Intentional islanding is used to limit cascading power failures by isolating
highly connected "islands" with local generating capacity. To efficiently
isolate an island, one should break as few power lines as possible. This is a
graph partitioning problem, and here we give preliminary results on islanding
of the Italian and Floridian high-voltage grids by spectral matrix methods.Comment: 4 pages, 2 figures

Andersson

Avramovic

Brett Israels

Chowdhury

Dale

Dobson

Fortunato

Ibrahim Abou Hamad

Newman

Peiravi

Per Arne Rikvold

Rosato

Seary

Svetlana V. Poroseva

Wang

Yang

English

arXiv

Crossref

Spectral matrix methods for partitioning power grids: Applications to the Italian and Floridian high-voltage networks

AbstractIntentional islanding is used to limit cascading power failures by isolating highly connected “islands” with local generating capacity. To efficiently isolate an island, one should break as few power lines as possible. This is a graph partitioning problem, and here we give preliminary results on islanding of the Italian and Floridian high-voltage grids by spectral matrix methods

Hamad, Ibrahim Abou

Israels, Brett

Rikvold, Per Arne

Poroseva, Svetlana V.

Elsevier - Publisher Connector 

Physics Procedia 00 (2010) 1–5
Physics Procedia
Spectral matrix methods for partitioning power grids:
Applications to the Italian and Floridian high-voltage networks
Ibrahim Abou Hamada, Brett Israelsa, Per Arne Rikvolda, Svetlana V. Porosevab
aDepartment of Physics and Center for Materials Research and Technology
Florida State University, Tallahassee, FL 32306-4350, U.S.A.
bCenter for Advanced Power Systems, Florida State University, Tallahassee, FL 32310, U.S.A.
Abstract
Intentional islanding is used to limit cascading power failures by isolating highly connected
“islands” with local generating capacity. To eﬃciently isolate an island, one should break as
few power lines as possible. This is a graph partitioning problem, and here we give preliminary
results on islanding of the Italian and Floridian high-voltage grids by spectral matrix methods.
Keywords: power grid, intentional islanding, cascading failure, blackout prevention, network
theory, graph partitioning, spectral matrix methods
1. Introduction
Large-scale blackouts have devastating eﬀects on the economy and welfare of any modern
society [1, 2]. One of the reasons [3, 4] that make such catastrophic events possible is the lack
of a pre-planned strategy for splitting a power grid into separate parts with independent gener-
ation, also called islands [5]. This defensive strategy, called planned, intentional, controlled, or
defensive islanding, is a last-resort, but eﬀective means to prevent cascading outages [2, 6].
Intentional islanding splits a power system into islands by breaking selected transmission
lines. Multiple approaches (see, e.g., [1, 2, 6, 7, 8, 9]) have been suggested for optimizing the
selection of the lines to be cut. Most analyze the system state, steady or dynamic. A useful con-
tribution to these studies can be an analysis of the system topology based on a representation of
the network as a graph [10, 11, 12, 13]. Identiﬁcation of “weak” links, whose removal can split a
given network into independent islands can be beneﬁcial for i) initiating fast predetermined inten-
tional islanding and ii) preventing unintentional islanding. If one knows in advance the minimal
set of links that must be broken to create a separate island, a decision on intentional islanding
can be made very fast. These links should also be closely monitored, as their removal (e.g., by
accident or sabotage) will result with certainty in unintentional islanding. The advantages of this
strategy are that i) the islanded areas can be planned and analyzed in advance with regard to their
