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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
A Unified Approach for the Optimal PMU Location
for Power System State Estimation
Nabil H. Abbasy and Hanafy Mahmoud Ismail
Abstract—Phasor measurement units (PMUs) are considered as
a promising tool for future monitoring, protection and control of
power systems. In this paper, a unified approach is proposed in
order to determine the optimal number and locations of PMUs
to make the system measurement model observable and thereby
can be used for power system state estimation. The PMU placement problem is formulated as a binary integer linear programming (BILP), in which the binary decision variables (0, 1) determine whether to install a PMU at each bus, while preserving the
system observability and lowest system metering economy. The
proposed approach integrates the impacts of both existing conventional power injection/flow measurements (if any) and the possibility of single or multiple PMU loss into the decision strategy
of the optimal PMU allocation. Unlike other available techniques,
the network topology remains unaltered for the inclusion of conventional measurements, and therefore the network connectivity
matrix is built only once based on the original network topology.
The mathematical formulation of the problem maintains the original bus ordering of the system under study, and therefore the solution directly points at the optimal PMU locations. Simulations
using Matlab are conducted on a simple testing seven-bus system,
as well as on different IEEE systems (14-bus, 30-bus, 57-bus, and
118-bus) to prove the validity of the proposed method. The results
obtained in this paper are compared with those published before
in literature.
Index Terms—Integer programming, optimal location, PMUs,
state estimation.
I. INTRODUCTION
T
HE commercialization of the global positioning system
with accuracy of timing pulses in the order of 1 microsecond has made possible the commercial production of
phasor measurement units (PMUs). The PMU is a power
system device capable of measuring voltage and current phasor
in a power system. Synchronism among phasor measurements
is achieved by same-time sampling of voltage and current
waveforms using a common synchronizing signal from the
global positioning satellite (GPS) [1]–[3].
The way power systems are controlled is ought to be revolutionized by such a new technology. However, the overall cost
of the metering system will limit the number and locations of
PMUs. In addition to the cost factor, different criteria are suggested for the proper allocation of PMUs in a given system.
Network observability, state estimation accuracy and robustness
present samples of such criteria.
Manuscript received May 27, 2008; revised September 10, 2008. Current version published April 22, 2009. Paper no. TPWRS-00426-2008.
The authors are with the Department of Electrical Engineering, College of
Technological Studies, Shuwaikh, Kuwait 70654, Kuwait (e -mail: abbassyna@
hotmail.com; [email protected]).
Digital Object Identifier 10.1109/TPWRS.2009.2016596
Network observability determines whether a state estimator
will be able to determine a unique state solution for a given
set of measurements, their location, and a specified network
topology. The problem is independent of the measurement errors (or even the measurement values), branch parameters as
well as the operating state of the system. There are three basic
approaches to conduct network observability analysis; namely
numerical, topological, and hybrid approaches. The numerical
observability approach is based on the fact that a unique solution for the state vector can be estimated if the gain matrix is
nonsingular or equivalently if the measurement Jacobian matrix
has a full column rank and well conditioned. The topological observability approach is based on the fact that a network is fully
observable if the set of measurements can form at least one measurement spanning tree of full rank [5].
The placement of a minimal set of phasor measurement units
to make the system measurement model observable is the main
objective of this paper. Due to the high cost of having a PMU
at each node, some of the studies performed in the mid 1980s
focused on PMU placement and pseudo measurements for
complete or partial observability of the system for static and
dynamic state estimators [4]–[7]. One of these studies suggested
a gradual placement of PMU and provided a methodology for
PMU placement with a selected depth of unobservability [7].
This study defined the depth of unobservability as the distance
of an unobserved bus to its observed neighbors and provided
a methodology for a phased installation of PMUs. A methodology for PMU placement for voltage stability analysis in
power system was developed in [8]. Reference [9] introduced
a strategic PMU placement algorithm to improve the bad data
processing capability of state estimation by taking advantage of
PMU technology. In addition, the optimum number of PMUs
placement, which makes the power system observable, has been
reached through a genetic-based algorithm [10]. An algorithm
based on integer programming was successfully implemented
to find the optimum number and the locations of the installed
PMUs of the power system [11], [12].
In this paper, a unified approach is developed to find the minimum number (least cost) and locations of PMUs, such that the
power network is observable, as a priori step for power system
state estimation. The developed method accounts for the existing conventional measurements in the mathematical model
of the optimal PMU placement strategy; while considering the
chance of single or multiple PMU loss in its decision making.
