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806 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009

A Unified Approach for the Optimal PMU Locationfor Power System State Estimation

Nabil H. Abbasy and Hanafy Mahmoud Ismail

Abstract—Phasor measurement units (PMUs) are considered asa promising tool for future monitoring, protection and control of power systems. In this paper, a unified approach is proposed inorder to determine the optimal number and locations of PMUsto make the system measurement model observable and therebycan be used for power system state estimation. The PMU place-ment problem is formulated as a binary integer linear program-ming (BILP), in which the binary decision variables (0, 1) deter-mine whether to install a PMU at each bus, while preserving thesystem observability and lowest system metering economy. Theproposed approach integrates the impacts of both existing con-ventional power injection/flow measurements (if any) and the pos-

sibility of single or multiple PMU loss into the decision strategyof the optimal PMU allocation. Unlike other available techniques,the network topology remains unaltered for the inclusion of con-ventional measurements, and therefore the network connectivitymatrix is built only once based on the original network topology.The mathematical formulation of the problem maintains the orig-inal bus ordering of the system under study, and therefore the so-lution directly points at the optimal PMU locations. Simulationsusing Matlab are conducted on a simple testing seven-bus system,as well as on different IEEE systems (14-bus, 30-bus, 57-bus, and118-bus) to prove the validity of the proposed method. The resultsobtained in this paper are compared with those published beforein literature.

Index Terms—Integer programming, optimal location, PMUs,state estimation.

I. INTRODUCTION

THE commercialization of the global positioning systemwith accuracy of timing pulses in the order of 1 mi-

crosecond has made possible the commercial production of phasor measurement units (PMUs). The PMU is a powersystem device capable of measuring voltage and current phasorin a power system. Synchronism among phasor measurementsis achieved by same-time sampling of voltage and currentwaveforms using a common synchronizing signal from theglobal positioning satellite (GPS) [1]–[3].

The way power systems are controlled is ought to be revolu-tionized by such a new technology. However, the overall costof the metering system will limit the number and locations of PMUs. In addition to the cost factor, different criteria are sug-gested for the proper allocation of PMUs in a given system.Network observability, state estimation accuracy and robustnesspresent samples of such criteria.

Manuscript received May 27, 2008; revised September 10, 2008. Current ver-sion 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: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TPWRS.2009.2016596

Network observability determines whether a state estimatorwill be able to determine a unique state solution for a givenset of measurements, their location, and a specified network topology. The problem is independent of the measurement er-rors (or even the measurement values), branch parameters aswell as the operating state of the system. There are three basicapproaches to conduct network observability analysis; namelynumerical, topological, and hybrid approaches. The numericalobservability approach is based on the fact that a unique solu-tion for the state vector can be estimated if the gain matrix isnonsingular or equivalently if the measurement Jacobian matrix

has a full column rank and well conditioned. The topological ob-servability approach is based on the fact that a network is fullyobservable if the set of measurements can form at least one mea-surement spanning tree of full rank [5].

The placement of a minimal set of phasor measurement unitsto make the system measurement model observable is the mainobjective of this paper. Due to the high cost of having a PMUat each node, some of the studies performed in the mid 1980sfocused on PMU placement and pseudo measurements forcomplete or partial observability of the system for static anddynamic state estimators [4]–[7]. One of these studies suggesteda gradual placement of PMU and provided a methodology forPMU placement with a selected depth of unobservability [7].This study defined the depth of unobservability as the distanceof an unobserved bus to its observed neighbors and provideda methodology for a phased installation of PMUs. A method-ology for PMU placement for voltage stability analysis inpower system was developed in [8]. Reference [9] introduceda strategic PMU placement algorithm to improve the bad dataprocessing capability of state estimation by taking advantage of PMU technology. In addition, the optimum number of PMUsplacement, which makes the power system observable, has beenreached through a genetic-based algorithm [10]. An algorithmbased on integer programming was successfully implementedto 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 min-imum number (least cost) and locations of PMUs, such that thepower network is observable, as a priori step for power systemstate estimation. The developed method accounts for the ex-isting conventional measurements in the mathematical modelof the optimal PMU placement strategy; while considering thechance of single or multiple PMU loss in its decision making.The problem is formulated as a binary integer linear program-ming (BILP) problem. Only the branch-bus model of the net-work is needed to obtain the minimum number of the PMUsand their locations. Simulation results conducted on a simpletesting 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 807

