at node nine.
epancy in control
ighbours of node
es not only the
s between nodes,
generation (green nodes). Increasing q7 and q9 in this ratio
provides the desired control action, allowing capacitor banks to
be switched out, and we find that D ¼ 0.52 after control. A simple
heuristic control has therefore reduced network stress by
(0.64 À 0.52)/(0.52)C23%, while the voltage profile of the grid
(dotted black) is essentially unchanged.
In summary, the stability condition (7) can be simply and
intuitively used to select control policies which increase grid
stability margins with minimal control effort; additional details
on eigenvector-based control directions40 and on the simulation
setup are available in Supplementary Note 5 and the Supplemen-
tary Methods, respectively.
Discussion
The stability condition (7) provides a long sought-after connec-
tion between network structure, reactive loading and the resulting
voltage profile of the grid. As such, the condition (7) can be used
to identify weak network areas and trace geographical origins of
1
0
0.5
1
Loadingmargin∆
l)
n for high power
Vbase ¼ 345 kV. The
magnitude at node
V Ã
4 1 À dÀð Þ, where
D is shown in
Node number
0 2 4 6 8 10 12 14 16 18 20
Nodevoltage
1.1
1.05
1
0.95
0.85
0.8
0.75
0.7
0.9
a
b
9
8
7
q7
q9
Vi
Vi /Vi*
Figure 5 | Corrective action results for the reduced New England
39-node network. (a) Depiction of the reduced New England grid. Load
nodes {1,y,30} are red circles, while generators {31,y,39} are green
squares. Shunt capacitors are present at all load nodes, but shown explicitly
at nodes 7, 8 and 9. (b) Results of corrective action study. Voltage profile Vi
(black) and scaled voltages Vi=VÃ
i (red), before (solid) and after (dashed)
corrective action. All voltages were scaled by the grid’s base voltage
Vbase ¼ 345 kV. Horizontal dashed lines are operational limits for Vi of
±5% from base voltage. For clarity only nodes {1,y,20} are plotted.
Map by freevectormaps.com.
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10790 120 Per Arne Rikvold et al. / Physics Procedia 34 (2012) 119 – 123
0
0
0
0
0
0
0
2
4
2
2
3
1
2
1 7
2
3
2
2
4
2
3
5
3
3
4
4
4
2
2
2
2
2
4
4
4
4
4
4
2
3
5
3
3
33
3
3
3
5
3
3
3
3
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3
4
3 3
6
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4
4
4
4
4
4 4
4
Figure 1: The Florida high-voltage power grid (left) and a representative model network (right). Generators are repre-
sented by squares and loads by ovals. The partitions shown are in each case the “best” ones obtained in ten independent
runs with the bottom-up partitioning algorithm of Ref. [6]. Islands are identified by different colors and by numbers that
can be seen if viewed at high magnification. See text for discussion.
N × N symmetric weight matrix W, whose elements wij ≥ 0 represent the “conductances” of the
edges (transmission lines) between vertices (generators or loads) i and j,
wij = (number of lines between vertices i and j)/( normalized geographical distance), (1)
where the “geographical distance” is the length of the edge connecting i and j. To obtain a model
independent of any specific systems of length units, the distances are normalized such that the
areal density of vertices is unity. In Fig. 1 we show a map of the Florida network together with a
representative model network. Model networks were produced by the following procedure.
1. We placed the N = 84 vertices randomly in a square of area N.
2. Following the standard “stub” method [8], we attached 2M = 400 stubs or half-edges
randomly to the N vertices. (Actually, to ensure that no vertices in this small network
should be totally isolated, we first attached one stub to each vertex, and then distributed the
remaining 2M − N stubs randomly between the vertices.) The resulting degree distribution
for the particular model grid discussed in this paper is shown together with that of the real
Florida grid in Fig. 2(a).
3. We connected the stubs randomly in pairs, with the restriction that self-loops (two mutually
connected stubs at the same vertex) were forbidden.
4. To obtain an edge-length distribution with the same average as that of the real Florida
grid (≈ 1.09 in our dimensionless units), we employed a Monte Carlo (MC) “cooling”
procedure using a “Hamiltonian” in which the total edge length L plays the role of the
n read
dK sinðh1 À h0Þ;
K sinðh0 À h1Þ:
(12)
tion jNj ¼ jdj always holds,
tate is determined by
dition for the existence of a
¼ P0; (13)
ust be able to transmit the
it. Fig. 2(b) shows a different
umer units arranged on a
nections between the central
est consumers (h1) and d2 ¼ 2
mers with phase h1 and those
of the problem, we have to
hases. The reduced equations
À h0Þ;
À h1Þ þ K sinðh0 À h1Þ;
h2Þ:
(14)
d the relations
ðNP0Þ=ðd1KÞ;
P0=K:
(15)
now be higher than the criti-
NP0
d1
(16)
or the example shown in Fig.
r critical coupling strength
ous motif for the existence of
tely clear from physical rea-
eading away from the power
mer units instead of just one.
OWER GRIDS
ctive behavior of large net-
and consumers and analyze
small decentralized power stations, which contribute
PR ¼ 2.5 P0 each. This means that, for instance, for NP ¼ 5
we have five large power sources and 5 Á 4 ¼ 20 small power
sources, thus 50% of the total power is produced by decen-
tralized power sources. Consumers and generators are con-
nected by transmission lines with a capacity K, assumed to
be the same for all connections.
We consider three types of networks topologies, sche-
matically shown in Fig. 3. In a quasi-regular power grid, all
consumers are placed on a squared lattice. The generators
are placed randomly at the lattice and connected to the adja-
cent four consumer units (cf. Fig. 3(a)). In a random net-
work, all elements are linked completely randomly with an
average number of six connections per node (cf. Fig. 3(b)).
A small world network is obtained by a standard rewiring
algorithm15
as follows. Starting from ring network, where
every element is connected to its four nearest neighbors, the
connections are randomly rewired with a probability of 0.1
(cf. Fig. 3(c)).
B. The synchronization transition
The stable operation of a power grid requires that all
machines run at the same frequency. The phases of the
machines will generally be different but the phase differen-
ces are constant in time. This global phase locking is related
to phase cohesiveness.16,17
Phases are called phase cohesive
if their differences are below an upper bound, which is auto-
matically the case if every phase has a fixed value. Thus, if
the system is phase-locked it is automatically phase-cohesive