generating capacity and necessary load-shedding if an island has to be formed, ii) islands do not
c© 2010 Published by Elsevier B.V.
Physics Procedia 4 (2010) 125–129
www.elsevier.com/locate/procedia
1875-3892 c© 2010 Published by Elsevier B.V.
doi:10.1016/j.phpro.2010.08.016
Open access under CC BY-NC-ND license.
Open access under CC BY-NC-ND license.
I. Abou Hamad et al. / Physics Procedia 00 (2010) 1–5 2
depend on the system disturbance and coherency of generators [14], iii) depending on the scale
of the event, several islands can be formed. Furthermore, this approach is fully compatible with
other techniques [1, 2, 6, 7, 8, 9].
Here we present some preliminary results on using spectral matrix methods for intentional
islanding of utility power grids, illustrated by applications to the Italian and Floridian high-
voltage grids. The methods are brieﬂy outlined in Sec. 2, numerical results are presented in
Sec. 3, and some conclusions are drawn in Sec. 4.
2. Methods
We represent a power grid by an undirected graph [10, 12], deﬁned by the N × N symmetric
weight matrix W, whose elements wi j ≥ 0 represent the capacities of the transmission lines
(edges) between the N locations (vertices) i and j. (If all wi j are either 0 or 1, W is known as the
adjacency matrix.) Examples of the graph representations of the Italian [13] and Floridian [15]
high-voltage grids at various levels of islanding are shown in Figs. 1 and 2, respectively.
The row sums of W, wi =
∑
j wi j, are the vertex strengths, and w =
∑
i wi is the total strength
of the graph. (In the unit-weight case, the wi are known as vertex degrees, and w is twice the
total number of edges.) The list of vertex strengths, {wi}, deﬁnes a diagonal matrix D. Spectral
graph analysis is usually not performed directly on W, but rather on one of several matrices de-
rived from it. The most common ones are the Laplacian matrix and the Normal matrix [10, 11].
The Laplacian is deﬁned as L = D −W and is symmetric with vanishing row sums. It embodies
Kirchhoﬀ’s laws and represents a simple resistor network with conductances wi j. Multiplied with
a column vector |φ〉 of vertex potentials, it yields the vector of currents entering the circuit at each
vertex. The eigenvalue problem, L|φ〉 = λ|φ〉, has (at least) one zero eigenvalue, whose eigenvec-
tor corresponds to equal potentials at each vertex. If the zero eigenvalue is k-fold degenerate, the
graph has k disjoint parts. The signs of the components of the eigenvector corresponding to the
smallest nonzero eigenvalue λ1 (the Fiedler vector |φ1〉 [10]) provide a partition of the network
into two almost disconnected parts (“min-cut theorem” [10, 13]).
The normal matrix N is deﬁned by its elements, ni j = wi j/wi, so that all its row-sums equal
unity (i.e., it is a row-stochasticmatrix). Left multiplication by a vector representing a probability
distribution, 〈p(t)|N = 〈p(t + 1)|, describes a discrete-time random walk along the edges of the
graph. An eigenvalue problem is now given by 〈ψ|N = 〈ψ|μ. The largest eigenvalue of N
equals unity, and the corresponding left eigenvector (properly normalized) corresponds to the
equilibrium distribution, p0i = wi/w. (If the unit eigenvalue is k-fold degenerate, the graph has
k disjoint parts.) The eigenvector corresponding to the second-largest eigenvalue represents the
most slowly relaxing perturbation away from the equilibrium distribution, and the signs of its
components identify two almost disconnected sets. While the Laplacian depends only on the oﬀ-
diagonal part of W, the Normal matrix also depends on the diagonal terms, wii, which represent
self loops in the graph that may endow vertices with internal structure.
The islanding problem is one of partitioning the power grid into communities of vertices
