1

Abstract—In this paper graph theory is used to identify

coherent groups of generators and to locate PMUs for Inter-Area

monitoring. Initially, the problem of identifying coherent groups

is presented as a graphical problem. Then, three graph clustering

methods are implemented to group coherent generators. Finally,

when the electric network is separated in coherent regions, a

placement of PMU based on centrality criteria is proposed. This

can be used as a first stage in the implementation of a plan of

monitoring in a large electric power system. Dynamic simulations

and phasorial representation of simulations are done with the

reduced order equivalent of the interconnected New England

system (NETS) and New York power system (NYPS). The results

show that graph theory can be applied to identify coherent

groups and to locate PMUs to Inter-area monitoring.

Index Terms-- Coherency recognition, graph partitioning,

graph theory, inter-area monitoring, observability, Phasor

Measurement Unit (PMU), PMU placement.

I. INTRODUCTION

NTER-AREA mode oscillations consist of a swinging

group of generators against another group of generators.

The characteristics of these oscillations are complex and differ

significantly from those of local mode oscillations [1]. Even in

a system, where the voltage phasor may be available at each

node in real-time, monitoring the dynamic state is only the

first step for understanding dynamic stability properties of the

complex power system. For instance, in the context of angle

stability, we need to know whether or not a generator or group

of generators can potentially become unstable based on

interactions with other components in the system [2]. Also,

dividing system in coherent areas, have several advantages

[3]: 1) improvement of security margins with coherency-based

controllers; 2) simplification of control methodologies when

decentralized procedures are implemented; 3) greater

effectiveness of corrective control actions based on dynamic

islanding procedures; 4) reduction of required measurements

when adopting phasor measurement units (PMUs) for each

coherent area. Consequently, it is not necessary to place a

PMU at all nodes belonging to a coherent group.

Oscar Gómez is with School of Engineering, Universidad de Los Andes,

Bogotá, Colombia and School of Electrical Technology, Universidad

Tecnológica de Pereira, Pereira, Colombia (e-mails:

o.gomez31@uniandes.edu.co, jr@utp.edu.co).

Mario A. Rios is with School of Engineering, Universidad de Los Andes,

Bogotá, Colombia (e-mail: mrios@uniandes.edu.co).

Several clustering algorithms have been used to identify

coherent groups of generators in power systems. In [4], a

method that consists of a time-domain based coherency

measure and a fuzzy clustering algorithm is presented. Since

the system dynamic behaviors are characterized by time-

domain responses, a coherency measure, which is derived

from the after-disturbance swing curves of generators, was

proposed to evaluate the coherency behaviors among

generators. The method is based on fuzzy c-means (FCM)

cluster analysis, using the coherency measures as a basis for

classification, and aims to group generators at different

prescribed number of coherent groups. As seen, a prescribed

number of coherent groups is necessary. In [5], the angular

speed deviations of synchronous generators have been used as

a criterion to perform the groups; a coherency quality index

and a tolerance to identify coherent groups is defined to reach

this purpose. [6] presents a technique to identify the coherent

groups of generators using fuzzy c-means clustering

algorithm. A recent time domain coherency identification

method introduced in the literature is the Principal Component

Analysis (PCA). The method transforms the system response,

e.g. generator angle variation, following a disturbance to

uncorrelated variables. In [7], PCA is supplemented by a

hierarchical clustering technique to accurately identify

coherent groups of generators in the system. Spatial distances

in a multi-dimensional space between transformed data are

used to construct a proximity measure describing the

coherency relationship between all generators.

Many real-world situations can be described by means of a

diagram consisting of a set of point together with lines joining

these points. Electrical power systems are an example of these

situations. These are composed of nodes that are linked by

mean of lines and can be represented graphically. This

representation allows the decomposition of the network into

several areas which main characteristic is that the elements in

the same area are strongly coupled, whereas elements in

different areas are weakly coupled. To the best of the authors'

knowledge there are few papers that use graph theory to

identify coherent generators. In [8], graph theory is used to

determine the boundary of each island when identification of

generator groups based on the slow coherency criterion has

been done. The proposed graph theoretic approach includes

two parts: a graph simplification method based on the

characteristic of the graph formed from a power system to

Identification of Coherent Groups and PMU

placement for Inter-Area monitoring Based on

Graph Theory Mario A. Rios, Member, IEEE, and Oscar Gómez

I

2

reduce the computational burden and a multi-level graph

partitioning method to solve the graph partitioning problem.

