# Optimal Multistage Scheduling of PMU Placement: An ILP Approach

**ABSTRACT** This paper addresses various aspects of optimal phasor measurement unit (PMU) placement problem. We propose a procedure for multistaging of PMU placement in a given time horizon using an integer linear programming (ILP) framework. Hitherto, modeling of zero injection constraints had been a challenge due to the intrinsic nonlinearity associated with it. We show that zero injection constraints can also be modeled as linear constraints in an ILP framework. Minimum PMU placement problem has multiple solutions. We propose two indices, viz, BOI and SORI, to further rank these multiple solutions, where BOI is bus observability index giving a measure of number of PMUs observing a given bus and SORI is system observability redundancy index giving sum of all BOI for a system. Results on IEEE 118 bus system have been presented. Results indicate that: (1) optimal phasing of PMUs can be computed efficiently; (2) proposed method of modeling zero injection constraints improve computational performance; and (3) BOI and SORI help in improving the quality of PMU placement.

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**ABSTRACT:**A genetic algorithm (GA) based approach for reliability placement of phasor measurement units (PMUs) in smart grid is proposed. The algorithm combines two conflicting objectives which are maximization of the reliability of observability and minimization of the number of PMU placements for ensuring full system observability. The multi-objective problem is formulated as a nonlinear optimization problem and genetic algorithm approach is employed for solving the large scale bus systems. The optimization model is solved for IEEE 14, 30, 57, 118, and 2383 standard bus systems. The effectiveness of the proposed approach has been demonstrated by comparing results with exact algorithms for smaller problem sizes. The results suggest that by employing genetic algorithm, the system reliability of observability is improved by approximately 48% as compared to traditional optimal PMU placement. According to results, the proposed approach achieve significant cost savings (~17%-~50%) compared to available reliability based models in literature.Journal of Network and Innovative Computing. 06/2014; 2(1):30-40. - SourceAvailable from: Sadegh Azizi[Show abstract] [Hide abstract]

**ABSTRACT:**Placement of phasor measurement units (PMUs) in power systems has often been formulated for achieving total network observability. In practice, however, the installation process is not implemented at once, but at several stages, because the number of available PMUs at each time period is restricted due to financial problems. This paper presents a novel integer linear programming framework for optimal multi-stage PMU placement (OMPP) over a preset schedule, in order to improve the network observability during intermediate stages, in addition to its complete observability by the end of the PMU placement process. To precisely evaluate the network observability and fully exploit the potential of circuit rules, the observability function is formulated by Boolean algebra. Next, it will be utilized to formulate the optimal PMU placement problem and then construct a unified optimization model for OMPP. As such, among various strategies for installing PMUs over the scheduled horizon, the one achieving the maximum total observability of the system during intermediate stages is found. Moreover, the unified optimization model allows for considering the network expansion scenarios as well as the diverse importance of critical buses during the installation process so that these buses may be equipped with PMUs at primary stages. Finally, to illustrate the applicability and efficiency of the proposed method, it is applied to several IEEE test systems and two practical large-scale power systems, and the results are studied. Copyright © 2012 John Wiley & Sons, Ltd.International Transactions on Electrical Energy Systems. 04/2014; 24(4). - SourceAvailable from: Gaurav Khare[Show abstract] [Hide abstract]

**ABSTRACT:**Power system state estimation with exclusive utilization of synchronous phasor measurements demands that the system should be completely observable through PMUs only. To have minimum number of PMUs, the PMU placement issue in any network is an optimization problem. In this paper, a model for the optimal placement of phasor measurement units (PMUs) in electric power networks is presented. Optimal measurement set is determined to achieve full network observability during normal conditions, i.e. no PMU failure or transmission line outage is considered. Observability analysis is carried out using topological observability rules. A connectivity matrix modification based integer linear programming is used as an optimization tool to obtain the minimal number of PMUs and their corresponding locations while satisfying associated constraint. Simulation results for IEEE 30-bus and New England 39-bus test systems are presented and compared with the existing techniques, to justify the effectiveness of proposed method. In All investigations, the zero-injected buses are considered to obtain the best answers. Keywords—Phasor measurment unit; integer linear programming; network observability; optimal placement; network connectivity.International Conference on Circuit, Power and computing technologies, Noorul Islam University, Kumaracoil, Thuckley, Tamilnadu, India; 03/2014

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1812IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008

Optimal Multistage Scheduling of PMU

Placement: An ILP Approach

Devesh Dua, Sanjay Dambhare, Rajeev Kumar Gajbhiye, Student Member, IEEE, and S. A. Soman, Member, IEEE

Abstract—This paper addresses various aspects of optimal

phasor measurement unit (PMU) placement problem. We propose

a procedure for multistaging of PMU placement in a given time

horizon using an integer linear programming (ILP) framework.

