IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 3, AUGUST 2008
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Generalized Integer Linear Programming
Formulation for Optimal PMU Placement
Bei Gou, Member, IEEE
Abstract—Based on the integer linear programming formulation
proposed for optimal PMU placement, this paper presents a generalized integer linear programming formulation for cases including
redundant PMU placement, full observability and incomplete observability. Due to accurate voltage phasor measurement and current phasor measurements provided by PUM units, the accuracy,
redundancy and thus the robustness of state estimation will be enhanced with the integration of PMU measurements. The problem
of optimal placement of PMU for the redundant PMU placement,
full observability and incomplete observability analysis needs to
be studied, for various purposes and considerations. The proposed
modeling approach by the author in another paper, which models
PMU placement as an integer linear programming problem, is extended and generalized to satisfy different needs. Cases with and
without zero injection measurements are considered. Simulation
results on different power systems show that the proposed algorithm can be used in practice.
Index Terms—Integer linear programming, observability analysis, phasor measurement units.
I. INTRODUCTION
P
HASOR measurement units (PMUs) become more and
more attractive to power engineers because they can
provide synchronized 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 needed to be solved.
Several algorithms and approaches have been published in the
literature.
An algorithm which finds the minimal set of PMU placement needed for power system state estimation has been developed in [2] and [3] where the graph theory and the simulated
annealing method have been used to achieve the goal. In [4]
a strategic PMU placement algorithm is developed to improve
the bad data processing capability of state estimation by taking
advantage of the PMU technology. Techniques for identifying
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 pragmatic communication–constrained PMU placement problem. In [5] a special tailored nondominated sorting genetic algorithm is developed for the PMU placement problem. The authors in [5] developed an optimal placement algorithm for PMUs by using integer
programming. However, the proposed integer programming becomes a nonlinear integer programming under the existence of
conventional power flow or power injection measurements.
In this paper optimal PMU placement problem is re-studied
for the cases of redundant PMU placement, full observability
and incomplete observability. Based on the author’s newly proposed integer linear programming algorithm in [7], the PMU
placement problem is re-studied and a generalized integer linear
programming formulation is presented in this paper. Numerical
tests show that this formulation is efficient and able to obtain the
identical results as those in previous publications.
II. PRELIMINARY RESULTS OF FULL OBSERVABILITY
Unlike traditional measurement units, the PMU is able to
measure the voltage phasor of the installed bus and the current
phasors of all the lines connecting to the bus. That is, a PMU can
make the installed bus and its neighboring buses observable. The
objective of placing PMUs in power systems is to determine a
minimal set of PMUs such that the whole system is observable.
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.
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
.
are defined as follows:
The entries in
if
if and are connected
otherwise.
The optimal placement of PMUs is formulated as follows [7]:
where
is the PMU placement variable and
Manuscript received October 30, 2007; revised March 17, 2008. Paper no.
TPWRS-00794-2007.
The author is with the Energy System Research Center, University of Texas
at Arlington, Arlington, TX 76019 USA (e-mail: bgou@uta.edu).
Digital Object Identifier 10.1109/TPWRS.2008.926475
0885-8950/$25.00 © 2008 IEEE
..
.
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B. With Conventional Measurements
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 detail discussion, the following three cases need to be
analyzed.
1) If a power flow measurement is on line , then the following needs to be held:
which means one bus voltage can be solved from this measurement and the other needs to be covered by PMU.
2) Let us assume that bus connects buses , and .
Suppose that an injection measurement is at bus , then
the following inequality needs to be held:
3) The above power flow measurements and the injection measurements are associated. According to the approaches introduced in 1) and 2), we have the following
two inequalities:
Fig. 1. Example for illustration.
C. Example I
In order to clearly explain the above formulation of integer
linear programming for the case of full observability, we use
the example in [6] to illustrate it.
Suppose that a branch flow measurement be on line 2–3 and
an injection measurement at bus 2. Bus 1, 2, 3, 6 and 7 are associated to these two conventional measurements. According to
the definition given above, we have
and
In order to satisfy
, the first inequality needs to
be subscribed from the second inequality corresponding to the
injection measurement
and consider that its right hand side needs to be reduced
by one due to the injection , the second inequality be.
comes
Therefore, the inequality for this case is
If bus is not associated to any conventional measurements,
then the corresponding constraint of the minimization problem
.
in Section II-A is still kept
When considering the conventional measurements, the optimal placement of PMUs can be formulated as a problem of
integer linear programming as follows:
Buses 4 and 5 are not associated to these two conventional
measurements. The two inequality constraints corresponding to
these two conventional measurements are
Therefore, we have
Branch Measurement
Injection Measurement
Then matrix
can be written as follows:
The permutation matrix
where the matrix
; the matrix
is
a permutation matrix and is the number of buses not associated to conventional measurements; the column vector
is
defined in [7]. The details of forming these matrices are given
in the following examples.
is
GOU: GENERALIZED INTEGER LINEAR PROGRAMMING FORMULATION FOR OPTIMAL PMU PLACEMENT
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Therefore, the integer linear programming for this PMU optimal placement can be written as
Fig. 2. Simple example.
two terminals of any branch, is larger than 1. Therefore, the
depth-of-one unobservability PMU placement can be formulated as the following integer linear programming problem:
The optimal solution of this integer linear programming is
, which means the PMU needs
to be placed as buses 2 and 5. This example shows that the conventional measurements do not affect the decision of optimal
placement because of the system configuration and the locations
of conventional measurements.