that are highly interconnected among themselves, but only sparsely connected to the rest or the
graph. So, ideally one would like to ﬁnd a partitioning into a “suitable” number of communities
while maximizing the number of intra-community edges and minimizing the number of inter-
community edges. This problem is NP-hard [11], and so one has to resort to heuristics producing
“reasonably good,” approximate solutions. For the islanding to be useful, each island should
contain at least one generating plant.
126 I.A. Hamad et al. / Physics Procedia 4 (2010) 125–129
I. Abou Hamad et al. / Physics Procedia 00 (2010) 1–5 3
0 5 10 15 20 25 30
Eigenvector
0
0.2
0.4
0.6
0.8
M
od
ul
ar
it
y
Soft bisection by L
Soft bisection by N
Hard bisection by N
2
2
2
2
2
2
2
2
2
2
6
6
6
2
2
2
2
6
6
6
6
8
8
6
6
6
6
6
66
66
8
6
8
8
8
6
6
8
8
6
8
8
8
8
8
7
7
3
6
6
8
6
5
7
77
7
3
7
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
5
55
5
5
1
33
1 1
3
3
3
3
3
3
1
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1 1
1 1
1 1
1 1
3
3
7
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
1 3
1 3
1 3
1 3
1 3
1 31 3
1 31 3
1 3
1 3
1 3
1 3
1 3
1 3
1 3
1 3
1 3
1 3
1 3
1 3
1 3
1 3
1 3
1 0
1 0
1 0
1 3
1 3
1 4
1 3
1 6
1 4
1 41 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
2
2
2
2
1 0
2
1 0
2
2
2
2
2
2
2
2
2
1 6
1 61 6
1 6
8
1 2
1 01 0
1 2 1 2
1 0
1 0
1 0
1 0
1 0
1 2
1 2
1 2
1 2
8
8
1 2
1 2
4
4
4
4
4
8
8
8
8
8
8
8
4
4
8
8
8
8
8
8
8
2
2
2
2
2
2
2
4
4
4
4
4
6
4
4
4
4
4
4
4
4
4
8
6
6
6
6
6
66
66
8
6
6
6
6
6
6
8
8
6
8
8
8
8
8
1
1
1
6
6
6
6
7
1
11
1
1
1
1
1
1
5
5
1
5
5
5
5
1
5
1
5
5
1
5
5
5
5
5
5
7
77
3
7
3
11
3 3
1
1
1
1
1
1
3
1
3
7
7
3
3
3
3
3
3
3
3
7
7
7
7
7
7
3
3
3
7
7
7
7
7
7
(a)
0 10 20 30 40 50 60
Eigenvector
0
0.2
0.4
0.6
0.8
M
od
ul
ar
it
y
2
2
2
2
1
1
1
1
2
1
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
12
11
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
1
1
1
2
2
2
2
2
2
22
1
1
2
2
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
2
2
2
2
2
2
2
22
2
2
2
22
2 2
2
2
1
2
2
2
2
2
2
1
1
2
1
1
1
1
1
2
1
2
1
2
2
1
2
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
1
2
2
1
1
1
1
1
1
2
1
1
2
2
1
1
1
1
1
21
21
1
2
2
2
2
1
1
1
2
1
2
2
2
2
1
1
1
1
1
1
1
1
2
1
12
1
2
2
2
1
1
1
1
1
2
2
1
2
2
2
1
2
2
2
2
2
1
1
1
1
2
22
2
1
2
22
2 2
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
2
1
1
1
1
2
1
2
2
2
2
2
1
1
2
2
1
1
2
2
2
1
2
2
2
2
1
1
1
1
2
2
1
1
2
1
2
1
22
12
1
1
2
2
2
1
2
1
2
2
2
1
2
2
2
1
1
2
2
2
1
1
1
2
22
2
1
1
1
2
2
1
1
1
2
1
1
2
2
2
2
1
1
1
1
2
2
1
1
1
2
21
2
2
2
11
1 1
2
1
1
1
1
1
2
1
2
2
2
2
1
2
2
2
2
1
2
1
2
1
1
1
2
2
1
1
2
1
1
1
1
1
(b)
Figure 1: Modularities and islanding conﬁgurations for the Italian grid, shown vs. eigenvector number, as obtained by
bisection methods (a) and from single eigenvectors (b). See details in the text.
There are several ways that the eigenvectors of L or N can be used to partition a graph in
such an approximate fashion. Two are based on successive bisections. At the ﬁrst step they are
identical: one divides the graph into two parts or “communities” according to the signs of the
components of the ﬁrst nontrivial eigenvector (i.e., the one with the smallest nonzero eigenvalue
for L, or the one with the eigenvalue closest below unity for N). In the following steps one can
either a) continue to successively bisect the network into quadrants, octants, etc., according to the
signs of the components of the next following eigenvectors (“soft bisection”), or b) remove the
edges connecting the two parts obtained in the ﬁrst step, calculate the ﬁrst nontrivial eigenvectors
of each part separately, bisect each according to the signs of their respective eigenvector com-
ponents, and then repeat this procedure with the individual parts as often as desired [12] (“hard