The problem is formulated as a binary integer linear programming (BILP) problem. Only the branch-bus model of the network is needed to obtain the minimum number of the PMUs
and their locations. Simulation results conducted on a simple
testing seven-bus system, as well as the IEEE 14-bus, 30-bus,
57-bus and 118-bus systems are presented.
0885-8950/$25.00 © 2009 IEEE
ABBASY AND ISMAIL: A UNIFIED APPROACH FOR THE OPTIMAL PMU LOCATION FOR POWER SYSTEM STATE ESTIMATION
II. PMUS AND POWER SYSTEM STATE ESTIMATION
PMUs are now used in power systems for many potential applications. The PMUs receive their synchronized signals form
the GPS Satellite and are now being manufactured commercially. Their importance comes from the fact that they can provide synchronized measurements of real-time phasor of voltage
and currents to the state estimator [2]. A PMU located at any
bus can measure the phasor voltage of that bus (magnitude and
angle) and as many as needed phasor currents (magnitude and
angle) of branches emanating from that bus. Applications of the
phase measuring units include; measuring frequency and magnitude of phasors, state estimation, instability prediction, adaptive relaying, and improved control.
State estimators are extensively used in modern electric
power system utilities control systems to monitor the state of
the power system. Various measurements such as complex
powers and voltage and current magnitudes received from
different substations are fed into the state estimator. Using an
iterative nonlinear estimation procedure, the state estimator
calculates the power system state. The state (vector) is a
collection of all the positive sequence voltage phasors of the
network and, from the time the first measurement is taken to
the time when the state estimate is available, several seconds
or minutes may have elapsed. Because of the time skew in
the data acquisition process, as well as the time it takes to
converge to a state estimate, the available state vector is at best
an averaged quasi-steady-state description of the power system.
Consequently, the state estimators available, in this way, in
control centers are restricted to steady state applications only.
If voltages at all substations are measured at the same instant
by using synchronized phasor measurement units, true simultaneous measurements of the power system state could be obtained. It is also sensible to use the positive sequence currents,
which provide data redundancy. This leads to a linear estimation
of the power system state, which uses both current and voltage
measurements.
A dynamic state estimator is also obtained by using synchronized phasor measurements. This can be achieved by
maintaining a continuous stream of phasor data (voltage and
current) from the substations to the control center. In this case,
a state vector that can follow the system dynamics can be
constructed [2]. In fact, by using PMUs, the state estimator
can play an important role in the security of power system
operations.
III. STATEMENT OF THE PROBLEM
A. Without Considering Conventional Measurements [11],
[12]
A PMU placed at bus will measure the phasor voltage of
bus and a predetermined number of phasor currents of the outgoing branches of that bus. The number of the measured phasor
currents depends on the number of PMU channels made available. In this paper, we assume that a PMU placed at bus will
measure all phasor currents of the branches connected to that
bus, in addition to the phasor voltage of bus . Therefore, with
the absence of any conventional measurements in the system,
bus will be observable if at least one PMU is placed within the
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set formed by bus and all buses incident to it. Therefore, the
problem of optimal PMU placement is one where the objective
is to minimize the number of PMUs utilized while preserving
the system observability. This objective can be formulated as
(1)
where
is a binary decision variable vector, defined as
(2)
is a binary network connectivity matrix defined as in (3)
(3)
. is the vector of PMU cost coefficients, is a vector whose
is a
entries are all ones, and is the total number of buses.
vector function whose entries are non-zeros if the corresponding
bus voltage is observable using the given measurement set and
zeros otherwise.
B. Considering Conventional Measurements
In practice, PMUs need to be installed in real systems which
are already monitored by conventional injection and/or power
flow measurements, in order to enhance the state estimator performance. Therefore, the model presented in the above section
needs to be modified to account for the existence of such conventional measurements in the network under study. Reference
[12] introduced a method to include conventional measurements
in the optimal PMU placement strategy. This method will be
referred to as the Individual Bus Merging (IBM) method. A
brief description of this method, along with its merits, is given
in the following section. Next an alternative proposed method,
which will be referred to as the Augmented Bus Merging (ABM)
method, will be introduced.
1) The Individual Bus Merging (IBM) Method: This method
proposes an approach for determining the optimal PMU locations for systems equipped with conventional measurements.