as well. This must be distinguished from partial
FIG. 3. Small size cartoons of different network topologies: (a) Quasi-
regular grid, (b) random network, and (c) small-world network.
e article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
115.145.205.240 On: Wed, 19 Nov 2014 06:15:32
Y. Xu et al. / Physica A 401 (2014) 130–140 131
a b
Fig. 1. The Florida Grid and its network representation. (a) Map of the Floridian high-voltage grid [8]. Despite being referred to as the ‘‘Florida Grid’’
(FLG), this map does not cover power stations in the far northwest corner of Florida state, while including 2 power plants and 4 transmission substations
in southern Georgia. (b) FLG in our network representation, in which power lines are represented as straight lines between vertices. The distances are
normalized such that the number of vertices per unit area is unity, and the total area is illustrated as the shaded region. Generators are represented as
squares and loads as ovals. Thicker, darker links represent multiple power lines between vertices.
electric power-grid science [4], we study the Floridian (high-voltage) power transmission grid as a complex network [5–7]
by characterizing its topology and investigating its structural architecture. The purpose of our investigation is to explore
organizational principles of power grids, using the Floridian high-voltage grid as a case study.
The organization of this paper is as follows. In Section 2, we view the Florida Grid as a spatial network, focusing on the
structural organization of geographic lengths and edge conductance weights, as well as the mixing patterns of generators and
loads. In Section 3, we propose network optimization models that exhibit desirable properties, and show that they coincide
with the real Florida Grid on several key network measures. In Section 4, we compare the box-counting fractal dimensions
of the power-grid networks generated by our models. Our conclusions and some suggestions for future research are given
in Section 5.
2. The Floridian power grid as a complex weighted network
The Floridian high-voltage power grid (hereafter referred to as ‘‘FLG’’, see map in Fig. 1), is a relatively small network
of the damage of single transmission lines on the sta
of the power grid. An example is shown in Fig. 4(a) f
0
0
0.5
1
|r∞
|
c=2b
0 5
0
1
2
3
ω∞
(1/s)
Fraction
FIG. 4 (color online). Decentralizing power plants de
damage. This figure illustrates simulated future developm
distributed ones. (a) First step of decentralizing: One powe
to zero (Pj ¼ 0). Instead, ten new small generators are ad
illustrates their relevance for the global structural stability
power outage: Two links (blue solid arrows) are crucial f
plant is replaced by distributed generators. Six links (red d
To increasingly decentralize the grid, randomly chosen
(c) Promotion of self-organized synchronization due to the
and mean frequency !1 as a function of coupling strength
strength Kc for the onset of synchronization due to the rep
stability. Panel shows a number of critical links in the ne
outage, discarding bridges. Here, coupling strength is fi
realizations, as in Fig. 3. The shaded areas in (d) and (e)
PRL 109, 064101 (2012) P H Y S I C A L
3. Self-Organising sync.2. Cascading failure1. Voltage collapse
4. Decentralisation 5. Topology characteristics
/ 47
Previous studies
6
2004 - Albert, R. et al., Structural vulnerability of the North American power grid. Phys Rev E. 69
- Crucitti, P. et al., A topological analysis of the Italian electric power grid. Physica A. 338
2005 - Crucitti, P. et al., Locating critical lines in high-voltage electrical power grids. Fluc. and Noise Letters. 5
- Kinney, R. et al., Modeling cascading failures in the North American power grid. Eur. Phys. J. B. 46
2008 - Filatrella, G. et al., Analysis of a power grid using a Kuramoto-like model. Eur. Phys. J. B. 61
2009 - Arianos, S. et al., Power grid vulnerability: A complex network approach. Chaos. 19
- Xu, S. et al., Vulnerability assessment of Power Grid Based on Complex Network Theory. APPEEC 2009
- Bompard, E. et al., Assessment of Structural Vulnerability for Power Grids by Network Performance
Based on Complex Networks. CRITIS 2008
2010 - Chen, G. et al., Attack structural vulnerability of power grids: A hybrid approach based on complex
networks. Physica A 389
- Wei, Z. et al., The networks data mining of power grid based on complex networks theory. IRE
Professional Group on.
- Hamad, I.A. et al., Floridian high-voltage power-grid network partitioning and cluster optimization using
simulated annealing.CSP2011. 15
- Galli, S. et al., For the Grid and Through the Grid: The Role of Power Line Communications in the Smart
Grid. Proceedings of the IEEE. 99
- Rikvold, P.A. et al., Modeling power grids. Proceedings of CSP2011. 34
2011
/ 47
Previous studies
7
- Simpson-Porco, J.W. et al., Voltage collapse in complex power grids. Nat Comms. 7
2013
2014
2016
2015
- Motter, A.E. et al., Spontaneous synchrony in power-grid networks. Nat Phys. 9 (2013).
- Pagani, G.A., Aiello, M.: The Power Grid as a complex network: A survey. Physica A. 392 (2013).
- Schultz, P. et al., A random growth model for power grids and other spatially embedded
infrastructure networks. Eur. Phys. J. Spec. Top. 223
- Xu, Y. et al., Architecture of the Florida power grid as a complex network. Physica A. 401
- Oliver, J.M. et al., Electrical Power Grid Network Optimisation by Evolutionary Computing. Computer
Science. 29
- Nardelli, P.H.J. et al., Models for the modern power grid. Eur. Phys. J. Spec. Top.
- Pagani, G.A., Aiello, M.: Power grid complex network evolutions for the smart grid. Physica A. 396
- Pahwa, S. et al., Abruptness of Cascade Failures in Power Grids. Sci. Rep. 4
- Rohden, M. et al., Impact of network topology on synchrony of oscillatory power grids. Chaos. 24
- Ouyang, M. et al., Comparisons of complex network based models and direct current power flow
model to analyze power grid vulnerability under intentional attacks. Physica A. 403
- Menck, P.J. et al., How dead ends undermine power grid stability. Nat Comms. 5
- Schultz, P. Detours around basin stability in power networks. New J. Phys. 16
- Nishikawa, T., Motter, A.E.: Comparative analysis of existing models for power-grid synchronization. New
J. Phys. 17
2012 - Menck, P.J., Kurths, J.: Topological Identification of Weak Points in Power Grids. Presented at the
Nonlinear Dynamics of Electronic Systems, Proceedings of NDES 2012
- Rohden, M. et al., Self-Organized Synchronization in Decentralized Power Grids. Phys. Rev. Lett. 109
/ 47
Research question 1
8
How stable is each node in a power grid