Hitherto, modeling of zero injection constraints had been a

challenge due to the intrinsic nonlinearity associated with it. We

show that zero injection constraints can also be modeled as linear

constraints in an ILP framework. Minimum PMU placement

problem has multiple solutions. We propose two indices, viz, BOI

and SORI, to further rank these multiple solutions, where BOI is

Bus Observability Index giving a measure of number of PMUs ob-

serving a given bus and SORI is System Observability Redundancy

Index giving sum of all BOI for a system. Results on IEEE 118

bus system have been presented. Results indicate that: 1) optimal

phasing of PMUs can be computed efficiently; 2) proposed method

of modeling zero injection constraints improve computational

performance; and 3) BOI and SORI help in improving the quality

of PMU placement.

Index Terms—Integer linear programming (ILP), optimal

phasor placement (OPP), phasor measurement unit (PMU), zero

injection measurement.

I. INTRODUCTION

P

Synchronicity in PMU measurements is achieved by time

stamping of voltage and current waveforms using a common

synchronizing signal available from the global positioning

system (GPS). The ability to calculate synchronized phasors

makes PMU one of the most important measuring devices in

future of power system monitoring and control.

Throughout this paper, we presume that a PMU placed on a

bus measures the following parameters:

1) voltage magnitude and phase angle of the bus;

2) branch current phasor of all branches emerging from the

bus.

PMU placement at all substations allows direct measurement

of the state of the network. However, PMU placement on each

bus of a system is difficult to achieve either due to cost factor or

duetononexistenceofcommunicationfacilitiesinsomesubsta-

tions. Moreover, as a consequence of Ohm’s Law, when a PMU

is placed at a bus, neighboring busses also become observable

HASOR Measurement Units (PMUs) provide time syn-

chronized phasor measurements in a power system [1].

ManuscriptreceivedJune12,2007;revisedOctober14,2007.Firstpublished

April 15, 2008; current version published September 24, 2008. This work was

supported by PowerAnser Labs, IIT Bombay, India. Paper no. TPWRD-00354-

2007.

The authors are with the Indian Institute of Technology, Bombay 400076,

India (e-mail: ddua@ee.iitb.ac.in; sanjay_dambhare@iitb.ac.in; rajeev81@ee.

iitb.ac.in; soman@ee.iitb.ac.in).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRD.2008.919046

Fig. 1. IEEE 14 Bus Test System (seventh bus is a zero injection bus).

[2], [3]. This implies that a system can be made observable with

a lesser number of PMUs than the number of busses.

Reference [4] has shown that minimum PMU placement

problem is NP-complete. This implies that no polynomial time

algorithm can be designed to solve the problem exactly. Work

on optimal PMU placement using an integer linear program-

ming (ILP) approach has been pioneered by Abur [5], [6].

The following example illustrates the ILP approach to PMU

placement.

Consider the IEEE-14 bus system shown in Fig. 1. Let

a binary decision variable associated with the bus . Variable

is set to one if a PMU is installed at bus , else it is set to

zero. Then minimum PMU placement problem for IEEE-14 bus

system (Fig. 1) can be formulated as follows:

be

(1)

subject to bus observability constraints defined as follows:

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

0885-8977/$25.00 © 2008 IEEE

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DUA et al.: OPTIMAL MULTISTAGE SCHEDULING OF PMU PLACEMENT: AN ILP APPROACH1813

Bus

Bus

Bus

Bus

(12)

(13)

(14)

(15)

The objective function in (1) is the total number of PMUs re-

quired for complete system observability, which has to be min-

imized. Solution of problem [(1)–(15)] shows that for complete

system observability, a minimum of four PMUs are required at

busses 2, 6, 7, and 9. In fact, the set of busses where the PMUs

have to be installed correspond to a dominating set of the graph.