III. PMU PLACEMENTS FOR INCOMPLETE
OBSERVABILITY ANALYSIS
A novel concept, incomplete observability, has been introduced in [1], where incomplete observability refers to the
network condition in which the number and location of the
PMUs are insufficient to determine the complete set of bus voltages (the state). Also, the depth of unobservability is a measure
of the distance of an unobserved bus from its observed neighbors [1]. Based on the depth of unobservability, the concepts
depth-of-one unobservability, depth-of-two unobservability,
and so on are also defined in [1]. The details of the definitions
can be found in [1].
Based on the idea proposed in [7], formulation of the
incomplete observability PMU placement problem can be
studied using the similar modeling approach and solution
method, without and with the considerations of zero injections.
In this paper we will only study the cases of depth-of-one
unobservability and depth-of-two unobservability with the
understanding that depth of higher degrees can be dealt with in
the same manner.
where
,
is the total number of
is the PMU placement variable, and
is the
branches,
branch-to-node incident matrix.
2) Depth-of-Two Unobservability: Depth-of-two unobservability means that the at most two unobservable buses can be
connected in the system, which means that the sum of , corresponding to any three connecting buses, is larger than 1. Therefore, this problem can be formulated as follows:
where
,
is the total number of posis the PMU
sible combinations of three connecting buses,
placement variable, and is the matrix, each row of which corresponds to three connecting buses and contains all of the possible combinations of three connecting buses.
B. With Zero Injection Measurements
If a bus has neither generation nor load, the sum of flows on all
the associated branches to the bus is zero. This equality normally
corresponds to a zero injection measurement and can be used in
state estimation.
1) Depth-of-One Unobservability: Let us use Fig. 2 as an
example.
Zero injection measurement at bus implies that
A. Without Zero Injection Measurements
1) Depth-of-One Unobservability: Depth-of-one unobservability is defined as a situation in which all of the neighboring
buses of any unobservable bus must be observable. The definition implies that it is impossible for two unobservable buses to
connect together, which provides us the way to develop the integer linear programming for this case.
To obtain the formulation, a matrix
needs to be deare given in Section II-A.
fined. The details of
If we define a vector
, the element
of indicates the number of times that bus can be
is the th row of
.
calculated by PMUs, where
So that the depth-of-one unobservability can be modeled as a
set of linear inequalities, the sum of , corresponding to the
The equalities
and
as well as
imply that
Since
and
is the vector of binary variables
is always true. Therefore, it is not necessary that this be listed
in the formulation, as illustrated below.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 3, AUGUST 2008
Also, since branch 1, 2, 3 and 4 are associated to the zero
is
injection measurement, matrix
The final integer linear programming for this example is
Fig. 3. Example for incomplete observability.
Based on the problem
, we have
where
is the matrix that keeps the branches that are not
associated with zero injection measurements and removes the
branches that are associated with zero injection measurements.
.
Others are the same as in
2) Depth-of-Two Unobservability: Let us use the same example as in the previous section. Depth-of-two unobservability
implies that we have the following inequalities:
and
The solution is
. So if a
PMU is placed at bus 3, then we can have depth-of-one unobservability for this system.
2) Depth-of-Two Unobservability : For this system, the total
number of all possible three connecting buses is 15. Since a
defined
zero injection measurement is at bus 2, the matrix
in Section III-B can be formed as follows:
which imply
when the zero injection measurement at bus
is considered.
Therefore, the problem can be formulated as follows:
Matrix
is formed as follows:
where
is the matrix that keeps the combinations that are not
associated with zero injection measurements and removes the
branches that are associated with zero injection measurements.
.
Others are the same as in
C. Example II
Let us use the same system as in Fig. 1. Here we only show
the case with zero injection measurement. We assume that a zero
injection measurement is at bus 2 (check Fig. 3 for details).
is
1) Depth-of-One Unobservability: For this case,
the same as in example I. The branch-to-node incident matrix
is given as
Branch
Branch
Branch
Branch
Branch
Branch
Branch
Branch
Then the integer linear programming can be written as
GOU: GENERALIZED INTEGER LINEAR PROGRAMMING FORMULATION FOR OPTIMAL PMU PLACEMENT
The
solution
of
this problem is
,
which
means
a
PMU
needs to be placed at bus 4 to reach the depth-of-two
unobservability.