bisection”). A third possible method could use each eigenvector separately, labeling each vertex
+1 or −1 according to the sign of the corresponding eigenvector component, and then identify
communities with the separate “Ising clusters” generated by that particular eigenvector.
The quality of a particular partitioning of the graph into M communities, C = {C1, ...,CM}
can be quantiﬁed by Newman’s modularity [12]. It gives the diﬀerence between the proportion
of edges that are internal to a community in the particular graph, and the average of the same
proportion in a null-model that preserves the individual vertex strengths, wi (and consequently
also the total strength, w), but is otherwise randomly connected. It is deﬁned as follows:
Q =
1
w
∑
i j
(
wi j − wiwjw
)
δ (C(i),C( j)) , (1)
where δ (C(i),C( j)) = 1 if vertices i and j are in the same community, and vanishes otherwise.
3. Results
3.1. Italy
The Italian 380 kV grid [13] was modeled as an undirected graph of 127 vertices and 169
edges of unit weight. We had no information to distinguish between vertices representing gener-
ating plants, substations, etc., so they are all treated as equivalent. International connections are
I.A. Hamad et al. / Physics Procedia 4 (2010) 125–129 127
I. Abou Hamad et al. / Physics Procedia 00 (2010) 1–5 4
0 5 10 15 20 25
Eigenvector
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
M
od
ul
ar
ity
Soft bisection by L
Soft bisection by N
Hard bisection by N
3
3
3
3
3
3
1 5
1 5
7
7
1 1
1 1
9
8
7
8
8
8
1 5
7
1 1
5
1 3
5
1 5
1 5
8
7
1 5
1 5
8
1 5
1 3
5
5
5
1 3
5
5
1 5 1 5
5
7
8
8
2
5
5
5
1 5
1 5
1 5
1 3
1 3
1 3
5
5
5
1 5
1 3
1 3
1 3
1 3
2
2
1 0
2
2
2
1 3
1 4
2
2
2
1 4
1 0
1 0
1 4
1 0
1 0
1 0
1 0
1 0
2
5
5
5
5
5
5
3
3
4
4
3
3
7
2
4
2
2
2
3
4
3
7
7
8
3
4
2
4
4
4
2
3
3
8
8
8
8
8
8
4 4
8
4
2
2
6
8
8
8
4
4
4
3
8
7
8
8
8
4
8
8
8
8
6
6
6
6
6
6
8
6
6
6
6
6
6
6
6
6
6
6
6
6
6
2
2
2
2
2
2
6
6
6
6
2
2
5
5
5
5
6
6 6
8
8
4
6
6
6
6
7
6
5
6
6
6
4
4
4
4
4
4
6 6
4
6
7
7
4
4
4
6
6
6
8
8
4
4
4
6
8
8
8
8
8
8
1
1
1
1
1
1
8
1
1
1
3
3
3
3
3
3
3
3
3
3
3
1
(a)
0 2 4 6 8 10
Eigenvector
0
5
10
15
20
# 
co
m
un
iti
es
 w
ith
ou
t g
en
er
at
or
s
no self loops
selfloops and neighbors additive
equal self loops
selfloops by degree
self loops 6, neighbors 3 constant
4
4
4
4
4
4
8
8
8
8
8
4
2
7
8
7
7
7
8
8
6
2
2
2
8
8
7
7
6
8
7
6
6
2
2
2
2
2
2
8 8
2
8
7
7
3
2
2
2
6
6
6
6
2
6
2
2
2
6
6
2
2
2
3
3
1
3
3
3
2
1
3
3
3
1
3
1
1
3
3
3
3
1
3
(b)
Figure 2: Results for the Floridian grid, shown vs. eigenvector number. (a) Modularities and islanding conﬁgurations
obtained by bisection with all vertices equivalent. (b) The number of islands without generators for several schemes that
distinguish the generators by self loops. Circles: no self loops. Diamonds: 6 self loops per generator. Triangles: number
of self loops equal to generator vertex strength. Squares: self loops for generators and their nearest neighbors according
to strength. Star: 6 self loops for generators and 3 for each of their nearest neighbors. The inset represents the three-level
soft bisection with 6 self loops per generator. See details in the text.
ignored. Modularity and partitionings for the Italian grid are shown vs. eigenvector number in
Fig. 1. The results of soft bisection based on L and N and hard bisection based on N are shown
in Fig. 1(a), and partitionings based on higher-order, single eigenvectors are shown in Fig. 1(b).
A maximum modularity of Q ≈ 0.72 was obtained at the third level of hard bisection based
on N (8 islands with at least 7 vertices each). Among the bisection methods, this is followed by
Q ≈ 0.69 at the fourth level of soft bisection based on L (12 islands including one single vertex),
and Q ≈ 0.65 at the third level of soft bisection based on N (7 “islands” including two that