Hence, minimum PMU placement problem maps to smallest

dominating set problem on the graph. A generic ILP formula-

tionforminimumPMUplacementproblem,henceforthreferred

as OPP, is summarized in Appendix A.

Reference [7] proposed an approach for PMU placement

which requires complete enumeration of trees. Formulation

based on meta heuristics like simulated annealing (SA), genetic

algorithm, tabu search, etc., have been considered in [8]–[10].

A genetic algorithm method using adaptive clonal algorithm

has been proposed in [11].

Beyond the number of PMUs required to make a system ob-

servable, a good PMU placement algorithm must also consider

following additional issues:

1) loss of a PMU or communication line;

2) modeling of zero injection busses;

3) phasing of PMU placement.

When a system is made observable with minimum number of

PMUs, lack of communication channels or a PMU outage itself

will lead to unobservable busses in the system. Hence, loss of

PMU has to be considered in the design stage. In particular, [3],

[6],and[12]considertherequirementthatsystemshouldremain

observable even with one PMU outage/branch outage.

Zero injections busses, which are analogous to transshipment

nodes, have a potential to reduce number of PMUs for complete

system observability. Reference [6] considers modeling of zero

injection constraints in an otherwise ILP framework. In the re-

sulting formulation, observability constraints arising out of zero

injectionbussesturnouttobenonlinear.Thisincreasesthecom-

plexity of discrete optimization problem.

Further pragmatic approach to PMU placement requires

staging/phasing the PMU placement in time. This requirement

arises due to high cost of PMUs, including that of communi-

cation facilities, which has financial implications and do effect

budgetary allocations. To the best of the authors’ knowledge,

this problem has only been addressed in [7]. It uses the concept

of reducing depth of unobservability to phase PMU place-

ment. Depth of unobservability model is not amenable to LP

framework. It is difficult to model constraint related to depth of

unobservability in traditional optimization, i.e., LP framework.

Consequently, the authors have used enumeration of trees

for finding optimal PMU placement and SA formulation for

solving pragmatic phased installation of PMUs. Using this

technique, more PMUs are needed in phasing technique in

comparison to the nonphased installation technique. This is

because intermediate PMU placement solutions are not related

to “one-shot optimal PMU placement solution.”

This paper develops an optimization (ILP) approach for

staging/phasing of PMU placement over a given time horizon.

We propose maximization of number of observed busses at

each stage as an objective during phasing such that at the end

of phasing, the final solution is identical to optimal solution

obtained without phasing. This is a primary contribution of this

paper.

Further, we show that modeling of zero injection busses

in the optimal PMU placement problem can be achieved by

using linear constraints. This is a noteworthy contribution of

this paper. This implies that optimal PMU placement problem

with zero injection busses can now be solved by standard

ILP solvers. Subsequently, we also develop a simple and an

elegant methodology to handle single PMU outage as well as

different system topologies arising due to single line outage in

the system.

Our investigations show that for optimal PMU placement

problem, multiple solutions with same cost exist. To compare

these solutions qualitatively, we introduce a performance index

SORI.Ifa bus isobservedby

observability redundancy index (SORI) is given by

If in a system, multiple optimal solutions exist, then it is worth-

while to choose that optimal solution which further maximizes

observability redundancy index. This important objective is

also introduced in this paper.

This paper is organized as follows. Section II covers the for-

mulation of phasing of PMUs. Section III deals with modeling

ofzeroinjectionbusses,andSectionIVexplainsmaximizingre-

dundancyinobservability.InSectionV,wepresentcasestudies

involving IEEE 14, 57, and 118 bus systems. Section VI con-

cludes the paper.

numberofPMUs,thensystem

.