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TABLE I
NUMBER OF PMU PLACEMENT FOR INCOMPLETE
OBSERVABILITY WITHOUT ZERO INJECTIONS
IV. PMU PLACEMENT FOR REDUNDANT
MEASUREMENT PLACEMENT
The formulation of optimal PMU placement under the cases
of full observability and incomplete observability has been presented in Sections II and III. However, the measurements redundancy is critical for bad data detection and identification in state
estimation. Different levels of measurements redundancy can be
requested for optimal PMU placement. As in previous sections,
the two cases with and without zero injection measurements, are
considered.
The requirements of the measurement redundancy can
be formulated through the column vector in the right hand
side of the linear inequality constraints in the integer linear
programming formulation for the cases with and without zero
injection measurements. The measurement redundancy can
be equal at all buses or different at different buses, which
completely depends on the requirements of the system and the
budget.
V. GENERALIZED INTEGER LINEAR
PROGRAMMING FORMULATION
The above linear programming formulation can be further
generalized for the cases of redundant PMU placement, full observability and incomplete observability. The generalized form
is given as follows:
TABLE II
NUMBER OF PMU PLACEMENTS FOR INCOMPLETE
OBSERVABILITY WITH ZERO INJECTIONS
In Table II, we consider the zero injection measurements and
not nonzero injection measurements. The number of zero injection measurements is listed in Table II, respectively, for three
power systems.
It should be noted that the number of required PMUs of the
proposed integer linear programming algorithm for different
systems is identical to that in [1]. However, the locations of
required PMUs are different, which means that the problem
of PMU placement with incomplete observability has multiple
solutions.
VII. CONCLUSION
where matrix it the transformation matrix which transforms
into matrix the requirements of the cases of redundant PMU
placement, full observability or incomplete observability; the
indicates the redundancy requirements for
column vector
all buses. and
take different values according to different
cases, based on the above results in previous sections.
VI. SIMULATION RESULTS
We have tested the proposed formulation on IEEE 14-, 30-,
and 57-bus systems. We use the binary integer programming of
MatLab to solve this problem.
The proposed integer linear programming algorithm has been
tested on different systems. The results without and with zero
injection measurement are displayed in Tables I and II.
In Table I, we do not consider any zero and nonzero injections; that means, we only consider the placement of pure phasor
measurements provided by PMUs.
This paper proposes a generalized integer linear programming formulation for optimal PMU placement under different
cases of redundant PMU placement, full observability and incomplete observability. This generalized formulation, considering the situations with and without zero injection measurements, shows that the problem of optimal PMU placement can
be modeled linearly and can be solved by integer linear programming. The generalized formulation paves an efficient way
for future research in PMU placement and related topics. Simulation results show that the proposed algorithm is computationally efficient and can be used in practice.
REFERENCES
[1] R. F. Nuqui and A. G. Phadke, “Phasor measurement unit placement
techniques for complete and incomplete observability,” IEEE Trans.
Power Del., vol. 20, no. 4, pp. 2381–2388, Oct. 2005.
[2] L. Mili, T. Baldwin, and R. Adapa, “Phasor measurement placement
for voltage stability analysis of power systems,” in Proc. 29th Conf.
Decision and Control, Honolulu, HI, Dec. 1990.
[3] T. L. Baldwin, L. Mili, M. B. Boisen, and R. Adapa, “Power system observability with minimal phasor measurement placement,” IEEE Trans.
Power Syst., vol. 8, no. 2, pp. 707–715, May 1993.
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[4] J. Chen and A. Abur, “Placement of PMUs to enable bad data detection in state estimation,” IEEE Trans. Power Syst, vol. 21, no. 4, pp.
1608–1615, Nov. 2006.
[5] B. Milosevic and M. Begovic, “Nondominated sorting genetic algorithm for optimal phasor measurement placement,” IEEE Trans. Power
Syst., vol. 18, no. 1, pp. 69–75, Feb. 2003.
[6] B. Xu and A. Abur, “Observability analysis and measurement placement for systems with PMUs,” in Proc. 2004 IEEE Power Eng. Soc.
Conf. Expo., Oct. 10–13, 2004, vol. 2, pp. 943–946.
[7] B. Gou, “Optimal placement of PMUs by integer linear programming,”
IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1525–1526, Aug. 2008.
Bei Gou (S’97–M’00) received the B.S. degree in electrical engineering from
North China University of Electric Power, Beijing, China, in 1990, the M.S.
degree from Shanghai JiaoTong University, Shanghai, China, in 1993, and the
Ph.D. degree from Texas A&M University, College Station, in 2000.
From 1993 to 1996, he taught in the Department of Electric Power Engineering at Shanghai JiaoTong University. He worked as a research assistant at
Texas A&M University beginning in 1997. Subsequently, he worked at ABB
Energy Information Systems, Santa Clara, CA, for two years, and at ISO New
England for one year as a senior analyst. He is currently an assistant professor
in the Energy Systems Research Center (ESRC) at the University of Texas at
Arlington. His main interests are power system state estimation, power market
operations, power quality, power system reliability, and distributed generators.