II. PMUS AND POWER SYSTEM STATE ESTIMATION

PMUs are now used in power systems for many potential ap-

plications. The PMUs receive their synchronized signals form

the GPS Satellite and are now being manufactured commer-

cially. Their importance comes from the fact that they can pro-

vide synchronized measurements of real-time phasor of voltage

and currents to the state estimator [2]. A PMU located at anybus 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 mag-

nitude of phasors, state estimation, instability prediction, adap-

tive 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 simul-

taneous measurements of the power system state could be ob-tained. 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 syn-

chronized 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 out-going branches of that bus. The number of the measured phasor

currents depends on the number of PMU channels made avail-

able. 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

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 whoseentries areall ones, and is the total number of buses. is a

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 per-

formance. Therefore, the model presented in the above section

needs to be modified to account for the existence of such con-

ventional measurements in the network under study. Reference

[12] introduced a method to include conventional measurementsin 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 loca-

tions 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 mea-

surement at a particular bus allows one to calculate the phasor

voltage of only one bus among its associate set of buses, pro-

viding that the phasor voltages of all the remaining buses in that

set are known. Therefore, the IBM method suggests that the in-

jection bus to be merged with any one of its associate buses, and

to resolve the BILP problem defined by (3) for finding the op-timal PMU locations. With this merging process, the number of

system buses will be lowered by 1 for each available injectionmeasurement. In addition, the network topology will be altered

and the network connectivity matrix will need to be reestab-

lished accordingly. Similarly, the flow measurement in a par-

ticular branch allows one to calculate the phasor voltage of onebranch terminal bus, providing that the phasor voltage of the




Fig. 1. Seven-bus system.

other branch terminal bus is known. Therefore, the IBM method

suggests combining the observability constraint equations [in-

cluded in (3)] of the two terminal buses of the flow branch into

one constraint equation. Accordingly, the number of constraintequations will be reduced by one for each existing flow mea-

surement, 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 so-

lution 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. Re-

sults of this case study (presented in Section IV) show that dif-

ferent solutions may be obtained for different selections of the

bus merged with the injection bus. The network will be reconfig-

ured as many times as the number of existing injection measure-

ments. 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 topolog-ical 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 incor-

porates the effect of existing conventional measurements in the

formulation of the optimal PMU selection problem. The pro-

posed method makes use of the relaxation provided by the con-

ventional 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 observ-

ability conditions imposed by the PMU placement strategy atthat particular bus. First, for each bus having an injection mea-

surement, a set of associate buses will be formed by the

injection bus and all its connected buses. Therefore, as ex-

plained in Section III-B.1, the phasor voltage of only one bus

belonging to can be calculated using the known injection

at bus and thus need not be directly observed by a PMU. Sim-

ilarly, 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 prop-

erly 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

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 mea-

surement comprises the two terminal buses of that particular

flow measurement. The rows of the permutation matrix repre-

sent the new bus-order of the system whereas the columns of

the permutation matrix represent the original system bus-order.A new system branch connectivity matrix will then be

formed as a linear transformation of the original system con-nectivity matrix via the developed permutation matrix .

Adopting the following abbreviations: of in-

jection measurements; of flow measurements;

of buses not associated to conventional mea-

surements; number of conventional measure-ments ; and of buses associ-

ated to conventional measurements, we get

(4)

where is bus-bus incidence matrix for the

buses not incident to conventional measurements, is

bus-bus incidence matrix for the augmented

buses. is bus-bus connectivity

matrix for buses not incident to conventional measurements,

and is bus-bus connectivity

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

are all equal to 1, is and is .