First, an associate set of buses will be defined for each available
zero/nonzero injection measurement in the system. This set will
be formed by the injection bus and all its associate (connected)
buses. Using network equations, the available injection measurement at a particular bus allows one to calculate the phasor
voltage of only one bus among its associate set of buses, providing that the phasor voltages of all the remaining buses in that
set are known. Therefore, the IBM method suggests that the injection bus to be merged with any one of its associate buses, and
to resolve the BILP problem defined by (3) for finding the optimal PMU locations. With this merging process, the number of
system buses will be lowered by 1 for each available injection
measurement. In addition, the network topology will be altered
and the network connectivity matrix will need to be reestablished accordingly. Similarly, the flow measurement in a particular branch allows one to calculate the phasor voltage of one
branch terminal bus, providing that the phasor voltage of the
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Fig. 1. Seven-bus system.
other branch terminal bus is known. Therefore, the IBM method
suggests combining the observability constraint equations [included in (3)] of the two terminal buses of the flow branch into
one constraint equation. Accordingly, the number of constraint
equations will be reduced by one for each existing flow measurement, while the problem variables will be unaltered.
We have extensively tested the approach explained above for
a number of systems and case studies. It is found that the solution of the IBM is sensitive to the selection of the bus to be
merged with the injection bus, especially in the presence of
flow measurements. For example, refer to the seven-bus system
shown in Fig. 1 with one zero injection measurement (denoted
by ) exists at bus 3 and a flow measurement (denoted by X)
in branch 1–2. Bus 3 was merged with one of its associated
buses at a time and the BILP was solved for each case. Results of this case study (presented in Section IV) show that different solutions may be obtained for different selections of the
bus merged with the injection bus. The network will be reconfigured as many times as the number of existing injection measurements. Another limitation of this approach is that if the solution
results in a PMU to be placed at the merged bus, it will mean
that the PMU should be placed at the original injection bus or at
its associate merged bus, or at both. In such a case, a topological observability test must be conducted to determine the final
decision regarding that particular PMU allocation.
2) Proposed Augmented Bus Merging (ABM) Method: A
newly developed method is proposed in this section that incorporates the effect of existing conventional measurements in the
formulation of the optimal PMU selection problem. The proposed method makes use of the relaxation provided by the conventional measurements on the observability constraints, while
avoiding the possibility of getting different solutions provided
by other available methods. The key fact is that the existence
of a conventional measurement at a bus would relax the observability conditions imposed by the PMU placement strategy at
that particular bus. First, for each bus having an injection measurement, a set of associate buses
will be formed by the
injection bus and all its connected buses. Therefore, as explained in Section III-B.1, the phasor voltage of only one bus
can be calculated using the known injection
belonging to
at bus and thus need not be directly observed by a PMU. Similarly, the phasor voltage of one terminal bus of a branch having
a flow measurement can be calculated providing that the other
bus terminal phasor voltage is known. In order to implement
these observations, the network buses are reordered, via a properly constructed permutation matrix . Bus reordering is made
such that the buses which are not incident to any conventional
measurement come first, followed by a set of augmented buses.
Each augmented bus corresponds to an available conventional
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
measurement. An augmented bus corresponds to an injection
measurement comprises the injection bus and all its associate
buses, whereas an augmented bus corresponds to a flow measurement comprises the two terminal buses of that particular
flow measurement. The rows of the permutation matrix represent the new bus-order of the system whereas the columns of
the permutation matrix represent the original system bus-order.
will then be
A new system branch connectivity matrix
formed as a linear transformation of the original system convia the developed permutation matrix .
nectivity matrix
of inAdopting the following abbreviations:
jection measurements;
of flow measurements;
of buses not associated to conventional meanumber of conventional measuresurements;
; and
of buses associments
ated to conventional measurements, we get
(4)
where
is
bus-bus incidence matrix for the
is
buses not incident to conventional measurements,
bus-bus incidence matrix for the augmented
buses.
is
bus-bus connectivity
matrix for buses not incident to conventional measurements,
is
bus-bus connectivity
and
matrix for the augmented buses.
The right-hand side vector must be modified accordingly.