in terms of synchronization?
Kim, H., Lee, S.H., Holme, P.

Community consistency determines the stability transition window of power-grid nodes

New J. Phys. 17, 113005 (2015).
/ 47
Synchronisation
9
✓Synchronised fireflies
t
t
✓Synchronised power generators’ motion
Event happening at the same time
Video clipVideo clip
/ 47
distri-
lower
merical
ngth of
s. (13)
imates
nsition
nchro-
erators
transi-
critical
NP 6¼ 0.
order
ps to a
wer sta-
sumers
plants
upling
diately
quasi-
differ-
nected
fferent
ther as
mount
re, we
inverse of the stability exponent k (cf. the discussion in Sec.
III A).
Fig. 7(c) shows how the synchronization time depends
on the structure of the network and the mixture of power
generators. For several paradigmatic systems of oscillators,
it has been demonstrated that the time scale of the relaxation
process depends crucially on the network structure.19,20
Here, however, we have a network of damped second order
oscillators. Therefore, the relaxation is almost exclusively
given by the inverse damping constant aÀ1
. Indeed, we find
FIG. 7. Relaxation to the synchronized steady state: (a) Illustration of the
relaxation process (K/P0 ¼ 10 and Np ¼ 10). We have plotted the dynamics
of the phases hj only for one generator (red) and one consumer (blue) for the
sake of clarity. (b) Exponential decrease of the distance to the steady state
(blue line) and a fit according to dðtÞ $ eÀt=ssync
(black line). (c) The synchro-
Synchronisation in power-grid network?
10
✓Synchronisation of coupled oscillators
Frequency harmony: ωsync = 0 (ωi-ωr =0)
Cohesive phases: |θi − θj| ≤ γ < π/2 for every edge {i, j} ∈ E.
Dörfler, F., Chertkov, M., Bullo, F. PNAS. 110, 2005–2010 (2013).
during a short time interval (Dt ¼ 10s) as illustrated in the
upper panels of Fig. 8. Therefore, the condition of (7) is
violated and the system cannot remain in its stable state.
After the perturbation is switched off again, the system
relaxes back to a steady state or not, depending on the
strength of the perturbation. Fig. 8 shows examples of the
dynamics for a weak (a) and strong (b) perturbation,
respectively.
FIG. 8. Weak and strong perturbation. The upper panels show the time-
dependent power load of the consumers. A perturbation of strength Ppert is
applied in the time interval t ʦ.5,6
The lower panels show the resulting dy-Rohden, M., Sorge, A., Witthaut, D., Timme, M. Chaos. 24, 013123 (2014).
FIG. 5. The synchronization transition as a function of the coupling strength
K: The order parameter r1 (left-hand side) and the phase velocity v1 (right-
Chaos 24, 013123 (2014)v(t) =
1
N
!θj (t)j∑r(t) =
1
N
e
iθj (t)
j∑
θj/2π
r(∞)
v(∞)
Angle ±10°
Voltage: 0–5%
Slip: ±0.067Hz
Michael J. Thompson, Fundamentals and Advancements in Generator Synchronising Systems
/ 47
Network representation of power grids
11
New J. Phys. 17 (2015) 015012 T Nishikawa and A E Motter
Nishikawa, T., Motter, A.E. New J. Phys. 17, 015012 (2015).
/ 47
Network representation of power grids
12
4 Structure and stability of power grids
✓Zhukov’s aggregation
Machowski, J., Bialek, J. W., and Bumby, J. R. (2008). Power System Dynamics: Stability and Control. John Wiley & Sons, Chichester.
A power-grid node represents net generator or net consumer.
With net power injection, nodes at which Pi >0 (Pi <0) are called net generators (net consumers).
a single equivalent generator i
All generators and consumers connected to
node i of the high-voltage grid
/ 47
i j
Power plant
(P>0)
Power plant
(P>0)
Consumer
(P<0)
Equation of dynamics
13
the phase angle of voltage at node i (relative to the reference frame)
i’s angular velocity (frequency) deviation from the reference frequency
adjacency matrix
the power input (P>0) or output (P<0)
the dissipation constant
the transmission capacity (transmission strength or coupling constant)
θi
ωi
Aij
Pi
α
K
!ωi = Pi −α !θi − K Aij sin(θi −θj )∑
!θi = ωi
Reference frameNode jNode i
/ 47
Measuring synchronization stability
14
Basin stability of a node ∈ [0,1] =
✓Basin stability measures how much a node recovers its synchronization.
P. J. Menck, J. Heitzig, N. Marwan, and J. Kurths, Nat Phys 9, 89 (2013).
https://youtu.be/dFjf_d69HtY
Synchronization Desynchronization
Total
phase
space
Recovered
From the given phase space, basin stability of a node is the fraction of the initial phase states
that lead to convergence after dynamic interaction between nodes.
Synchronization → Perturbation from phase space → Synchronization
↳ Desynchronization
+
=
/ 47
Basin stability: application example
15
P. J. Menck, J. Heitzig, J. Kurths, and H. Joachim Schellnhuber, Nat Comms 5, 3969 (2014).
<Northern European power grid>
/ 47
Sync stability, change!
16
1
0.75
0.50S
15
0
–15
Stable limit cycle
Trajectory 1
0–
/ncomms4969
S
15
0
–15
0 0
15
0
–15
–
( , )t1
( s, 0) = ( , )t0
Stable limit cycle
Trajectory 1
Trajectory 2
0–
S | DOI: 10.1038/ncomms4969
K=8 K=24 K=65
P. J. Menck, J. Heitzig, J. Kurths, and H. Joachim Schellnhuber, Nat Comms 5, 3969 (2014).
✓Synchronization stability changes according to the transmission strength
Basinstability(B)
Transmission strength (K)
/ 47
Sync stability, change, abruptly!
17
0
50
100
150
0 1
Numberofnodes
Basin stability
at K=1.2710
0 1
Basin stability
at K=1.2715
0 1
Basin stability
at K=1.2720
0 1
Basin stability
at K=1.2725
Numberofnodes
0
1
Basinstability
✓It is necessary to understand the entire transition
H. Kim, S. H. Lee, P. Holme, New J. Phys. 17, 113005 (2015).
/ 47
Various transition shapes
18
0
1
0 20 40
Producer Consumer
Basinstability
K
Node A, D
Node B, C
A B C D
✓The basin stability transition form varies in a network.
0
0.5
1
0 10 20
(a) (b)
Basinstability
K
Node 2
Node 5
1 2
3 4 5 6
7 8
0
0.5
1
0
(a) (b)
Basinstability
1 2
3 4 5 6
7 8
0
1
0 20 40
Producer Consumer
Basinstability
K
Node A, D
Node B, C
A B C D
KK
/ 47
Basin stability transition window
19
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
✓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
/ 47
Network generation
20
<Transmission system dada>
Node
(Poser plant)
Link
(Transmission line)
Agua
santa
Placilla
Node
(Substation)
CDEC-SIC Annual report (2014)
• 420 nodes
↳129 power plants
291 substations
• 543 edges
/ 4721
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
(a)Basinstability
K
Node A
Node B
Node C
0 5 10 15
(a) (b)
10
-3
−10
-2
10
-2
−10
-1
10
-1
−10
0
100
−101
10
1
−10
2
<K range>
K
0 1
Proportion
0 20
∆K
Kmid
20
0
6
3
0 5 10 15
(b)
10-3
−10-2
10
-2
−10
-1
10-1
−100
10
0
−10
1
10
1
−10
2
<K range>
K
0 1
Proportion
0 20
∆K
Kmid
0
∆K max
H. Kim, S. H. Lee, P. Holme, New J. Phys. 17, 113005 (2015).
✓Heterogeneous distribution of ∆K range
<ΔK>
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
(a)
Basinstability
K
Node 80
Node 286
Node 283
0 5 10 15
(a) (b)
10-3
−10-2
10
-2
−10
-1
10-1
−100
10
0
−10
1
10
1
−10
2
<K range>
K
0 1
Proportion
0 20
∆K
Kmid
10 15
K
0 20
∆K
Kmid
A
B
C
Transition windows of Chilean power grid
Transition windows of Chilean power grid
/ 47
Community detection
22
Mucha P J and Porter M A GenLouvain
http://netwiki.amath.unc.edu/GenLouvain/GenLouvain
✓Consistent vs inconsistent community membership Simulations
/ 47
Community consistency
23
φi : community consistency of node i.

φij : the fraction of the case that node i and j are assigned to the same

community for series of community detections.

N : the number of nodes.
i = 1
N 1
P
j6=i(1 2 ij)2
1
3
2
1
3
21
3
2 1
3
2
1 1 0.5
1 1 0.5
0.5 0.5 1
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
community membership matrix
trial #1 trial #2 trial #3 trial #4
Φi = 1: completely consistent
Φi = 0: inconsistent
H. Kim, S. H. Lee, P. Holme, New J. Phys. 17, 113005 (2015).
φij : Fraction of being the same community together
/ 47
1
0
∆K/∆Kmax
Result 1
24
1
0
Community
consistency
Φ: community consistency

k: degree

C: clustering coefficient

F: current flow betweenness centrality.
Table 1. Pearson correlation coefficient r of ΔK versus
community consistency (Φ), degree (k), clustering coeffi-
cient (C), and current flow betweenness (F) centrality.
Φ k C F
r −0.581 0.033 −0.054 0.072
p-value < 10−3
0.500 0.266 0.139
/ 47
Investigation on toy motifs
25
(a) (b)
(c) (d)
6
2
4
A 0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16
Basinstability
K
node A
node B
node C
node D
node E
node F
node G
B
C
D
E
F
G
1
5
3
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
Basinstability
K
node 1
node 2
node 3
node 4
node 5
node 6
/ 47
Research question 2
26
How does a network topology affect node’s stability?
Kim, H., Lee, S.H., Holme, P. 