[9] presents the identification of the dynamic equivalent of a

power system from on-line measurements by using knowledge

of coherent generators to determine the model structure. By

using the interpretation of coherency from the graph model of

a power system, the coherent generators can be identified

without the necessity of dynamic parameters.

This paper shows how to model the electrical power system

with an equivalent graph for coherence identification. Then,

techniques from graph theory are applied to identify coherent

generators without carry out a transient stability study. They

are named “inherent coherent groups”. Finally, when coherent

groups have been identified, graph theory is again used to

locate a PMU in each area for supervising inter-area stability

problems. A case study involving the application of the

algorithms is carried out. The results from the case study are

discussed and future work is provided

II. GRAPH MODELING

Converting the generation system into an equivalent graph

with vertices and edges could give a clear view of the mutual

influence between a network structure and a dynamic

behavior, especially the coherency phenomenon. This

representation could indicate how many equivalent generators

should there be and how they are interconnected.

The graph model built here is based on methodology

presented in [10] where coherency is determined by using the

synchronizing power between a generator i and a node k given

by: ' cos( )ik i k ik ikH E V B (1)

Where E’ is the generator transient emf (electromotive

force), Vk is the voltage at the node of generator k, Bik is the

imaginary part of the element i, j in the transfer admittance

matrix, δik is the angular difference between nodes i and k.

The transfer admittance matrix corresponds to a reduced

equivalent network where the load are eliminated retaining

only the generator nodes and equivalent branches linking

them.

A disturbance in node k which changes the voltage angle in

a generator with inertia coefficient Mi from the initial value δk

to a value δk + Δδk, causes the rotor acceleration,

iki k

i

H

M (2)

According to [10], generators i and j are

electromechanically exactly coherent generators if their rotor

accelerations γi and γj caused by the disturbance are the same,

i.e.

jkik

i j

HH

M M (3)

As it can be seen coherency depends not only of the

electrical distances between generators but also on generator

inertia constants. This definition permits to develop a

methodology for identifying coherent groups based on an

equivalent graph where edges represent how much can a

generator be affected when other generator suffer a

disturbance.

Therefore a complete graph is achieved where each

generator is connected to the others generators. To identify

coherency, it is necessary to find strongly connected groups of

generators because they have an inherent tendency to maintain

synchronism. Spectral clustering algorithms, k-means

algorithms, segmentation-based categorization, flow-based

graph partition, and others methodologies can be used to

achieve this purpose [11].

These methods to identify groups of coherent generators

divide the generation nodes in several disjoint sets. The

weight of the edges that stay within the same set should be as

high as possible and the weight of the edges that connect any

two set should be as low as possible. This problem is known

like multi-way graph partitioning problem.

The graph methodologies applied in the identifying

coherent generators, search generators that have similar

connections to the others generators and have a strong

connection between them. Let us consider the test system (Fig.

1) to expose the methodology that will be exposed later.

G07

05G04

G05

20

G06

2221

65

62

63

G03

03

64

58

G02

24G09

29

0928

26

G08 G01

G13

13

43

17

G12

12

36

61

30

53

47

4840

44

45

39

35

34

33

32

G11

1131

38

51

50

G10

10

46

49

G16

16

18

G15

15

42

G14

14

41

07

23

04

0668

19 66

67

57

56

52

55

54

27

37

25

08 01

0259

60

Fig. 1. Case Test

After load nodes have been eliminated, an equivalent graph

with only generators nodes and its equivalent branches are

obtained (Fig. 2). Only connections from generator G08 are

shown.