The new right-hand side vector
will be defined as
, where
is
, whose elements
is
and
is
.
are all equal to 1,
In order to reflect the relaxation provided by the existence of
is set to be equal
conventional measurements, each entry in
to the number of buses connected to the injection bus while the
are all equal to 1. Finally, the optimal PMU
elements of
placement problem, considering the existence of conventional
measurements can be stated as
(5)
To illustrate the above formulation, we refer again to the
seven-bus system shown in Fig. 1, with one conventional
injection measurement placed at bus 3 (referred to as ) and
one flow measurement placed in line 4–5. The original system
connectivity matrix is
In this case, the injection at bus 3 is incident to buses 2, 4, 6,
in addition to bus 3, whereas the flow in branch 4–5 is incident
ABBASY AND ISMAIL: A UNIFIED APPROACH FOR THE OPTIMAL PMU LOCATION FOR POWER SYSTEM STATE ESTIMATION
to buses 4 and 5. Therefore, the set of buses incident to conwhere the
ventional measurements will be
set of buses not incident to conventional measurements will be
. Next, buses 2, 3, 4, and 6 will be augmented into
whereas buses 4, 5, 3, and 7 will be
a newly created bus
The new bus order
augmented into a newly created bus
and permutation matrix will thus be
The right-hand side vector
will be
and the new connectivity matrix
One of the merits of the proposed approach is that it can be easily
programmed in a general format using an effective programming tool such as that of Matlab. Thus, it can be applied for systems of any size and topology. The network connectivity matrix
is built only once based on the original network topology and
need not be reestablished for the inclusion of the conventional
measurements. The mathematical formulation of the problem
maintains the original bus ordering of the system under study,
and therefore the solution directly points at the optimal PMU
locations.
C. Modeling Considering PMU Loss
The number of PMUs proposed by the integer programming
problem (5) represents the critical number required to make
the power system observable. So far, it has been assumed that
this number with its proposed locations will function perfectly.
Like any other measuring device, PMUs are prone to failures
although they are highly reliable. Therefore, it is necessary to
guard against such unexpected failures of PMUs. Reference [12]
proposed a method to account for considering single PMU loss
in the PMU placement problem. This method will be referred to
as the Primary and Backup (P&B) method. A brief description
of that method is given in the following section. Next, an alternative proposed method will be presented. This proposed method
will be referred to as the Local Redundancy (LR) method.
1) Primary and Backup (P&B) Method: In this method,
two independent PMU sets are determined, a primary set
and a backup set, where each of which can make the system
observable on its own. The primary set of PMUs is determined
by building the constraint functions according to the procedure
described in Section III and solving the BILP problem. To find
the backup set of PMUs, it is suggested that all the variables
to be removed in the constraint (3), where bus belongs to the
primary set, in order to avoid picking up the same bus which
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appears in the primary set. Then the BILP problem is solved to
obtain the backup set.
When trying to explore this approach, it is found that the
method maintains the system observability with a single PMU
loss either in the primary set or in the backup set. It is important to note that both the primary and backup sets are independent, and each of which, standing alone, can render the system
observable. Therefore, this method is also able to preserve the
system observability under multiple PMU loss, provided that
these multiple PMU loss occur in either the primary set or the
backup set, but not in both. Numerical experimentation conducted on this method shows that the method may fail if multiple
PMU loss occurs which combines lost PMUs from both the primary and backup sets simultaneously.
2) Proposed Local Redundancy (LR) Method: An alternative method to account for single or multiple PMU loss is proposed in this section. The proposed method attempts to provide
a local PMU redundancy to allow for the loss of PMUs while
preserving the global network observability. Recall that each
entry in the right-hand side vector of (3) is set to 1 in order
to guarantee the observability of each bus via at least one PMU.
constraint in (3). If the right-hand side of
Now, consider the
that constraint is changed to 2, it will practically mean that in
order for that bus to be observable, at least two PMUs must be
installed in the set of buses formed by all buses incident to bus
, including bus itself. In other words, we can say that the complex voltage of bus will be “reached” by at least two PMUs.
If this concept is extended for all constraints in (3), all elements
of the right-hand side vector will be changed to 2. This directly implies that each bus will be allowed to loose, at most,
one PMU in its vicinity (set of buses connected to this particular
bus including the bus itself) without sacrificing its observability.
When the concept is extended to all buses, the network global
observability will be directly maintained, with possible single
PMU loss. It should be admitted that when the BILP problem is
solved in this way, a higher number of PMUs will be expected,
and accordingly the cost of metering system will be significantly
affected. However; as it is the case in designing any metering
system, a tradeoff must be made between the economic restrictions and the required degree of metering system reliability.