Building blocks of basin stability of power grids
(Phys. Rev. E accepted)
/ 47
Topology vs transition
27
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
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
Node 7 Node 4 Node 8 Node 9
Node 16Node 2 Node 3 Node 10
Node 17Node 1
Node 18
0
1
0 25
Basin
stability
K
Node 6 Node 5 Node 14
Node 15
Node 7 Node 4 Node 8
Node 16Node 2 Node 3
Node 17Node 1
Node 18
0
1
0 25
Basin
stability
K
Node 6 Node 5
/ 47
2 / 4-nodes network motifs
28
0
1
0 20 40
Basinstability
K
Producer
Consumer
For ensembles of small networks
✓2-nodes network: 1 motif
✓4-nodes network: 11 motifs
✓Transition pattern analysis
/ 47
6-nodes network motifs
29
/ 47
6-nodes network motifs
30
…
≒584 motifs
/ 47
3-points classification
31
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
/ 47
3-points classification
32
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
/ 47
3-points classification
33
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
/ 47
3D cloud of BS of nodes
34
✓Nodes are clustered at certain regions in the 3D space
Network characteristics are correlated with transition patterns.
/ 47
Result 2
35
0
1
0 10 20 30 40
Basin
stability
K
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 40
1
0 10 20 30 40
Basin
stability
K
Node 1
Node 2 Node 3
Node 4
BS
K1
0.4
K2 K3
Betweenness explains the transition pattern well.
K (BS >0.4) K (BS >0.4)
/ 47
Research question 3
36
How much greenhouse gas is emitted during transmission?
Kim, H., Holme, P.

Network Theory Integrated Life Cycle Assessment for an Electric Power System
Sustainability 7, 10961–10975 (2015).
/ 47
Electric power system
37
Transmission
✓From resources to energy services
Consumption Generation
Power plantsUsers Infrastructures
/ 47
Electric power system
38
Consumption Generation
CO2
Infrastructures Power plantsUsers
✓Relationship between generation and consumption
/ 47
Electric power system
39
Power plantsUsers
TransmissionConsumption
Infrastructures
✓Relationship between transmission and consumption
Transmission
distance
? km
/ 47
Research purpose
40
✴ Allocate environmental impacts of transmission loss to regions
according to both electricity consumption and transmission
distance
✴ Integrate network theory into LCA
/ 47
Inventory analysis
41
SIC center of Economic Load Dispatch (CDEC-SIC)
✓the main national electricity company
✓serves 92% of country’s population
✓10 regions out of 15
✓42 provinces out of 57
Data collection
✓2007 to 2012
✓System boundary
/ 47
Developing Chilean conversion factor
42
g CO2/ kWh GWh
0.006 325
0.266 13,450
0.157 7,946
0.285 14,385
0.027 1,358
0.020 1,013
0.239 12,072
= 23.02 Mt CO2-eq
Greenhouse gas (GHG) emissions
of Chilean electric power system
/ 47
Transmission algorithm
43
Amount of electricity consumption × Transmission distance
i : a substation node
j : a power plant node
aij: electricity supply from j to i
dij : transmission distance from j to i
nhd(i) : neighbor nodes of i
Edi : energy distance of i
i
j Power plant
Substation
Transmission
distance dij
2
A
B
1
Greedy algorithm
↳the nearest substation has the top priority
and the others are supplied subsequently
Possible
pair
Transmission
distance
Optim
al
Electricity
supply aij
Edi =
f (aij ,dij )
j∈nhd(i)
k
∑
f (aij ,dij )
j∈nhd(i)
k
∑
i=1
n
∑
Energy distance
/ 47
Result 3
44
/ 47
Result 3
45
kt CO2
676
66000
30000
(b) Consumption-based

cartogram
(c) Energy distance-based

cartogram
(a) Map of Chile
Santiago 

Metropolitan
Region
Concepción
Chañaral
/ 47
Conclusion
46
Network analysis of power-grid nodes’ synchronisation
✓Transition width of synchronisation stability is correlated with community consistency
✓Transition Transition pattern of BS is affected by topology 

(structure of network and attributes of nodes).
✓Betweenness predicts the shape of patterns
Network theory is a good complement for power grid analysis
Network analysis + Environmental analysis
✓Re-allocate GHG emissions according to transmission load
✓Integrate network analysis on environmental impact assessment of power grids
/ 47
Acknowledgement
47
Thank you for your attention
sincerely appreciate your being my neighbour node.
and