are internally disconnected). The modularities obtained by all the bisection methods decrease
rapidly beyond the third or fourth level of bisection.
The modularities obtained from the “Ising clusters” deﬁned by individual eigenvectors are
somewhat irregular and also decrease much more slowly with the number of the eigenvector
used, than do the results from the bisection methods. The maxima are Q ≈ 0.69 for the 10th
eigenvector (8 islands with at least 4 vertices each), and Q ≈ 0.68 for the 15th eigenvector (11
islands with at least 3 vertices each), respectively. The 60th eigenvector still yields a partition
with Q ≈ 0.45, but it produces approximately 40 islands.
3.2. Florida
The map of the Floridian high-voltage grid [15] is a composite of three networks (500, 230,
and 138 kV) with 84 vertices, 31 of which are generating plants. We have modeled it as an
undirected graph with 137 edges. The edges have integer weights between 1 and 4, according to
the actual number of direct lines between pairs of connected vertices. Interstate connections are
ignored (except for including two generating plants and four substations in southern Georgia).
Modularity and partitionings for the Floridian grid based on bisection with all vertices equiv-
alent are shown vs. eigenvector number in Fig. 2(a). Maximum modularities of Q ≈ 0.66 were
obtained at the third level of hard bisection based on N (8 islands with at least 3 vertices each, all
128 I.A. Hamad et al. / Physics Procedia 4 (2010) 125–129
I. Abou Hamad et al. / Physics Procedia 00 (2010) 1–5 5
with generating plants) and the fourth level of soft bisection based on N [11 islands, including
one single vertex (not a generating plant) and one internally disconnected “island” in which the
smaller part has no generator]. This is followed by Q ≈ 0.64 at the third level of soft bisec-
tion based on L (7 islands with at least 6 vertices each, all with generators). As for Italy, the
modularities decrease rapidly beyond the third or fourth level of bisection.
Bisections according to N oﬀer the opportunity to give extra weight to generating plants by
attaching self loops. Several schemes were tested, as described in the caption of Fig. 2(b). In
some cases this enables us to increase the highest level of bisection that ensures generators in
all islands from two to three. An example for three-level soft bisection with six self loops per
generator (7 islands with a minimum of 6 vertices, Q ≈ 0.70) is shown as an inset.
4. Conclusions
Our results indicate that spectral matrix method can be used, at least to obtain an initial par-
titioning of a power grid into islands. As bisection methods (Fig. 1(a) and Fig. 2) require evalua-
tion of only a few, dominant eigenvectors, they are computationally much more economical than
methods based on higher-order, single eigenvectors (Fig. 1(b)). With the normal matrix method,
generating plants can be weighted by the introduction of self-loops to increase the probability
that each island has at least one generator.
Acknowledgments
This work was supported in part by U.S. National Science Foundation Grant No. DMR-
0802288, U.S. Oﬃce of Naval Research Grant No. N00014-08-1-0080, and the Institute for
Energy Systems, Economics, and Sustainability at Florida State University.
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Spectral matrix methods for partitioning power grids: Applications to the Italian and Floridian high-voltage networks 

Spectral matrix methods for partitioning power grids: Applications to
  the Italian and Floridian high-voltage networks

Spectral matrix methods for partitioning power grids: Applications to the Italian and Floridian high-voltage networks

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