From the theoretical point of view, the method explained
above can be extended for the consideration of multiple PMU
loss as well. For instance, the right-hand side of a particular
constraint can be set to 3 in order to account for the loss of,
at most, two PMUs among the set of PMUs responsible for
observing that particular bus. The method can also be made
adaptive. Different numbers can be assigned to the right-hand
side vector to account for different levels of PMU loss.
These numbers can be assigned according to the heaviness of
connectivity of each bus in the system.
D. Cost Consideration
As mentioned before, the PMU is a power system measurement device capable of measuring the synchronized voltage
phasor of the bus where it is installed and the current phasors
of some or all branches connected to that particular bus. In
of all PMUs are assumed to be
preceding section the cost
equal (at 1 p.u.). It is necessary to consider the unequal cost of
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
the PMUs in the formulation of the problem to demonstrate its
effect on the number and location of PMUs required keeping
the observability of the system.
The cost of a PMU depends on a number of factors, including
the number of measuring terminals (channels), CT and PT connections, power connection, station ground connection, and
GPS antenna connection. However, what really distinguishes
between different PMU costs is the number of channels, since
the remaining items are common to all PMU installations.
Available literature refer that some of the larger PMUs (measuring up to ten phasors plus frequency) cost approximately $30
to $40 thousands of dollars while the smaller ones (measuring
from one to three phasors plus frequency) cost considerably
less.
To account for the effect of unequal PMU cost, in (1) has to
be modified. The procedure suggested here is to assume that the
cost of a PMU placed at a particular bus connected via a branch
to one bus only is set to 1 p.u.. Since the number of PMU channels is dominant in determining the PMU cost, we may assume
that this cost increases by a decimal increment p.u., for each
additional channel. Then the PMU problem is formulated to determine the minimum number and locations of PMUs required
for the system to be observable. A reasonable selection for
may be taken as 0.1.
Combining the ideas presented in this section, the steps for
the implementation of the proposed unified PMU method can
be summarized as follows.
1) Read the network branch/bus data.
and PMU cost
2) Form the network connectivity matrix
coefficient vector .
3) Define the array of buses comprising zero injection mea, and the array of branches comprising
surements
; where
flow measurements
defines the from-to buses where flow measurements exist.
.
4) Establish the array of associated buses
; where
5) Establish the array of nonassociated buses
, and
is the set of all system buses.
;
6) Establish the new bus-order vector
is defined as
.
where
7) Establish the permutation matrix .
.
8) Establish the new connectivity matrix
9) Form the new right-hand side vector
,
.
10) If a single PMU loss is considered set
11) Solve the BILP problem
IV. RESULTS AND DISCUSSION
The unified approach presented in Section III is programmed
on Matlab and case studies are performed on the test seven-bus
system (Fig. 1), IEEE 14-bus system (Fig. 2), IEEE 30-bus,
57-bus, and 118-bus systems.
A. Validation of the Augmented Bus Merging (ABM) Method
The seven-bus system shown in Fig. 1 is used in order to validate the basic results of the present study. Table I summarizes
Fig. 2. IEEE 14-bus system.
results for different case studies, without considering PMU loss.
A comparison is made in Table I between the IBM method and
the ABM method proposed in this paper. When ignoring conventional measurements (Case 1) the two methods intuitively
yielded identical solution for the optimal PMU number and location. In case 2 the zero injection bus was considered. According to the IBM method, different bus merging for the injection bus 3 was performed. The IBM method in this case suggested buses 2 and 4 for locating the PMUs, for all different
selections of the bus merged with the injection bus. However,
it is noted that this solution is the same as that solution of case
1 (when the zero injection was ignored), which means that the
obtained solution in this case did not make benefit of the available zero injection at bus 3. On the contrary, the proposed ABM
method (case 2, columns 4 and 5) provided different optimal
locations for PMUs (buses 1 and 4), although it maintains the
same minimum number of PMUs. When exploring this latter
solution, it is found that the states of buses 1 and 2 will be observed by a PMU located at bus 1, while the states of buses 3,
4, 5, and 7 will be observed via the PMU located at bus 4. The
state of the remaining bus 6 can be calculated using the available injection measurement at bus 3.