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PhD Thesis examination

  • 1. Network Analysis of Power Grids: Synchronization Stability and Sustainability 10 June 2016, N-Center(86102) Department of Energy Science, Sungkyunkwan University Heetae Kim Thesis examination
  • 2. / 47 Thesis outline 2 Kim, H., Lee, S.H., Holme, P.
 Community consistency determines the stability transition window of power-grid nodes
 New J. Phys. 17, 113005 (2015). Kim, H., Holme, P.
 Network Theory Integrated Life Cycle Assessment for an Electric Power System Sustainability 7, 10961–10975 (2015). Kim, H., Lee, S.H., Holme, P. 
 Building blocks of basin stability of power grids (Phys. Rev. E accepted) Synchronization Community characteristics Sustainability (Greenhouse gas) Isomorphic motifs Network theory Power grids • Title: Network analysis of Power Grids: Synchronization Stability and Sustainability
  • 3. / 47 The seven bridges of Königsberg 3 L. Euler. Commentarii Academiae Scientiarum Petropolitanae, 8:128–140, 1741. Chapter 1 Introduction 1.1 Introduction to network theory C DA B fb g ✓Understanding connection or interaction with nodes (vertices) and links (edges)
  • 4. / 47 Applications of network theory 4 volution dominated by single links, whereas the co-authorship data have many dense, highly connected neighbourhoods. Furthermore, the links in the phone network correspond to instant communication events, cap- turing a relationship as it happens. In contrast, the co-authorship data a Co-authorship c d b Phone call 14 0.6 0.5 Vol 446|5 April 2007|doi:10.1038/nature05670 Palla, G., Barabási, A.-L., Vicsek, T. Nature. 446, 664–667 (2007). that the lack of necessary complexes does not tend to cause cancers, in contrast to the existence of unnecessary complexes in Pattern 1. It is known that one form of cancer can affect many tissues, not only the tissue from which it originated. The expression patterns of cancer-associated complexes may indicate the cancer-tissue rela- tions. One interesting way to verify the cancer-tissue relations from an external source is to use the Web search engine40 . Our basic assumption is that the more Web pages Google finds from the search query with ‘[cancer name][tissue name] ’, the more probably the tissue is related to the cancer. We measure cancer-tissue ‘‘Google correlation’’ (‘Google page’ column in Table S6). For a specific cancer A, most Google correlation values for ‘[cancer A][originated tissue of cancer A]’ pair are ranked on the top among all the ‘[cancer A][tissue name].’ More precisely, 14 of the 39 cancers have the largest number of Google correlation value with their originated tissues. This result validates our assumption. In addition, from Table S6, for each cancer, we calculated the Pearson correlation coefficient between columns ‘Google pages’ and column ‘Patterns 1–4,’ as shown in Table S8. The statistical significance test suggested that cancer-associated complexes are expressed according to Patterns 1, 2 or 4. Thus, we took the maximum values of Pearson correlation coefficient for Patterns 1, 2, and 4, and show them in the last column of Table S8. Most (about 3/4) of the Pearson correlation coefficients in the last column are positive, suggesting a positive correlation between can- cer-tissues relations from Google correlation and those from the number of cancer-associated complexes with differential abundance levels. Bipartite complex-cancer relations and common complexes associated with the same cluster of cancers. The previous subsec- tion suggests that most cancer-associated complexes are Pattern 1 complexes in the originated normal tissues, i.e., over-expressed in the cancer tissue but under-expressed in the originated normal tissue. Thus we focus on these Pattern 1 complexes, and investigate the bipartite network between cancers and Pattern 1 complexes in cancer tissues. We constructed a bipartite network between cancers and Pattern 1 complexes, in which a cancer node is connected to a complex node if and only if this complex is a Pattern 1 complex of this cancer. In the bipartite network, we measured the topological similarity of the vertices according to the following Jaccard similarity index: J(u,v)~ NuNvj j Nu|Nvj j , where Nu is the set of neighbors of node u. Then Ward’s clustering, a hierarchically agglomerative clustering method, was used to cluster the nodes in the network41 . The hierarchical clustering starts off with each node being its own cluster and the distance between nodes u and v is defined as d(u, v) 5 1 2 J(u, v). At each step, pair of clusters (u, v) with the smallest distance d(u, v) is selected to be merged as a single cluster and distance measures between clusters are updated as the weighted sum of distances according to the Lance-Williams algori- thm42 , and the process is repeated until all nodes have been combined into one cluster, represented as a dendrogram with a hierarchical structure. In our case, d(u, v) 5 2 is used as the threshold for cut- ting the hierarchical tree to yield the clustering structure. Figure 5 shows that some cancers are clustered because of their common over- expressed complexes, and also some complexes are clustered together. We classify the 39 cancers under study into six categories accord- ing to Medical Subject Headings (MeSH43 ) annotation of their origi- nated tissue categories: nerve tissue neoplasm, connective and soft tissue neoplasm, head and neck neoplasm, urogenital tissue neo- plasm, digestive system neoplasm, and respiratory tract neoplasm. Biologically, cancers originated from same tissue should be corre- lated to some extent. In Table 4, we list the cluster indexes of the cancers in Figure 5 and their originated tissues. It can be seen that cancers originated from the same tissue category are clustered together. Figure 5 shows that cancers in the clusters 4, 5, 6 tend to link with complexes in clusters 10, 20 and 18 respectively, suggesting Figure 5 | Bipartite network of cancers and protein complexes of Pattern 1. Triangles (circles) represent cancers (complexes), respectively. The numbers (and corresponding colors) on vertices show the clustering structure defined with the Jaccard similarity index (see the text). www.nature.com/scientificreports SCIENTIFIC REPORTS | 3 : 1583 | DOI: 10.1038/srep01583 6 Zhao, J., Lee, S.H., Huss, M., Holme, P. Sci. Rep. 3, (2013).Patric Hagmann, CHUV-UNIL, Lausanne, Switzerland ✓Brain network ✓Co-authors ✓Phone call ✓Protein network Wang, Pu et al., Sci. Rep. 2 (2012). ✓Urban road network
  • 5. / 47 Network analysis of power girds 5 1. Simpson-Porco, J.W., Doerfler, F., Bullo, F.Nat Comms. 7, (2016). 2. Rikvold, P.A., Hamad, I.A., Israels, B., Poroseva, S.V.:Proceed. on CSP2011. 34, 119–123 (2012). 4. Rohden, M., Sorge, A., Witthaut, D., Timme, M.Chaos. 24, 013123 (2014). 5. Xu, Y., Gurfinkel, A.J., Rikvold, P.A., Physica A 401, 130–140 (2014). 3. Rohden, M., Sorge, A., Timme, M., Witthaut, D., Phys. Rev. Lett. 109, 064101 (2012). demands in the e point of voltage overly optimistic. ysis to the coupled mentary Note 5. ility condition (7) goal of increasing rid in Fig. 5a is nt capacitors have to support voltage black in Fig. 5b) Node 8 is under er factor of 0.82, ough 9 have been le all voltages are calculate using the etwork is actually so apparent by r flow equations o the open-circuit e into account the ncing the greatest equipment being e of supplying an rid. Our goal is to stability margins. power electronic consumption; in tal control action. tability metric (7) mediately observes mation on where example, suppose s seven and nine, this example that ol action at node ing stress at node plied at node nine. epancy in control ighbours of node es not only the s between nodes, generation (green nodes). Increasing q7 and q9 in this ratio provides the desired control action, allowing capacitor banks to be switched out, and we find that D ¼ 0.52 after control. A simple heuristic control has therefore reduced network stress by (0.64 À 0.52)/(0.52)C23%, while the voltage profile of the grid (dotted black) is essentially unchanged. In summary, the stability condition (7) can be simply and intuitively used to select control policies which increase grid stability margins with minimal control effort; additional details on eigenvector-based control directions40 and on the simulation setup are available in Supplementary Note 5 and the Supplemen- tary Methods, respectively. Discussion The stability condition (7) provides a long sought-after connec- tion between network structure, reactive loading and the resulting voltage profile of the grid. As such, the condition (7) can be used to identify weak network areas and trace geographical origins of 1 0 0.5 1 Loadingmargin∆ l) n for high power Vbase ¼ 345 kV. The magnitude at node V Ã 4 1 À dÀð Þ, where D is shown in Node number 0 2 4 6 8 10 12 14 16 18 20 Nodevoltage 1.1 1.05 1 0.95 0.85 0.8 0.75 0.7 0.9 a b 9 8 7 q7 q9 Vi Vi /Vi* Figure 5 | Corrective action results for the reduced New England 39-node network. (a) Depiction of the reduced New England grid. Load nodes {1,y,30} are red circles, while generators {31,y,39} are green squares. Shunt capacitors are present at all load nodes, but shown explicitly at nodes 7, 8 and 9. (b) Results of corrective action study. Voltage profile Vi (black) and scaled voltages Vi=VÃ i (red), before (solid) and after (dashed) corrective action. All voltages were scaled by the grid’s base voltage Vbase ¼ 345 kV. Horizontal dashed lines are operational limits for Vi of ±5% from base voltage. For clarity only nodes {1,y,20} are plotted. Map by freevectormaps.com. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10790 120 Per Arne Rikvold et al. / Physics Procedia 34 (2012) 119 – 123 0 0 0 0 0 0 0 2 4 2 2 3 1 2 1 7 2 3 2 2 4 2 3 5 3 3 4 4 4 2 2 2 2 2 4 4 4 4 4 4 2 3 5 3 3 33 3 3 3 5 3 3 3 3 4 6 3 5 3 3 3 3 4 3 3 6 4 3 3 4 3 3 4 3 4 4 4 4 4 4 4 4 4 Figure 1: The Florida high-voltage power grid (left) and a representative model network (right). Generators are repre- sented by squares and loads by ovals. The partitions shown are in each case the “best” ones obtained in ten independent runs with the bottom-up partitioning algorithm of Ref. [6]. Islands are identified by different colors and by numbers that can be seen if viewed at high magnification. See text for discussion. N × N symmetric weight matrix W, whose elements wij ≥ 0 represent the “conductances” of the edges (transmission lines) between vertices (generators or loads) i and j, wij = (number of lines between vertices i and j)/( normalized geographical distance), (1) where the “geographical distance” is the length of the edge connecting i and j. To obtain a model independent of any specific systems of length units, the distances are normalized such that the areal density of vertices is unity. In Fig. 1 we show a map of the Florida network together with a representative model network. Model networks were produced by the following procedure. 1. We placed the N = 84 vertices randomly in a square of area N. 2. Following the standard “stub” method [8], we attached 2M = 400 stubs or half-edges randomly to the N vertices. (Actually, to ensure that no vertices in this small network should be totally isolated, we first attached one stub to each vertex, and then distributed the remaining 2M − N stubs randomly between the vertices.) The resulting degree distribution for the particular model grid discussed in this paper is shown together with that of the real Florida grid in Fig. 2(a). 3. We connected the stubs randomly in pairs, with the restriction that self-loops (two mutually connected stubs at the same vertex) were forbidden. 4. To obtain an edge-length distribution with the same average as that of the real Florida grid (≈ 1.09 in our dimensionless units), we employed a Monte Carlo (MC) “cooling” procedure using a “Hamiltonian” in which the total edge length L plays the role of the n read dK sinðh1 À h0Þ; K sinðh0 À h1Þ: (12) tion jNj ¼ jdj always holds, tate is determined by dition for the existence of a ¼ P0; (13) ust be able to transmit the it. Fig. 2(b) shows a different umer units arranged on a nections between the central est consumers (h1) and d2 ¼ 2 mers with phase h1 and those of the problem, we have to hases. The reduced equations À h0Þ; À h1Þ þ K sinðh0 À h1Þ; h2Þ: (14) d the relations ðNP0Þ=ðd1KÞ; P0=K: (15) now be higher than the criti- NP0 d1 (16) or the example shown in Fig. r critical coupling strength ous motif for the existence of tely clear from physical rea- eading away from the power mer units instead of just one. OWER GRIDS ctive behavior of large net- and consumers and analyze small decentralized power stations, which contribute PR ¼ 2.5 P0 each. This means that, for instance, for NP ¼ 5 we have five large power sources and 5 Á 4 ¼ 20 small power sources, thus 50% of the total power is produced by decen- tralized power sources. Consumers and generators are con- nected by transmission lines with a capacity K, assumed to be the same for all connections. We consider three types of networks topologies, sche- matically shown in Fig. 3. In a quasi-regular power grid, all consumers are placed on a squared lattice. The generators are placed randomly at the lattice and connected to the adja- cent four consumer units (cf. Fig. 3(a)). In a random net- work, all elements are linked completely randomly with an average number of six connections per node (cf. Fig. 3(b)). A small world network is obtained by a standard rewiring algorithm15 as follows. Starting from ring network, where every element is connected to its four nearest neighbors, the connections are randomly rewired with a probability of 0.1 (cf. Fig. 3(c)). B. The synchronization transition The stable operation of a power grid requires that all machines run at the same frequency. The phases of the machines will generally be different but the phase differen- ces are constant in time. This global phase locking is related to phase cohesiveness.16,17 Phases are called phase cohesive if their differences are below an upper bound, which is auto- matically the case if every phase has a fixed value. Thus, if the system is phase-locked it is automatically phase-cohesive as well. This must be distinguished from partial FIG. 3. Small size cartoons of different network topologies: (a) Quasi- regular grid, (b) random network, and (c) small-world network. e article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 115.145.205.240 On: Wed, 19 Nov 2014 06:15:32 Y. Xu et al. / Physica A 401 (2014) 130–140 131 a b Fig. 1. The Florida Grid and its network representation. (a) Map of the Floridian high-voltage grid [8]. Despite being referred to as the ‘‘Florida Grid’’ (FLG), this map does not cover power stations in the far northwest corner of Florida state, while including 2 power plants and 4 transmission substations in southern Georgia. (b) FLG in our network representation, in which power lines are represented as straight lines between vertices. The distances are normalized such that the number of vertices per unit area is unity, and the total area is illustrated as the shaded region. Generators are represented as squares and loads as ovals. Thicker, darker links represent multiple power lines between vertices. electric power-grid science [4], we study the Floridian (high-voltage) power transmission grid as a complex network [5–7] by characterizing its topology and investigating its structural architecture. The purpose of our investigation is to explore organizational principles of power grids, using the Floridian high-voltage grid as a case study. The organization of this paper is as follows. In Section 2, we view the Florida Grid as a spatial network, focusing on the structural organization of geographic lengths and edge conductance weights, as well as the mixing patterns of generators and loads. In Section 3, we propose network optimization models that exhibit desirable properties, and show that they coincide with the real Florida Grid on several key network measures. In Section 4, we compare the box-counting fractal dimensions of the power-grid networks generated by our models. Our conclusions and some suggestions for future research are given in Section 5. 2. The Floridian power grid as a complex weighted network The Floridian high-voltage power grid (hereafter referred to as ‘‘FLG’’, see map in Fig. 1), is a relatively small network of the damage of single transmission lines on the sta of the power grid. An example is shown in Fig. 4(a) f 0 0 0.5 1 |r∞ | c=2b 0 5 0 1 2 3 ω∞ (1/s) Fraction FIG. 4 (color online). Decentralizing power plants de damage. This figure illustrates simulated future developm distributed ones. (a) First step of decentralizing: One powe to zero (Pj ¼ 0). Instead, ten new small generators are ad illustrates their relevance for the global structural stability power outage: Two links (blue solid arrows) are crucial f plant is replaced by distributed generators. Six links (red d To increasingly decentralize the grid, randomly chosen (c) Promotion of self-organized synchronization due to the and mean frequency !1 as a function of coupling strength strength Kc for the onset of synchronization due to the rep stability. Panel shows a number of critical links in the ne outage, discarding bridges. Here, coupling strength is fi realizations, as in Fig. 3. The shaded areas in (d) and (e) PRL 109, 064101 (2012) P H Y S I C A L 3. Self-Organising sync.2. Cascading failure1. Voltage collapse 4. Decentralisation 5. Topology characteristics