G07

07

05G04

04

G05

G06

06

G03

03G02

02

G09

09

G08

08

G01

01

G13

13

G12

12

G11

11

G10

10

G16

16

G15

15

G14

14

Fig. 2. Equivalent graph with only generator nodes (Connections from G08)

3

The graph G has a set of elements V called vertexes and a

set of elements E called edges. Each vertex represents a

generator of the system and they do not have weight. Each

edge represents how the connection is between two

generators. Particularly, it is a complete graph because every

pair of distinct vertexes is adjacent. According to (3) the

connection from generator i to generator k is ik

i

H

M and

connection from generator k to generator i is ki

k

H

M, so to

obtain a undirected graph, the edge between two generators is

defined by:

min ,ik ki

i k

H H

M M

(4)

III. COHERENCY IN TERM OF THE GRAPH MODEL

Two generators are said to be coherent if they tend to

swing together. Coherency among a group of generators

implies coherency between each pair of generators in the

group. Further, coherency of generators is not dependent on

the magnitude of disturbance. In general, the process of

partitioning a given graph consists of finding a group of edges

which when removed will split the original graph into desired

numbers of connected sub-graphs.

The groups of coherent generators can be identified by

analyzing for the particular structure of the graph, in which

the groups of coherent generators have the strong connections

between generators within their group and have the weak

connections among generators of other groups. Some

algorithms for graph partitioning are used as follow.

A. Recursive Spectral Bisection

The spectrum of a graph is the set of eigenvalues associated

with the matrices that represent the graph. A graph with n

vertices has n eigenvalues. Specifically, the spectra from the

different matrices contain information about the graph

connectivity, sizing, and degree of the nodes [12],[13].

The procedure to split a graph is carried out by the matrix

spectral analysis of the Laplacian matrix which provides

structural information about the graph. Initially the adjacency

matrix A is calculated and the degree matrix D to obtain the

Laplacian matrix; thus:

min , , ,( , )

0,

ik ki

i k

H Hif i k E

A i k M M

otherwise

(5)

deg 1, ,deg 2G GD diag (6)

Where degG N is the number of edges incident to the vertex N.

The Laplacian matrix of a graph G is defined as:

L D A (7)

Although, it is also possible to use the normalized

Laplacian matrix defined as:

1 1

2 2L I D A D

(8)

The eigenvalues of L will always be enumerated in the

increasing order λ1 ≤ λ2 ≤, … , ≤ λn repeated according to their

multiplicity.

The properties of the second eigenvalue and the

corresponding eigenvector ν2 of the Laplacian matrix reveal

connectivity properties of a graph. The eigen pair (,ν2) was

called the Algebraic Connectivity and the Characteristic

Valuation of G [14], [15]. The components of the second

eigenvector are assigned to the vertices of G; so that each

node of the graph is associated to a value of the second

eigenvector of the Laplacian matrix (that is, the node k is

associated to the kth component of the second eigenvector).

Positive and negative values of the second eigenvector

components, which are associated with each node, give the

required partition of the graph in two sets. This method is

applied recursively to get more cut-sets. In each step, it is

necessary to calculate the new adjacency matrix

corresponding to each sub-graph obtained (Fig. 3.).

Calculate Adjency

and Degree Matrix

Calculate

Laplacian

Matrix

Calculate

Second

Eigenvector

Positive

eigen-

values?

Subgraph

2

NoSubgraph

1

New

partition?

End End

Graph

New

partition?

Yes

No No

Yes

Yes

Fig. 3. Flow chart of recursive spectral bisection

B. Maximum Spanning Tree Clustering

Given a connected, undirected graph G=(V, E), where V is

the set of nodes, E is the set of edges between pairs of nodes,

and a weight w(u,v) specifying weight of the edge (u,v) for

each edge (u, v)∈ E. A spanning tree is an acyclic subgraph of

a graph G, which contains all vertices from G. The Maximum

Spanning Tree (MST) of a weighted graph is maximum

weight spanning tree of that graph.

Once the MST is built for a given graph, there are two

different ways to produce groups of clusters (Fig. 4). If the

number of clusters k is given in advance, the simplest way to

obtain k clusters is to sort the edges of maximum spanning

tree in ascending order of their weights, and remove edges

with first k-1 weights [16]. If the number of clusters k is

unknown, it is possible to eliminate edges whose weights are

significantly smaller than the average weight of the edges in

the tree [17].