One more advantage in the solution of the proposed ABM
method in this case is that the number of utilized PMU channels
is 4 compared to 6 channels if PMUs were located at buses 2 and
4. Next, in addition to the injection measurement at bus 3, a flow
measurement was then added to branch 1–2 in case 3. Again,
different bus merging for the injection bus 3 was performed and
the BILP was solved for each one of these bus merging. As
shown in Table I case 3, the IBM method gave different PMU
locations (buses 3 and 4 and buses 2 and 4) with different bus
merging. The results of this case study indicate that the IBM
method is sensitive, in terms of the location of PMUs, to the
selection of bus to which the injection bus is merged with. On the
contrary, the solution provided by the proposed ABM method in
this case (buses 1 and 4) is unique; again with a fewer number
of utilized PMU channels.
ABBASY AND ISMAIL: A UNIFIED APPROACH FOR THE OPTIMAL PMU LOCATION FOR POWER SYSTEM STATE ESTIMATION
TABLE I
RESULTS FOR THE SEVEN-BUS SYSTEM WITHOUT CONSIDERING PMU LOSS
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TABLE III
RESULTS FOR THE 14-BUS SYSTEM WITHOUT AND WITH CONSIDERING
SINGLE PMU LOSS (IGNORING ZERO INJECTION)
TABLE IV
RESULTS FOR THE 14-BUS SYSTEM CONSIDERING UNEQUAL COSTS OF PMUS
(WITHOUT CONSIDERING SINGLE PMU LOSS AND IGNORING ZERO INJECTION)
TABLE II
RESULTS FOR THE SEVEN-BUS SYSTEM CONSIDERING
SINGLE PMU LOSS (IGNORING ZERO INJECTION)
TABLE V
RESULTS FOR THE 14-BUS SYSTEM WITH CONVENTIONAL MEASUREMENTS
B. Results Without Conventional Measurements-Without and
With Considering PMUs Loss
As stated before, the PMUs placed by the proposed unified
approach are assumed to function perfectly. Sometimes, one or
more of these PMUs may fail to operate and therefore, it is necessary to guard against such unexpected failures. A part of this
paper is to apply the proposed method for single PMU loss to
systems with and without conventional measurements. Table II
demonstrates results for the seven-bus system, in case of ignoring zero injections, using both the P&B and the proposed LR
methods. Results of Table II indicate that the number of PMUs
required to guard against such single PMU loss using the proposed LR method is less by 1 than that number obtained by using
the P&B method.
The optimal PMU placement algorithm, using both the P&B
and the LR method, is applied to the IEEE 14-bus system shown
in Fig. 2. Zero injections are ignored in this case, and results are
shown in Table III. In this case, a complete agreement between
the two approaches, regarding the optimal number and locations
of PMUs is achieved.
The effect of considering unequal costs of PMUs in the placement strategy problem is conducted in this study. As proposed
in Section III, the cost of a PMU may be increased by 0.1 p.u. for
each additional branch (channel) emanating from its respective
bus. Results of this case study are shown in Table IV, without
considering single PMU loss and ignoring zero injections. For
each case, the number of utilized channels is shown in the same
table. These results indicate that the number of PMUs in case
of equal and unequal PMU costs is the same (4 PMUs) but the
locations are different. Since the overall cost of PMU metering
system increases with increasing number of measuring channels, therefore, results of Table IV indicate that the overall cost
of PMU metering system may be substantially affected by considering different (unequal) costs for PMUs.
C. Results With Conventional Measurements-Without and
With Considering PMUs Loss
The application of the proposed unified approach to IEEE
14-bus system with conventional measurements is carried out.
Results are shown in Table V, without and with single PMU loss
consideration. The P&B and proposed LR methods for single
PMU loss are applied for the purpose of comparison. The system
has only one injection measurement at bus 7 and one flow measurement in branch 5–6. With no PMU loss considered, the optimal number of PMUs is 3 with their locations as indicated in
Table V. In case of considering single PMU loss, both the P&B
and the proposed LR methods possess the same optimal number
of PMUs (which is 7 in this case). However, a slight difference
in their locations is depicted (bus 1 in the P&B method is interchanged with bus 5 in the proposed LR method).