In order to reflect the relaxation provided by the existence of

conventional measurements, each entry in is set to be equal

to the number of buses connected to the injection bus while the

elements of are all equal to 1. Finally, the optimal PMU

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 conventionalinjection 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




to buses 4 and 5. Therefore, the set of buses incident to con-

ventional measurements will be where the

set of buses not incident to conventional measurements will be

. Next, buses 2, 3, 4, and 6 will be augmented into

a newly created bus whereas buses 4, 5, 3, and 7 will be

augmented into a newly created bus The new bus order

and permutation matrix will thus be

The right-hand side vector and the new connectivity matrix

will be

One of the merits of the proposed approach is that it can be easily

programmed in a general format using an effective program-

ming tool such as that of Matlab. Thus, it can be applied for sys-tems of any size and topology. The network connectivity matrix

is built only once based on the original network topology andneed 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 lossin 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 alterna-

tive 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 theprimary set, in order to avoid picking up the same bus which

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 impor-

tant to note that both the primary and backup sets are indepen-

dent, and each of which, standing alone, can render the systemobservable. Therefore, this method is also able to preserve the

system observability under multiple PMU loss, provided thatthese multiple PMU loss occur in either the primary set or the

backup set, but not in both. Numerical experimentation con-ducted on this method shows that the method may fail if multiple

PMU loss occurs which combines lost PMUs from both the pri-

mary and backup sets simultaneously.

2) Proposed Local Redundancy (LR) Method: An alterna-

tive method to account for single or multiple PMU loss is pro-posed in this section. The proposed method attempts to provide

a local PMU redundancy to allow for the loss of PMUs whilepreserving the global network observability. Recall that each

entry in the right-hand side vector of (3) is set to 1 in orderto guarantee the observability of each bus via at least one PMU.Now, consider the constraint in (3). If the right-hand side of

that constraint is changed to 2, it will practically mean that inorder 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 com-

plex voltage of bus will be “reached” by at least two PMUs.

If this concept is extended for all constraints in (3), all elementsof the right-hand side vector will be changed to 2. This di-

rectly 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 restric-

tions 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 forobserving 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 measure-

ment 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

preceding section the cost of all PMUs are assumed to beequal (at 1 p.u.). It is necessary to consider the unequal cost of




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 con-

nections, power connection, station ground connection, and

GPS antenna connection. However, what really distinguishesbetween 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 (mea-

suring 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 chan-

nels is dominant in determining the PMU cost, we may assumethat this cost increases by a decimal increment p.u., for each

additional channel. Then the PMU problem is formulated to de-

termine 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.

2) Form the network connectivity matrix and PMU cost

coefficient vector .

3) Define the array of buses comprising zero injection mea-

surements , and the array of branches comprising

flow measurements ; where

defines the from-to buses where flow measurements exist.

4) Establish the array of associated buses .

5) Establish the array of nonassociated buses ; where

, and is the set of all system buses.

6) Establish the new bus-order vector ;

where is defined as .

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 val-

idate 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 andthe ABM method proposed in this paper. When ignoring con-

ventional measurements (Case 1) the two methods intuitively

yielded identical solution for the optimal PMU number and lo-

cation. In case 2 the zero injection bus was considered. Ac-

cording to the IBM method, different bus merging for the in-

jection bus 3 was performed. The IBM method in this case sug-

gested 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 avail-

able 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 ob-

served 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 avail-

able 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 and4. 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.