II. PHASING OF PMUS

Let the minimum number of PMUs required for guaranteeing

system observability be

. This number is obtained by solving

the problem OPP (refer to Appendix A). Let the set of nodes

which have been selected for PMU installation be given by

In other words,

for every bus belonging to set

for every busnot in

of phasing PMU placement into

the th time horizon,

PMUs are installed.1The problem of

PMU phasing involves finding

i.e.,

, such that their union generates the set

i.e.,

.

and

. Now consider the problem

time horizon such that in

nonintersecting subsets of,

(16)

In other words, all PMUs identified in set

end of

time horizons. Also, the set of PMUs available for in-

stallation at the th time horizon is given by2

are installed at the

(17)

It is obvious that if PMUs obtained from minimum PMU place-

ment formulation are phased over multiple time horizon, then

1? ? ? ? ??? ? ? ? ? .

2Initially?? ? ?.

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1814IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008

the power system becomes completely observable only in the

last time horizon . In the intermediate horizons, the system is

only partially observable.This suggests that, in theintermediate

period,weshouldchoose

insuchawaysoastomaximizethe

number of observable busses in the system. Also, we notice that

constraint set given by (2)–(15) will be infeasible in interme-

diate phases. Hence, to model partial observability constraints,

let us introduce a variable

known as bus observability confir-

mation variable with bus . If

is observable while if , then it implies that the bus is un-

observable. Now for the first

phasing can be obtained by solving

lems as explained below.3

We consider the IEEE 14 bus system (Fig. 1). Recall that for

complete system observability, a minimum of four PMUs are

required at busses 2, 6, 7, and 9. Let the placement be phased

over a three-year horizon with, say, one, two, and one PMUs to

be installed in first, second, and third year, respectively.4Then

,,

Formulation for various phases is as follows.

Phase-I: We define objective function (18) to maximize the

number of observable busses

, it implies that the bus-

time horizons, the optimal

subsidiary ILP prob-

, , and.

(18)

Placement of the only PMU in phase-I can be on any one of

the four busses, i.e., bus number 2, 6, 7, or 9. Thus

(19)

All other

are set to zero. Therefore, constraints governing the observ-

ability for phase-I can be obtained by modifying inequalities

(2)–(15) as follows:

’s, i.e.,,,,,,,,, , and

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

3PMUs in the optimal set which are not installed in the first ??? phases must

be installed in the last phase. Hence, no separate optimization problem has to

be solved for the last phase.

4In practice, a better phasing scheme would be 2-1-1.

Forexample,inequality(20)impliesthatif

Thus, bus-1 will remain unobservable if PMU is not installed at

bus 2. However, if

, i.e., PMU is installed at bus 2, then

can take a value of unity, i.e., bus-1 becomes observable.

Similarly, inequality (23) implies that bus-4 can be made ob-

servable by placement of one or more PMUs among the busses

2,7,and9.Thus,constraints(20)–(33)signifythatallthebusses

of the system can be made observable from PMU locations on

busses 2, 6, 7, and 9. Also, variables

nary values, i.e.,

,then.

andcan take only bi-

Solving the optimization problem (18), (19), (20)–(33) using

the ILP solver leads to the PMU placement on bus 2, i.e.,

.

Phase-II: After phase-I, busses 1, 2, 3, 4, and 5 become

observable. Hence, in phase-II these busses are omitted from

the objective function; for maximizing observability for the re-

maining busses, the objective function is as follows:

(34)

TheconstraintthatonlytwoPMUscanbeplacedinthisphase

among the three busses 6, 7, or 9 can be modeled by

the following equality constraint:

(35)

Further, we can ignore the constraints (20)–(24) which are

anyway satisfied after the phase I. Solving the ILP formulation

(34), (35), and (25)–(33) leads to the PMU placement on busses

6 and 9, i.e.,

.

Phase-III: All remaining PMUs from set

phase. We conclude that

henceforth referred as phasing for the subsidiary ILP formula-

tion is given in Appendix B.

are placed in this

. A generic ILP formulation,

III. MODELING OF ZERO INJECTIONS

Zero injection correspond to the transhipment nodes in the

system. At zero injection busses, no current is injected in to the

system. If zero injection busses are also modeled in the PMU

placement problem, the total number of PMUs can further be

reduced. To understand this issue, consider a four bus example

shown in Fig. 2. Fig. 2(a) ignores information about the zero in-

jectionbusseswhileFig.2(b)showsthebus2asa zeroinjection

bus.ForsysteminFig.2(a),itcaneasilybeseenthataminimum

of two PMUs are required to make the system completely ob-

servable. These can be placed on any two of the four busses.