4

Build Maximum

Spanning tree

No

Graph

Yes Predefined

clusters?Define k

Calculate

average

weight

Sort edges iIn

ascending order

Remove edges with

first k-1 weight

Eliminate edges whose

weights are smaller

than the average weight

End

End

Fig. 4. Flow chart of recursive spectral bisection

Prim's algorithm can be used to find the maximum

spanning tree for the connected weighted undirected graph. It

finds a subset of the edges that forms a tree that includes every

vertex, where the total weight of all the edges in the tree is

maximized.

C. Minimum Cut Tree Clustering

Methodology is based on maximum flow techniques, in

particular minimum cut trees. The main idea behind maximum

flow (or equivalently, minimum cut) clustering techniques, is

to create clusters that have small inter-cluster cuts and

relatively large intra-cluster cuts [19].

Every graph has a weighted graph, which is named the

minimum cut tree (or simply, min-cut tree). The min-cut tree

has the property that it is possible to find the minimum cut

between two nodes by inspecting the path that connects the

nodes. The edge of minimum capacity on that path

corresponds to the minimum cut. The capacity of the edge is

equal to the minimum cut value, and its removal yields two

sets of nodes.

The clustering algorithm is based on inserting an artificial

sink into a graph that gets connected to all nodes in the

network through an undirected edge of capacity α. Maximum

flows are then calculated between all nodes of the network

and the artificial sink [19].

Insert an artificial

sink

Graph

End

Find the min-cut tree

of expanded graph

Remove the artificial sink

from the min-cut tree

Connected vertexes are

the cluster of the graph

Fig. 5. Flow chart of Minimum Cut Tree Clustering

Methodology finds the min-cut tree of expanded graph,

next it removes the artificial sink from the min-cut tree, and

the resulting connected vertexes form the cluster of the graph

(Fig. 5).

As α goes to 0, the min-cut between the sink and any other

node will be the cut, which isolates the sink from the rest of

the nodes. So, the cut-clustering algorithm will produce only

one cluster, namely the entire graph. On the other extreme, as

α goes to infinity it will cause that the graph to become a star

with the artificial sink at its center. Thus, the clustering

algorithm will produce n clusters, all singletons. For values of

α between these two extremes the number of clusters will be

between 1 and n, but the exact value depends on the structure

of the graph and the distribution of the weights over the edges.

IV. PMU PLACEMENT FOR INTER-AREA OBSERVABILITY

Once the sets or groups of coherent generators have been

defined, it is obtained the real network that connects generator

nodes in each group. This subnetwork is built considering

only the imaginary part of the elements in the admittance

matrix. Therefore a small graph is obtained and admittance

matrix establishes the edge weights.

Then the single-source shortest-path problem is solved to

get the eccentricity and subsequently the central vertex in the

subnetwork of each coherent group. Dijkstra’s algorithm

permits calculate the distance between two vertices in a graph.

Distance is defined as the sum of the weights of the numbers

of edges in a shortest path connecting two vertices. This is

known as the geodesic distance [18]. This distance must be

calculated from each node toward the other nodes. The

eccentricity of a node is the greatest geodesic distance

between the node and any other node. Finally, the central

vertex is one that has minimum eccentricity. The central

vertex is the node where PMU will be located to monitor the

coherent group (Fig. 6).

For group i, find

subnetwork that

link generators

No

N= Number of

coherent groups

i=1

Yes

i <= N ?

Coherent

groups are

observed

End

For each node in the

subnetwork, calculate

eccentricity

Node with mimimun

eccentricity will be

the central vertex

Place a PMU at

central vertex

i=i+1

Fig. 6. Flow chart of PMU placement algorithm

5

V. STUDY CASE

The methodologies has been tested on the 16-machine, 68

bus test system (Fig. 7). This is a reduced order equivalent of

the interconnected New England test system (NETS) and New

York power system (NYPS). There are five geographical

regions out of which NETS and NYPS are represented by a

group of generators whereas, import from each of the three

other neighboring areas #3, #4 and #5 are approximated by

equivalent generator models [21].