TABLE IRESULTS FOR THE SEVEN-BUS SYSTEM WITHOUT CONSIDERING PMU LOSS

TABLE IIRESULTS FOR THE SEVEN-BUS SYSTEM CONSIDERING

SINGLE PMU LOSS (IGNORING ZERO INJECTION)

B. Results Without Conventional Measurements-Without and

With Considering PMUs Loss

As stated before, the PMUs placed by the proposed unifiedapproach are assumed to function perfectly. Sometimes, one or

more of these PMUs may fail to operate and therefore, it is nec-

essary 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 ig-

noring 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 pro-

posed 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&Band 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 place-

ment strategy problem is conducted in this study. As proposed

in Section III,the costof a PMU may beincreasedby 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 sametable. These results indicate that the number of PMUs in case

TABLE IIIRESULTS FOR THE 14-BUS SYSTEM WITHOUT AND WITH CONSIDERING

SINGLE PMU LOSS (IGNORING ZERO INJECTION)

TABLE IVRESULTS FOR THE 14-BUS SYSTEM CONSIDERING UNEQUAL COSTS OF PMUS

(WITHOUT CONSIDERING SINGLE PMU LOSS AND IGNORING ZERO INJECTION)

TABLE VRESULTS FOR THE 14-BUS SYSTEM WITH CONVENTIONAL MEASUREMENTS

of equal and unequal PMU costs is the same (4 PMUs) but thelocations are different. Since the overall cost of PMU metering

system increases with increasing number of measuring chan-

nels, therefore, results of Table IV indicate that the overall cost

of PMU metering system may be substantially affected by con-

sidering 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 singlePMUloss are applied for the purpose of comparison. Thesystem

has only one injection measurement at bus 7 and one flow mea-

surement in branch 5–6. With no PMU loss considered, the op-

timal 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 inter-

changed 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 inTable VI. Each system has a number of injection buses but no




TABLE VIDATA FOR 30-, 57-, AND 118-BUS SYSTEMS

TABLE VIIOPTIMAL NUMBER OF PMUS REQUIRED FOR THE

30-, 57-, AND 118-BUS SYSTEMS

TABLE VIIISIMULATION RESULTS FOR THE 118-BUS SYSTEM

(WITHOUT CONSIDERING SINGLE PMU LOSS)

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 sys-

tems when using the P&B method. As expected, results of thiscase 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 mea-

surements. Three cases are studied. In the first case, five flow

measurements are considered. Case 2 contains ten flow mea-

surements (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 arecarried out 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 measure-

ments increases, the number of PMUs required keeping the

system observable decreases. It can also be noticed, when com-

paring 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 considering15 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 measure-

ments and the possibility of single or multiple PMU loss into thedecision 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 loca-

tions 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 re-

sults are compared with those published in literature with very

good agreement.

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Nabil H. Abbasy was born 1956. He received theB.Sc. (Hons.) and M.Sc. degrees from the Universityof Alexandria, Alexandria, Egypt, in 1979 and 1983,respectively, and the Ph.D. degree from Illinois Insti-tute of Technology, Chicago, in 1988.

He was an Assistant Professor at Clarkson Univer-sity, Potsdam, NY, from 1988 to 1989, and then at theUniversity of Alexandria from 1989 to 1994, where

he has been a Full Professor since 2000. He has beenon leave of absence with the College of Technolog-ical Studies, Kuwait, since 1994. His research inter-

ests include power systems operation, dynamics, and transients.

Hanafy Mahmoud Ismail was born in Cairo, Egypt,in 1956. He received the B.S. and M.S. degrees inelectrical engineering from Electrical EngineeringDepartment, Faculty of Engineering at Ain-ShamsUniversity, Cairo, in 1979 and 1984, respectively.He received the Ph.D. degree in 1989 from Elec-trical Engineering Department at the University of Windsor, Windsor, ON, Canada.

Since graduation, he has been working at theElectrical Engineering Department, Faculty of Engineering, Ain-Shams University. He is now

a Professor, on leave of absence from Ain-Shams University, joining theElectrical Engineering Department at the College of Technological Studies inKuwait since 1997. He deals mainly with the high voltage power transmissionand their associated electrostatics and electromagnetic fields. He is alsoworking in the area of power systems, under ground cables, and fault detectionon transmission lines.

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