1. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
GHEORGHE GRIGORAS
GHEORGHE CARTINA
MIHAI GAVRILAS
Technical University “Gh. Asachi” of Iasi,
Electrical Engineering Faculty, Iasi, Romania

2. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
1. Introduction
The paper proposes an algorithm based on the clustering techniques for
solving optimal measurement placement in power system state estimation.
The phasor measurement unit (PMU) is considered to be one of the most
important measuring devices in the future of power systems.
The distinction comes from its unique ability to provide synchronized
phasor measurements of voltages and currents from widely dispersed
locations in an electric power grid.

3. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
1. Introduction
There are two important methods for optimal placement of PMUs:
topology based methods and numerical methods.
Topology methods use the decoupled measurement model and graph
theory. In these methods decision is based on logical operations.
Numerical methods use either fully coupled or decoupled measurement
models. These methods are based on numerical factorization of the
measurement Jacobian or measurement information gain matrix.

4. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
2. Power System State Estimation
State estimator provides the optimal estimate of the system state based on
the received measurements and the knowledge of the network model.
The finding of the best locations for PMUs, in order of state estimation, can
be formulated as an optimization problem and solved using an appropriate
numerical method.
For the objective function can be used the weighted sum of the
measurement residual squares, which leads to the well known Weighted
Least Squares (WLS) state estimation solution.

5. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
2. Power System State Estimation
The WLS state estimator equations are:
(1)
where:
z - the measurement vector, dimension m;
h - the non-linear function vector, dimension m;
x - the system state vector, dimension 2N;
w – the measurement error vector, dimension m;
m - the number of measurements;
N - the number of buses.
The WLS state estimation problem is solved as an iterative calculus:
(2)
(3)

6. AN IMPROVED METHOD FOR POWER LOSSES DETERMINATION IN RURAL DISTRIBUTION NETWORKS
Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
3. Clustering techniques r s 5 4 3 1 2 r s r s
A. Single linkage clustering
B. Complete linkage clustering
D(r,s) = Min {d(i,j) : i  r and j  s}
D(r,s) = Max { d(i,j) : i  r and j  s}
C. Average linkage clustering.
D. Centroid Method
D(r,s) = Trs / ( Nr * Ns)
E. K-means clustering

7. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
4. Optimal PMU Placement Method with Clustering Techniques
The task of area identify of PMUs is formulated as a problem of nodes connectivity
identification from analysis power system.
A hierarchical clustering method is used, which well overcomes problems
concerning formation of coherent and representative groups (named below zones).
Step 1. Units (that represent the links of every node with other) from binary connectivity matrix A compare, and to gather them progressively in coherent groups (zones) in a way that the nodes in the same group to belong same zone.
The elements of the binary connectivity matrix A as defined below:

8. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
4. Optimal PMU Placement Method with Clustering Techniques
After that, they are linked together in to the new group (zone) Z. The newly formed
zones are grouped into larger zones until a hierarchical tree is formed. The process
is repeated until there is only one zone if all nodes meet the required criterion.
Step 2. The hierarchical tree is divided into coherent zones by cutting off the
hierarchical tree at an arbitrary point, α. The threshold of inconsistency
coefficient strongly influences the final number of zones. Thus, it is defined so
that every zone to contain a nodes number smaller than maximum links number of
a node from system.
Step 3. The PMUs placement in nodes of system. The PMUs number which can be
installed in system is equal with number of obtained zones. These PMUs will be
installed in node that have the more links with other nodes from respective zone.
If the links number among nodes is equal, then the PMU is placement in the first
node that it belong zone from dendogram of clustering process.

9. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
5. Study Case 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Fig. 1. IEEE 14-bus system

10. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
5. Study Case
Fig. 2. Dendogram of clustering process

11. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
5. Study Case
Table 1 – Results of IEEE 14 – bus system 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Zone 2
Zone 1
Zone 4
Zone 3
Fig. 3.The zones of IEEE 14-bus system
and optimal placement of PMUs
Table 2. Synthesis of results for different size systems
Table 3 – Optimum number and location of PMUs using
Branch and Bound approach

12. Using of Clustering Techniques in Optimal Placement of Phasor Measurements Units
6. Conclusions
In this paper a new approach, based on the clustering techniques, is
proposed for determination of the minimum number of PMUs in order to
make the system fully observable
The results obtained using different systems demonstrate that the
methodology can be used with the success in solving optimal measurement
placement in power system state estimation.