5/24/2018 Optimal Placement of PMUs by Integer Linear
Programming
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 3, AUGUST 2008
1525
Optimal Placement of PMUs by Integer Linear Programming
Bei Gou, Member, IEEE
AbstractThis letter presents a simple optimal placement
algorithm of phasor measurement units (PMU) by using
integerlinear programming. Cases with and without conventional
powerflow and injection measurements are considered. The
measure-ment placement problems under those cases are formulated as
aninteger linear programming which saves the CPU computationtime
greatly. Simulation results show that the proposed algorithmcan be
used in practice.
Index TermsInteger linear programming, observability anal-ysis,
phasor measurement units.
I. INTRODUCTION
P
HASOR measure units (PMU) become more and more at-
tractive to power engineers because they can provide
syn-chronized measurements of real-time phasors of voltage and
currents [1]. As the sole system monitor, state estimator
plays
an important role in the security of power system
operations.
Optimal placement of PMUs in power systems to enhance state
estimation is a problem that needs to be solved. Several
algo-rithms and approaches have been published in the
literature.
An algorithm which finds the minimal set of PMU place-
ment needed for power system state estimation has been de-
veloped in [2] and [3], where the graph theory and the simu-
lated annealing method have been used to achieve the goal.
In
[4], a strategic PMU placement algorithm is developed to im-
prove the bad data processing capability of state estimation
by
taking advantage of the PMU technology. Techniques for
iden-tifying placement sites for phasor measurement units in a
power
system based on incomplete observability are presented in
[1],
where simulated annealing method is used to solve the prag-
matic communication-constrained PMU placement problem. In
[5], a special tailored nondominated sorting genetic algorithm
is
developed for the PMU placement problem. The authors in [6]
developed an optimal placement algorithm for PMUs by using
integer programming. However, the proposed integer program-
ming becomes a nonlinear integer programming under the exis-
tence of conventional power flow or power injection measure-
ments.
In this letter, a similar formulation of optimal PMU place-ment
problem is proposed by integer linear programming. The
contribution of this letter is that the proposed formulation
is
linear with and without conventional power flow and power
in-
jection measurements. Therefore, the solution of the optimal
PMU placement problem is more efficient and can be used in
practice.
Manuscript received January 5, 2007; revised April 23, 2007.
Paper no.PESL-00108-2006.
The author is with the Energy System Research Center, University
of Texasat Arlington, Arlington, TX 76019 USA (e-mail:
[email protected]).
Color versions of one or more of the figures in this paper are
available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRS.2008.926723
II. FORMULATION OF THEPROBLEM
A. Without Conventional Measurements
A PMU, different from traditional meters, is able to measure
the voltage phasor of the installed bus and the current
phasors
of all the lines connecting to this bus. That is to say, a
PMU
can make the installed bus and its neighboring buses
observable.
The objective of placing PMUs in power systems is to decide
a
minimal set of PMUs such that the whole system is
observable.
Therefore, the placement of PMUs becomes a problem that
finds a minimal set of PMUs such that a bus must be reached
at least once by the set of the PMUs. This gives us an idea
to
define a matrix (it is matrix in [6] and see the detailsin
[6]).
Now the optimal placement of PMUs can be formulated as a
problem of integer linear programming, as follows:
where and is the PMU placement
variable.
B. With Conventional Measurements
In this letter, we only consider power flow andpower
injection
measurements, and we assume they are in pairs.
Let us define a vector . The element
of indicates the number of times for
bus reached by PMUs, where is the th row of
and is the th element of . For either a power
flow or a power injection measurement, one of the associate
buses can be solved by the measurement. In another word, for
a power flow or an injection measurement, the element
ofcorresponding to one of the associated bus of the measurement
can be zero, while the remainder has to be at least one.
For detail discussion, the following three cases need to be
analyzed.
1) If a power flow measurement is on line , then the fol-
lowing needs to be held:
which means one bus voltage can be solved from this mea-
surement and the other needs to be covered by PUM.
0885-8950/$25.00 2008 IEEE
5/24/2018 Optimal Placement of PMUs by Integer Linear
Programming
1526 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 3, AUGUST
2008
2) Suppose that an injection measurement is at bus as fol-
lows.
Then the following inequality needs to be held:
3) The powerflow measurements and the injection measure-
ments are associated.
According to the approaches introduced in 1) and 2), we have
the following two inequalities:
and
In order to satisfy , thefirst inequality needs to
be subscribed from the second inequality corresponding to
theinjection measurement , and
consider that its right-hand side needs to be reduced by one
due
to the injection , the second inequality becomes .
If bus is not associated to any conventional measurements,
then the corresponding constraint of the minimization
problem
in section is still kept .
Therefore, based on the three cases given above, if we first
order the buses without associated conventional
measurements,
then the constraint considering the conventional
measurements
becomes
where and are formed as introduced in the above
three cases, is a permutation matrix, and is the number of
buses not associated to conventional measurements.
Therefore, when considering the conventional measurements,
the optimal placement of PMUs can be formulated as a problem
of integer linear programming, as follows:
III. SIMULATION RESULTS
Due to space limitations, we only use the IEEE 14-bus system
as the example (see Fig. 1). We use the binary integer
program-
ming of Matlab to solve this problem.
A. Without Conventional Measurements
The optimal problem in Section II-A generates the results as
follows:
Fig. 1. IEEE 14-bus system.
which means we need to place PMUs at bus 2, 6, 8, and 9 such
that the whole system is observable.
B. With Conventional Measurements
We add a zero injection measurement at bus 7 and no
otherpowerflow and injection measurements. We form the
optimiza-
tion problem formulation with the constraint for this pair of
zero
injection measurement
where the four elements corresponding to buses 4, 7, 8, and
9
The solution indicates that PMUs need to be installed at
buses
2, 6, and 9, which is identical with the solution in [6].
IV. CONCLUSION
This letter proposes a simple algorithm of optimal placement
of PMUs in power systems by using integer linear program-
ming. Besides the placement of mere PMUs, this letter also
considers the placement of PMUs when conventional measure-
ments are present in the system. Simulation results show
that
the proposed algorithm is computational efficiency and can
be
used in practice.
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