For example, if a PMU is placed on bus 1, another PMU is re-

quired to make bus 4 observable. In contrast consider system in

Fig. 2(b). With a PMU at bus 1, current in branch 2–4 also be-

comes knownas the bus 2is a zero injection bus,i.e.,

.

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DUA et al.: OPTIMAL MULTISTAGE SCHEDULING OF PMU PLACEMENT: AN ILP APPROACH1815

Fig. 2. Optimal PMU placement for a four-bus system (a) neglecting zero in-

jection constraints and (b) considering bus-2 as zero injection bus.

Hence, knowing the line parameters, the voltage at bus 4 can be

calculated5as:

Hence, a separate PMU is not required at bus 4 for system of

Fig. 2(b). Therefore, it is seen that presence of zero injections

can help in reducing total number of PMUs required to observe

the system.

Modeling of zero injection busses in ILP framework has

remained a challenge. Reference [6] follows an approach re-

quiring non linear framework. We now propose a methodology

to model these constraints within a linear framework. Consider

a zero injection bus as shown in Fig. 3. If busses 1 to

are observable, i.e., their voltage phasors are known, then

either current

is available directly from a PMU or it can be

calculated as follows:

where

quently, bus

voltage as follows:

isthelineadmittancebetweenbus1andbus .Conse-

can also be made observable by calculating bus

where

zero injection node leads to one additional constraint. Hence, in

the best case, the minimum number of PMUs required to ob-

serve the system can be further reduced by total number of zero

injection busses in the system.

The way to model zero injection busses in the ILP frame-

work is to selectively allow for existence of some unobserv-

ablebusseseveninthetraditionalsingle-stageformulation(e.g.,

OPP). However, we impose following additional constraints.

1) Unobservable busses, if any, must belong to the cluster of

zero injection busses and busses adjacent to zero injection

busses.

isthelineimpedancebetweenbusses1and.Every

5For simplicity, we have assumed series branch model. However, the descrip-

tion applies equally for ? model of transmission line.

Fig. 3. Modeling of zero injection busses.

2) For a zero injection bus- , let

adjacent to bus- . Let

unobservable busses in each cluster defined by set

a zero injection bus- and its adjacent busses) is at most

one.

For the sake of illustration, consider the IEEE 14 bus system

(Fig. 1) which has bus-7 as a zero injection bus. Thus, set

and set

to be modeled in the ILP formulation becomes

indicate the set of busses

. Then, number of

(i.e.,

. Thus, additional constraint

i.e., out of the four busses 4, 7, 8, and 9, we must have, at least,

three busses observable. Thus, the modified ILP formulation in-

corporating zero injection constraints is given by

(36)

subjecttobusobservabilityandzeroinjectionconstraintsasfol-

lows:

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

Bus

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

Solving the ILP problem (36), (37)–(51) we identify busses

2, 6, and 9 for PMU placement. Hence, it is seen that when

zeroinjectionconstraintsaremodeled,completesystemobserv-

ability is achieved with just three PMUs, i.e., minimum number

of PMUs required for system observability has reduced by one.

A generic formulation for modeling zero injection constraints,

referred as OPP-Z, is detailed in Appendix C.

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1816IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008

Fig. 4. A six-bus system with system observability (a) 6 and (b) 8.

IV. MAXIMIZING REDUNDANCY IN OBSERVABILITY

If the minimum PMU placement problem defined by formu-

lation-OPP has multiple number of optimal solutions, then the

question of superiority of a particular solution vis-a-vis other

optimal solution arises. In this section, we propose Bus Observ-

ability Index (BOI) as a performance indicator on quality of the

optimization. Let us define BOI for bus-

of PMUs which are able to observe a given bus. Consequently,

maximum bus observability index is limited to maximum con-

nectivity

of a bus plus one, i.e.,

as the number

Now we define SORI as the sum of bus observability for all the

busses of a system. Then

where

in Fig. 4. It is seen that a minimum of two PMUs are required

to ascertain system observability. Consider two such optimal

solutions shown in Fig. 4.

For the PMU placement as given in Fig. 4(a), BOI

busses 1 to 6 are 1, 2, 1, 1, 2, and 1, respectively. This makes

SORI,

. Alternatively, for PMU placement in Fig. 4(b),

BOIforbusses1to6areunity,making

placement with maximum SORI in Fig. 4(a) should be chosen

for final placement.