G07

07

23 05G04

04

G05

19

20

G06

06

22

68

21

65

62

63

G03

03

64

66

67

58

G02

02

60

59

57

56

52

37

27

24G09

29

09

28

2625

G08

08

54

G01

01

55

G13

13

43

17

G12

12

36

61

30

53

47

4840

44

45

39

35

34

33

32

G11

11

31

38

51

50

G10

10

46

49

G16

16

18

G15

15

42

G14

14

41

NETS NYPSAREA#5

AR

EA

#4

AREA# 3

Fig. 7. Test system

A. Recursive Spectral Bisection

Application of recursive spectral bisection to the test

system produced the results shown in Table I.

TABLE I

RECURSIVE SPECTRAL BISECTION APPLIED TO THE TEST SYSTEM Nodes

Initial Graph 1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16

Partition 1 1-2-3-4-5-6-7-8-9-10-11-12-13 14-15-16

Partition 2 1-2-3-4-5-6-7-8-9 10-11-12-13

Partition 3 1-2-3-8-9 4-5-6-7 10-11 12-13

Partition 4 2-3 1-8-9 4-5 6-7

Subgraphs 2-3 1-8-9 4-5 6-7 10-11 12-13 14-15-16

As can be seen, methodology requires a decision about if

the current partition should be subdivided. This is a subjective

decision and can lead to different clustering results.

B. Maximum Spanning Tree

Maximum spanning tree is built by using Prim’s algorithm.

Tree can be seen in Fig. 8 and edges organized in ascendant

order are shown in Table II.

G07

07 05G04

04

G05

G06

06

G03

03G02

02

G09

09

G08

08

G01

01

G13

13

G12

12

G11

11

G10

10

G16

16

G15

15

G14

14

Fig. 8. Maximum spanning tree of test system

TABLE II

EDGES ORGANIZED IN ASCENDANT ORDER

Edge Weight

From To

13 16 0,0093

16 15 0,0242

15 14 0,0256

1 10 0,0294

3 1 0,0349

3 6 0,0388

11 12 0,0559

8 9 0,0681

6 4 0,0876

10 11 0,0998

12 13 0,1195

1 8 0,1458

3 2 0,1539

6 7 0,2590

4 5 0,3051

1) Considering a number of predetermined clusters.

If seven (k) groups are defined, the first six (k-1) edges

must be eliminated. Fig. 9 shows clusters obtained when

edges are removed.

G07

07 05G04

04

G05

G06

06

G03

03G02

02

G09

09

G08

08

G01

01

G13

13

G12

12

G11

11

G10

10

G16

16

G15

15

G14

14

Fig. 9. Clusters obtained

2) Considering an unknown number of clusters.

Initially the mean weight is calculated and then edges

whose weight is smaller than mean weight are eliminated. Fig.

10 shows clusters obtained when edges are removed.

G07

07 05G04

04

G05

G06

06

G03

03G02

02

G09

09

G08

08

G01

01

G13

13

G12

12

G11

11

G10

10

G16

16

G15

15

G14

14

Fig. 10. Clusters obtained

6

C. Minimum Cut Tree Clustering

To build the min-cut tree an artificial node connected to all

nodes in the network through an edge of weight 0.13 was

used. Min-cut tree (Fig. 11) was obtained by using Gomory

and Hu algorithm [20].

G07

07 05G04

04

G05

G03

03G02

02

G09

09

G08

08

G01

01

G13

13

12

G11

11

G10

10

G16

16

G15

15

G14

14

17

G12

G06

Fig. 11. Min-cut tree with the artificial node

When artificial node is retired from the graph, connected

nodes form the clusters of graph (Fig. 12).

G07

07 05G04

04

G05

06

G03

03G02

02

G09

09

G08

08

G01

01

G13

13

12

G11

11

G10

10

G16

16

G15

15

G14

14

G12

G06

Fig. 12. Clusters obtained

D. PMU Placement for Inter-Area Observability

If generators G01, G08, and G09 from Fig. 7 are a coherent

group, the network that connects generators is as Fig. 13

shows.