  • 6. / 47 Previous studies 6 2004 - Albert, R. et al., Structural vulnerability of the North American power grid. Phys Rev E. 69 - Crucitti, P. et al., A topological analysis of the Italian electric power grid. Physica A. 338 2005 - Crucitti, P. et al., Locating critical lines in high-voltage electrical power grids. Fluc. and Noise Letters. 5 - Kinney, R. et al., Modeling cascading failures in the North American power grid. Eur. Phys. J. B. 46 2008 - Filatrella, G. et al., Analysis of a power grid using a Kuramoto-like model. Eur. Phys. J. B. 61 2009 - Arianos, S. et al., Power grid vulnerability: A complex network approach. Chaos. 19 - Xu, S. et al., Vulnerability assessment of Power Grid Based on Complex Network Theory. APPEEC 2009 - Bompard, E. et al., Assessment of Structural Vulnerability for Power Grids by Network Performance Based on Complex Networks. CRITIS 2008 2010 - Chen, G. et al., Attack structural vulnerability of power grids: A hybrid approach based on complex networks. Physica A 389 - Wei, Z. et al., The networks data mining of power grid based on complex networks theory. IRE Professional Group on. - Hamad, I.A. et al., Floridian high-voltage power-grid network partitioning and cluster optimization using simulated annealing.CSP2011. 15 - Galli, S. et al., For the Grid and Through the Grid: The Role of Power Line Communications in the Smart Grid. Proceedings of the IEEE. 99 - Rikvold, P.A. et al., Modeling power grids. Proceedings of CSP2011. 34 2011
  • 7. / 47 Previous studies 7 - Simpson-Porco, J.W. et al., Voltage collapse in complex power grids. Nat Comms. 7 2013 2014 2016 2015 - Motter, A.E. et al., Spontaneous synchrony in power-grid networks. Nat Phys. 9 (2013). - Pagani, G.A., Aiello, M.: The Power Grid as a complex network: A survey. Physica A. 392 (2013). - Schultz, P. et al., A random growth model for power grids and other spatially embedded infrastructure networks. Eur. Phys. J. Spec. Top. 223 - Xu, Y. et al., Architecture of the Florida power grid as a complex network. Physica A. 401 - Oliver, J.M. et al., Electrical Power Grid Network Optimisation by Evolutionary Computing. Computer Science. 29 - Nardelli, P.H.J. et al., Models for the modern power grid. Eur. Phys. J. Spec. Top. - Pagani, G.A., Aiello, M.: Power grid complex network evolutions for the smart grid. Physica A. 396 - Pahwa, S. et al., Abruptness of Cascade Failures in Power Grids. Sci. Rep. 4 - Rohden, M. et al., Impact of network topology on synchrony of oscillatory power grids. Chaos. 24 - Ouyang, M. et al., Comparisons of complex network based models and direct current power flow model to analyze power grid vulnerability under intentional attacks. Physica A. 403 - Menck, P.J. et al., How dead ends undermine power grid stability. Nat Comms. 5 - Schultz, P. Detours around basin stability in power networks. New J. Phys. 16 - Nishikawa, T., Motter, A.E.: Comparative analysis of existing models for power-grid synchronization. New J. Phys. 17 2012 - Menck, P.J., Kurths, J.: Topological Identification of Weak Points in Power Grids. Presented at the Nonlinear Dynamics of Electronic Systems, Proceedings of NDES 2012 - Rohden, M. et al., Self-Organized Synchronization in Decentralized Power Grids. Phys. Rev. Lett. 109
  • 8. / 47 Research question 1 8 How stable is each node in a power grid
 in terms of synchronization? Kim, H., Lee, S.H., Holme, P.
 Community consistency determines the stability transition window of power-grid nodes
 New J. Phys. 17, 113005 (2015).
  • 9. / 47 Synchronisation 9 ✓Synchronised fireflies t t ✓Synchronised power generators’ motion Event happening at the same time Video clipVideo clip
  • 10. / 47 distri- lower merical ngth of s. (13) imates nsition nchro- erators transi- critical NP 6¼ 0. order ps to a wer sta- sumers plants upling diately quasi- differ- nected fferent ther as mount re, we inverse of the stability exponent k (cf. the discussion in Sec. III A). Fig. 7(c) shows how the synchronization time depends on the structure of the network and the mixture of power generators. For several paradigmatic systems of oscillators, it has been demonstrated that the time scale of the relaxation process depends crucially on the network structure.19,20 Here, however, we have a network of damped second order oscillators. Therefore, the relaxation is almost exclusively given by the inverse damping constant aÀ1 . Indeed, we find FIG. 7. Relaxation to the synchronized steady state: (a) Illustration of the relaxation process (K/P0 ¼ 10 and Np ¼ 10). We have plotted the dynamics of the phases hj only for one generator (red) and one consumer (blue) for the sake of clarity. (b) Exponential decrease of the distance to the steady state (blue line) and a fit according to dðtÞ $ eÀt=ssync (black line). (c) The synchro- Synchronisation in power-grid network? 10 ✓Synchronisation of coupled oscillators Frequency harmony: ωsync = 0 (ωi-ωr =0) Cohesive phases: |θi − θj| ≤ γ < π/2 for every edge {i, j} ∈ E. Dörfler, F., Chertkov, M., Bullo, F. PNAS. 110, 2005–2010 (2013). during a short time interval (Dt ¼ 10s) as illustrated in the upper panels of Fig. 8. Therefore, the condition of (7) is violated and the system cannot remain in its stable state. After the perturbation is switched off again, the system relaxes back to a steady state or not, depending on the strength of the perturbation. Fig. 8 shows examples of the dynamics for a weak (a) and strong (b) perturbation, respectively. FIG. 8. Weak and strong perturbation. The upper panels show the time- dependent power load of the consumers. A perturbation of strength Ppert is applied in the time interval t ʦ.5,6 The lower panels show the resulting dy-Rohden, M., Sorge, A., Witthaut, D., Timme, M. Chaos. 24, 013123 (2014). FIG. 5. The synchronization transition as a function of the coupling strength K: The order parameter r1 (left-hand side) and the phase velocity v1 (right- Chaos 24, 013123 (2014)v(t) = 1 N !θj (t)j∑r(t) = 1 N e iθj (t) j∑ θj/2π r(∞) v(∞) Angle ±10° Voltage: 0–5% Slip: ±0.067Hz Michael J. Thompson, Fundamentals and Advancements in Generator Synchronising Systems
  • 11. / 47 Network representation of power grids 11 New J. Phys. 17 (2015) 015012 T Nishikawa and A E Motter Nishikawa, T., Motter, A.E. New J. Phys. 17, 015012 (2015).
  • 12. / 47 Network representation of power grids 12 4 Structure and stability of power grids ✓Zhukov’s aggregation Machowski, J., Bialek, J. W., and Bumby, J. R. (2008). Power System Dynamics: Stability and Control. John Wiley & Sons, Chichester. A power-grid node represents net generator or net consumer. With net power injection, nodes at which Pi >0 (Pi <0) are called net generators (net consumers). a single equivalent generator i All generators and consumers connected to node i of the high-voltage grid
  • 13. / 47 i j Power plant (P>0) Power plant (P>0) Consumer (P<0) Equation of dynamics 13 the phase angle of voltage at node i (relative to the reference frame) i’s angular velocity (frequency) deviation from the reference frequency adjacency matrix the power input (P>0) or output (P<0) the dissipation constant the transmission capacity (transmission strength or coupling constant) θi ωi Aij Pi α K !ωi = Pi −α !θi − K Aij sin(θi −θj )∑ !θi = ωi Reference frameNode jNode i
  • 14. / 47 Measuring synchronization stability 14 Basin stability of a node ∈ [0,1] = ✓Basin stability measures how much a node recovers its synchronization. P. J. Menck, J. Heitzig, N. Marwan, and J. Kurths, Nat Phys 9, 89 (2013). https://youtu.be/dFjf_d69HtY Synchronization Desynchronization Total phase space Recovered From the given phase space, basin stability of a node is the fraction of the initial phase states that lead to convergence after dynamic interaction between nodes. Synchronization → Perturbation from phase space → Synchronization ↳ Desynchronization + =