In order to check the validity of the proposed unified approach
for large systems applications, case studies are applied to the
30-, 57-, and 118-bus IEEE systems with the data shown in
Table VI. Each system has a number of injection buses but no
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
TABLE VI
DATA FOR 30-, 57-, AND 118-BUS SYSTEMS
TABLE VII
OPTIMAL NUMBER OF PMUS REQUIRED FOR THE
30-, 57-, AND 118-BUS SYSTEMS
TABLE VIII
SIMULATION RESULTS FOR THE 118-BUS SYSTEM
(WITHOUT CONSIDERING SINGLE PMU LOSS)
Simulation results for the three cases are shown in Table VIII.
It is clear from the table that as the number of flow measurements increases, the number of PMUs required keeping the
system observable decreases. It can also be noticed, when comparing these results with those previously published in literature
that the required number of PMUs is reduced from 29 when
considering no flow measurements to 24 when considering
15 flow measurements. A very good agreement between the
results obtained using the proposed unified approach and those
published before is achieved. As it is clear from the results and
as expected, conventional measurements generally reduces the
number of PMUs required to maintain the system observable.
V. CONCLUSION
In this study, a unified approach is proposed for determining
the optimal number and locations of PMUs required making the
entire power system observable. The proposed unified approach
considers the impacts of both existing conventional measurements and the possibility of single or multiple PMU loss into the
decision strategy of the optimal PMU allocation problem. The
proposed approach is easy in implementation using MATLAB
as an effective programming tool. Considering single PMU loss,
a new concept (method) is developed which permits single or
multiple PMU loss keeping the entire system observable. The
effect of PMU meters cost on their optimal number and locations is simulated in this unified approach through a suggested
procedure. The developed approach is applied to different IEEE
power systems (14-bus, 30-bus, 57-bus, and 118-bus) and results are compared with those published in literature with very
good agreement.
REFERENCES
flow measurements. Two case studies are considered; without
considering single PMU loss and with considering single PMU
loss. Results are shown in Table VII. As compared to the P&B
method, it is clear from Table VII that the proposed LR method
yields a lower number of PMUs to maintain the observability
of the system. It is also noticed that the number of PMUs in the
backup set is greater than that of the primary set in the three systems when using the P&B method. As expected, results of this
case study generally reveal that considering single PMU loss has
the effect of increasing the number of required PMUs.
Simulations using the proposed unified approach for PMU
meter locations are carried out on IEEE 118-bus system for fixed
number of injection buses and different number of flow measurements. Three cases are studied. In the first case, five flow
measurements are considered. Case 2 contains ten flow measurements (5 more than case 1), while case 3 contains 15 flow
measurements (5 more than case 2). The locations of the flow
measurements for these three cases are shown in Table VIII. The
number of injection buses is taken as 10 for the three cases and
their locations are as shown in Table VI. These simulations are
carried out without considering single PMU loss.
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ABBASY AND ISMAIL: A UNIFIED APPROACH FOR THE OPTIMAL PMU LOCATION FOR POWER SYSTEM STATE ESTIMATION
Nabil H. Abbasy was born 1956. He received the
B.Sc. (Hons.) and M.Sc. degrees from the University
of Alexandria, Alexandria, Egypt, in 1979 and 1983,
respectively, and the Ph.D. degree from Illinois Institute of Technology, Chicago, in 1988.
He was an Assistant Professor at Clarkson University, Potsdam, NY, from 1988 to 1989, and then at the
University of Alexandria from 1989 to 1994, where
he has been a Full Professor since 2000. He has been
on leave of absence with the College of Technological Studies, Kuwait, since 1994. His research interests include power systems operation, dynamics, and transients.
813
Hanafy Mahmoud Ismail was born in Cairo, Egypt,
in 1956. He received the B.S. and M.S. degrees in
electrical engineering from Electrical Engineering
Department, Faculty of Engineering at Ain-Shams
University, Cairo, in 1979 and 1984, respectively.
He received the Ph.D. degree in 1989 from Electrical Engineering Department at the University of
Windsor, Windsor, ON, Canada.
Since graduation, he has been working at the
Electrical Engineering Department, Faculty of
Engineering, Ain-Shams University. He is now
a Professor, on leave of absence from Ain-Shams University, joining the
Electrical Engineering Department at the College of Technological Studies in
Kuwait since 1997. He deals mainly with the high voltage power transmission
and their associated electrostatics and electromagnetic fields. He is also
working in the area of power systems, under ground cables, and fault detection
on transmission lines.