G09

29

09

26 25

G08

08

54

G01

01

PMU

Fig. 13. Subnetwork for generators G01, G08, and G09 and installed PMU

For the subnetwork shown in Fig. 13, node 25 is the central

vertex, so a PMU will be placed there to monitor the coherent

group.

In Fig. 14, it is possible to visualize phasors of coherent

group and monitored phasor for an operating point. Fig 14

shows graphically 7 phasors which it is possible to see 6; one

of them is under the bold line which represents the PMU

placed at node 25 to monitor the coherent group. Three

phasors are above bold line and others three phasors are below

the bold line. The nodes are respectively labeled in the figure.

Fig. 14. Phasor measurements of the coherent group

It should be noted that the central vertex voltage is located

between coherent group voltages; hence, voltage group can be

monitored by checking the central vertex voltage which

represent the group behavior.

E. Dynamic simulation

By using methodology presented in section IV, PMUs

were located at nodes 25, 22, 19, 63, 17, 30 and 41 to

monitor groups defined at Table I. A three-phase-to-ground

fault was created at 0.16 seconds into the simulation at bus

22 and the fault is cleared after 100 ms, followed by

opening line connecting buses 21 and 22.

Fig. 15 shows the dynamic behavior of the groups by

mean of phasors graphed in t=0 s, t=0.99 s and t=1.5 s.

Fig. 15. Phasor measurements of the coherent groups

The movement of the phasor located at node 22 (marked

with an asterisk) illustrates the angular separation of area of

generators 6 and 7 from rest of areas and the decrease in

voltage magnitude due to fault.

7

VI. CONCLUSIONS

In this paper graph theory was used to identify coherent

groups of generators and to locate PMUs for Inter-Area

monitoring. The problem of identifying coherent groups

was presented as a graphical problem and three graph

clustering methods were implemented to group coherent

generators. A placement of PMU based on centrality criteria

was proposed and dynamic simulations with a phasorial

representation of simulations were done in a test system.

The results showed that graph theory can be applied to

identify coherent groups without perform transient stability

studies which require large amounts of computational effort

in terms of time and memory. Although, some of them

require of subjective decisions to coherency identification,

they provide flexibility to choose the number of coherent

groups according to different needs and applications.

Recursive spectral bisection identifies groups with

similar characteristics according its connections. It uses

information of the entire graph and detects generators

strongly connected. However it require of external

decisions to define the number of groups.

Maximum spanning tree identifies generators that can be

isolated of the system, i.e. generators that represent external

equivalent systems. However it is not appropriated to find

groups with similar characteristics.

Minimum cut tree clustering, based on expanded graphs,

provide a means for finding good coherent groups. By

using the parameter α, it is possible to control the amount of

groups that can be built without to define exactly the groups

desired. It means the groups are a natural result of the

algorithm.

Concepts of graph theory were applied to get the

eccentricity and subsequently the central vertex of each

coherent group. It permit locate PMUs in each area to Inter-

area monitoring without calculate centers of inertia which

usually obtain an equivalent node that physically do not

exist in the system.

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VIII. BIOGRAPHIES

Oscar Gómez was born in Pereira, Colombia, on

April 4, 1979. He obtained his BScEE and MScEE

from Universidad Tecnológica de Pereira, Pereira, in

2003 and 2005, respectively. Currently, he is

Assistant Professor at the Department of Electrical

Technology at Universidad Tecnológica de Pereira

and he is pursuing the Ph.D. degree at the

Universidad de Los Andes, Bogotá. (e-mail:

o.gomez31@uniandes.edu.co).

Mario A. Ríos (M’89) received a degree in

electrical engineering in 1991 and an M.Sc. degree

in electrical engineering in 1992, both from

Universidad de los Andes, Bogotá, Colombia. He

received a Ph.D. degree in electrical engineering

from INPG-LEG, France, in 1998, and a Doctoral

degree in engineering from Universidad de los

Andes, in 1998. Currently, he is Associate Professor

at the Department of Electrical Engineering at

School of Engineering, Universidad de Los Andes,

Bogotá. (e-mail: mrios@uniandes.edu.co).