  • 15. / 47 Basin stability: application example 15 P. J. Menck, J. Heitzig, J. Kurths, and H. Joachim Schellnhuber, Nat Comms 5, 3969 (2014). <Northern European power grid>
  • 16. / 47 Sync stability, change! 16 1 0.75 0.50S 15 0 –15 Stable limit cycle Trajectory 1 0– /ncomms4969 S 15 0 –15 0 0 15 0 –15 – ( , )t1 ( s, 0) = ( , )t0 Stable limit cycle Trajectory 1 Trajectory 2 0– S | DOI: 10.1038/ncomms4969 K=8 K=24 K=65 P. J. Menck, J. Heitzig, J. Kurths, and H. Joachim Schellnhuber, Nat Comms 5, 3969 (2014). ✓Synchronization stability changes according to the transmission strength Basinstability(B) Transmission strength (K)
  • 17. / 47 Sync stability, change, abruptly! 17 0 50 100 150 0 1 Numberofnodes Basin stability at K=1.2710 0 1 Basin stability at K=1.2715 0 1 Basin stability at K=1.2720 0 1 Basin stability at K=1.2725 Numberofnodes 0 1 Basinstability ✓It is necessary to understand the entire transition H. Kim, S. H. Lee, P. Holme, New J. Phys. 17, 113005 (2015).
  • 18. / 47 Various transition shapes 18 0 1 0 20 40 Producer Consumer Basinstability K Node A, D Node B, C A B C D ✓The basin stability transition form varies in a network. 0 0.5 1 0 10 20 (a) (b) Basinstability K Node 2 Node 5 1 2 3 4 5 6 7 8 0 0.5 1 0 (a) (b) Basinstability 1 2 3 4 5 6 7 8 0 1 0 20 40 Producer Consumer Basinstability K Node A, D Node B, C A B C D KK
  • 19. / 47 Basin stability transition window 19 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 ✓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
  • 20. / 47 Network generation 20 <Transmission system dada> Node (Poser plant) Link (Transmission line) Agua santa Placilla Node (Substation) CDEC-SIC Annual report (2014) • 420 nodes ↳129 power plants 291 substations • 543 edges
  • 21. / 4721 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 (a)Basinstability K Node A Node B Node C 0 5 10 15 (a) (b) 10 -3 −10 -2 10 -2 −10 -1 10 -1 −10 0 100 −101 10 1 −10 2 <K range> K 0 1 Proportion 0 20 ∆K Kmid 20 0 6 3 0 5 10 15 (b) 10-3 −10-2 10 -2 −10 -1 10-1 −100 10 0 −10 1 10 1 −10 2 <K range> K 0 1 Proportion 0 20 ∆K Kmid 0 ∆K max H. Kim, S. H. Lee, P. Holme, New J. Phys. 17, 113005 (2015). ✓Heterogeneous distribution of ∆K range <ΔK> 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 (a) Basinstability K Node 80 Node 286 Node 283 0 5 10 15 (a) (b) 10-3 −10-2 10 -2 −10 -1 10-1 −100 10 0 −10 1 10 1 −10 2 <K range> K 0 1 Proportion 0 20 ∆K Kmid 10 15 K 0 20 ∆K Kmid A B C Transition windows of Chilean power grid Transition windows of Chilean power grid
  • 22. / 47 Community detection 22 Mucha P J and Porter M A GenLouvain http://netwiki.amath.unc.edu/GenLouvain/GenLouvain ✓Consistent vs inconsistent community membership Simulations
  • 23. / 47 Community consistency 23 φi : community consistency of node i.
 φij : the fraction of the case that node i and j are assigned to the same
 community for series of community detections.
 N : the number of nodes. i = 1 N 1 P j6=i(1 2 ij)2 1 3 2 1 3 21 3 2 1 3 2 1 1 0.5 1 1 0.5 0.5 0.5 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ community membership matrix trial #1 trial #2 trial #3 trial #4 Φi = 1: completely consistent Φi = 0: inconsistent H. Kim, S. H. Lee, P. Holme, New J. Phys. 17, 113005 (2015). φij : Fraction of being the same community together
  • 24. / 47 1 0 ∆K/∆Kmax Result 1 24 1 0 Community consistency Φ: community consistency
 k: degree
 C: clustering coefficient
 F: current flow betweenness centrality. Table 1. Pearson correlation coefficient r of ΔK versus community consistency (Φ), degree (k), clustering coeffi- cient (C), and current flow betweenness (F) centrality. Φ k C F r −0.581 0.033 −0.054 0.072 p-value < 10−3 0.500 0.266 0.139
  • 25. / 47 Investigation on toy motifs 25 (a) (b) (c) (d) 6 2 4 A 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 16 Basinstability K node A node B node C node D node E node F node G B C D E F G 1 5 3 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 Basinstability K node 1 node 2 node 3 node 4 node 5 node 6
  • 26. / 47 Research question 2 26 How does a network topology affect node’s stability? Kim, H., Lee, S.H., Holme, P. 
 Building blocks of basin stability of power grids (Phys. Rev. E accepted)
  • 27. / 47 Topology vs transition 27 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 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 Node 7 Node 4 Node 8 Node 9 Node 16Node 2 Node 3 Node 10 Node 17Node 1 Node 18 0 1 0 25 Basin stability K Node 6 Node 5 Node 14 Node 15 Node 7 Node 4 Node 8 Node 16Node 2 Node 3 Node 17Node 1 Node 18 0 1 0 25 Basin stability K Node 6 Node 5
  • 28. / 47 2 / 4-nodes network motifs 28 0 1 0 20 40 Basinstability K Producer Consumer For ensembles of small networks ✓2-nodes network: 1 motif ✓4-nodes network: 11 motifs ✓Transition pattern analysis
  • 29. / 47 6-nodes network motifs 29
  • 30. / 47 6-nodes network motifs 30 … ≒584 motifs
  • 31. / 47 3-points classification 31 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
  • 32. / 47 3-points classification 32 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
  • 33. / 47 3-points classification 33 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
  • 34. / 47 3D cloud of BS of nodes 34 ✓Nodes are clustered at certain regions in the 3D space Network characteristics are correlated with transition patterns.
  • 35. / 47 Result 2 35 0 1 0 10 20 30 40 Basin stability K 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 40 1 0 10 20 30 40 Basin stability K Node 1 Node 2 Node 3 Node 4 BS K1 0.4 K2 K3 Betweenness explains the transition pattern well. K (BS >0.4) K (BS >0.4)
  • 36. / 47 Research question 3 36 How much greenhouse gas is emitted during transmission? Kim, H., Holme, P.
 Network Theory Integrated Life Cycle Assessment for an Electric Power System Sustainability 7, 10961–10975 (2015).
  • 37. / 47 Electric power system 37 Transmission ✓From resources to energy services Consumption Generation Power plantsUsers Infrastructures
  • 38. / 47 Electric power system 38 Consumption Generation CO2 Infrastructures Power plantsUsers ✓Relationship between generation and consumption
  • 39. / 47 Electric power system 39 Power plantsUsers TransmissionConsumption Infrastructures ✓Relationship between transmission and consumption Transmission distance ? km
  • 40. / 47 Research purpose 40 ✴ Allocate environmental impacts of transmission loss to regions according to both electricity consumption and transmission distance ✴ Integrate network theory into LCA
  • 41. / 47 Inventory analysis 41 SIC center of Economic Load Dispatch (CDEC-SIC) ✓the main national electricity company ✓serves 92% of country’s population ✓10 regions out of 15 ✓42 provinces out of 57 Data collection ✓2007 to 2012 ✓System boundary
  • 42. / 47 Developing Chilean conversion factor 42 g CO2/ kWh GWh 0.006 325 0.266 13,450 0.157 7,946 0.285 14,385 0.027 1,358 0.020 1,013 0.239 12,072 = 23.02 Mt CO2-eq Greenhouse gas (GHG) emissions of Chilean electric power system
  • 43. / 47 Transmission algorithm 43 Amount of electricity consumption × Transmission distance i : a substation node j : a power plant node aij: electricity supply from j to i dij : transmission distance from j to i nhd(i) : neighbor nodes of i Edi : energy distance of i i j Power plant Substation Transmission distance dij 2 A B 1 Greedy algorithm ↳the nearest substation has the top priority and the others are supplied subsequently Possible pair Transmission distance Optim al Electricity supply aij Edi = f (aij ,dij ) j∈nhd(i) k ∑ f (aij ,dij ) j∈nhd(i) k ∑ i=1 n ∑ Energy distance
  • 45. / 47 Result 3 45 kt CO2 676 66000 30000 (b) Consumption-based
 cartogram (c) Energy distance-based
 cartogram (a) Map of Chile Santiago 
 Metropolitan Region Concepción Chañaral
  • 46. / 47 Conclusion 46 Network analysis of power-grid nodes’ synchronisation ✓Transition width of synchronisation stability is correlated with community consistency ✓Transition Transition pattern of BS is affected by topology 
 (structure of network and attributes of nodes). ✓Betweenness predicts the shape of patterns Network theory is a good complement for power grid analysis Network analysis + Environmental analysis ✓Re-allocate GHG emissions according to transmission load ✓Integrate network analysis on environmental impact assessment of power grids
  • 47. / 47 Acknowledgement 47 Thank you for your attention sincerely appreciate your being my neighbour node. and