18-09-2012, 11:07 AM
Phasor Measurement Unit Placement Techniques for Complete and Incomplete Observability
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Abstract
This paper presents techniques for identifying placement
sites for phasor measurement units (PMUs) in a power
system based on incomplete observability. The novel concept
of depth of unobservability is introduced and its impact on the
number of PMU placements is explained. Initially, we make use
of spanning trees of the power system graph and a tree search
technique to find the optimal location of PMUs. We then extend
the modeling to recognize limitations in the availability of communication
facilities around the network and pose the constrained
placement problem within the framework of Simulated Annealing
(SA). The SA formulation was further extended to solve the
pragmatic phased installation of PMUs. The performance of these
methods is tested on two electric utility systems and IEEE test
systems. Results show that these techniques provide utilities with
systematic approaches for incrementally placing PMUs thereby
cushioning their cost impact.
INTRODUCTION
PHASOR measurement units (PMUs) are power system
devices that provide synchronized measurements of
real-time phasors of voltages and currents. Synchronization
is achieved by same-time sampling of voltage and current
waveforms using timing signals from the Gobal Positioning
System Sattelite (GPS). Synchronized phasor measurements
elevate the standards of power system monitoring, control,
and protection to a new level [1]. The present and possible
future applications of phasor measurement units have been well
documented [2]. A number of PMUs are already installed in
several utilities around the world for various applications such
as post-mortem analysis, adaptive protection, system protection
schemes, and state estimation. One of the most important issues
that need to be addressed in the emerging technology of PMUs
is site selection. The intended system application influences
the required number of installations. The cost of PMUs limits
the number that will be installed although an increased demand
in the future is expected to bring down the cost.
INCOMPLETE OBSERVABILITY
Incomplete observability refers to the PMU placement scenario
when the number and location of the PMUs are not sufficient
to determine the complete set of bus voltages of a power
system (the state of the power network). Fig. 1 shows an incompletely
observed system. The voltage at buses B and F are directly
measured by PMU-1 and PMU-2 respectively, while voltages
at buses A, C, E, and G can be calculated using the measured
voltages and line currents. This leaves bus D as unobservable.
We define buses B and F as PMU buses where bus voltages
and line currents are directly measured. We define buses A, C,
E, and G as calculated buses because their voltages are calculated
from the PMU measurements of the buses linked to them.
Theoretical Formulation
To automate the whole procedure, we now proceed to develop
the theoretical foundation of this graph theoretic algorithm. Define
a spanning tree of the power system graph
with number of buses and number of lines. Let
the PMU placement set be whose elements ,
are the PMU buses, refers to the size of the placement set and
the instance when the placement set is incremented by a new
PMU. It is essential to keep track of the set of buses that have
been part of the set queried for possible placement. Let be the
instance when a node is visited. Define a vector whose elements
at any th instance are the set of -buses that have been
visited in the tree search. They shall be termed tagged buses.
Numerical Results of Tree Search Technique
The algorithm was tested on 5 systems for depths of one
to three unobservability placements. The required number of
PMUs in each system is compared with numbers associated for
complete observability. The latter objective is achieved by performing
a depth of zero placements. The results presented in
Table I confirm the reduction in the number of PMUs using
placement based on incomplete observability. Note that the set
of PMUs in lower depths of unobservability is not a superset of
the set of PMUs at the next higher depth of unobservability. The
PMUs at a given depth of unobservability placement strategy are
free to move to another buses in the next lower depth of unobservability
placement strategy. Fixing the location of PMUs at
any time constitutes phased installation, which will be modeled
in Section VI.
COMMUNICATION CONSTRAINED PLACEMENT RESULTS
A 444-bus, 574-line power system with only 37% (165) of
its buses with communication facilities is used to test the SA
formulated communication-constrained placement problem.
Because of space restrictions, the single line diagram is not
shown. The distribution of regions without communication
facilities in Fig. 10 gives the reader an indication of the relative
number and placement of communication facilities. Although
quite a large number of buses without communication facilities
are sandwiched between those with communication, a significant
number of regions containing 5 to 14 (contiguous) buses
without communication also exist.
The tree search technique was used to jumpstart SA. Table II
highlights the initial placement strategies from depth-of-one to
depth-of-five generated by the tree search technique. Note the
significant number of misplaced PMUs in each case; misplaced
here shall mean PMU placement on buses without communication
facilities.
CONCLUSION
A graph theoretic approach for placing PMUs based on incomplete
observability is presented. It makes use of spanning
trees of the power system graph to find the optimal locations
of PMUs based on a desired depth of unobservability. The concept
of depth of unobservability and how it affects the number of
PMUs is presented. Results have shown that this method guarantees
a dispersed placement of PMUs around the system and ensures
that the distance between the unobserved buses and those
observed is not too great.We have utilized Simulated Annealing
to solve the pragmatic communication-constrained PMU placement
problem. Slight modifications also resulted in modeling
the PMU phased installation problem as well as a technique
of locating new sites for communication facilities to make the
system observable.