Maximizing SORI has the advantage that a larger portion of

system will remain observable in case of a PMU outage. For

example, in Fig. 4(a), one PMU outage will result in loss of

observability of two busses, as against three busses remaining

unobservable for loss of single PMU for system in Fig. 4(b).

After the solution of the minimum PMU placement problem

given by formulation OPP, index SORI can be maximized by

solving a slave ILP problem where we maximize

constraints of OPP and additional linear equality constraint that

number of PMUs in the solution should be restricted to number

, whereis the minimum number of PMUs obtained for

complete observability as per master problem-OPP. This for-

mulation, referred as Max Obs is detailed in Appendix D. We

now consider the problem of modeling PMU outage.

represents SORI. Consider a six-bus system shown

for

.Hence,thePMU

subject to

Fig. 5. Four cases resulting from the formulated problems.

A. PMU Outage

To enhance the reliability of system monitoring, each bus

should be observed by at least two PMUs. This ascertains that

a PMU outage will not lead to loss of observability. In the ILP

framework, this can be achieved with ease by multiplying the

right hand side of theinequalities in (2)–(15) by2. Forexample,

in case of the IEEE 14 bus system, bus-1 will be made observ-

able from at least two busses if we replace inequality (2) in for-

mulation OPP by the following inequality:

(52)

The corresponding modifications in the generic formulation

OPP are described in Appendix E.

V. CASE STUDIES

Case studies for multistage scheduling of PMU placement

have been carried out for the IEEE 14 bus system, IEEE 57 bus

system, and IEEE 118 bus system. Tomlab’s ILP solver6has

been used for this purpose. The simulations are carried out for

various scenarios as summarized in Fig. 5.

A. Modeling of Zero Injection Busses

Zero injection busses considered for various systems are

6Tomlab Optimization, Inc. [Online]. Available: http://www.tomopt.com/

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DUA et al.: OPTIMAL MULTISTAGE SCHEDULING OF PMU PLACEMENT: AN ILP APPROACH1817

TABLE I

RESULTS-OPP: MINIMUM NO OF PMUS

TABLE II

RESULTS-Phasing: NUMBER OF OBSERVABLE BUSSES ?

? ?

TABLE III

RESULTS-Max Obs: SYSTEM OBSERVABILITY REDUNDANCY INDEX (INITIAL

SORI ? FINAL SORI)

TableIbringsoutresultsofformulation-OPPundertheback-

drop of zero injections and PMU outage. It is observed that the

number of PMUs almost doubles if the system observability is

to be maintained after single PMU loss. Considering zero injec-

tions with no PMU outage scenario, the number of PMUs re-

quired for system observability reduces, at most, by the number

of zero injection busses (Table I, s.no-1, column-3 and 4). Sim-

ilarly, considering zero injections while maintaining system ob-

servability for a single PMU outage, the number of PMUs re-

quired reduces, at most, by twice the number of zero injection

busses (Table I, s.no-1, column-5 and 6).

B. Phasing

For the optimal PMU placement scenario, given by column 3

of Table I, let the number of PMUs to be installed in the first,

second, and third phase be as follows:

bus system

bus system

bus system

The problem of optimal phasing (formulation Phasing in

Appendix B) is solved. Table II shows number of busses made

observable at the end of each phase. At the end of last phase,

the systems achieve completely observability. It is seen that

number of additional busses made observable is maximum for

initial phase and least for the final phase. This is consistent with

law of diminishing marginal utility.

Table III shows results after solving the slave problem-Max

Obs. It compares the results with those obtained from master

formulation-OPP. It is observed that redundancy in system

observability is enhanced significantly by solving the slave

problem.

Fig. 6. Results-Phasing: considering single PMU outage.

TABLE IV

COMPUTATIONAL TIME (IN SECONDS): 118 BUS SYSTEM

Fig. 6 brings out the results of formulation-Phasing while

considering a single PMU outage. For every phase, each of the

three bars correspond to SORI busses

observable busses

, and total number of busses that are

observable from at least two nodes. It is observed that complete

system observability is possible even before the last phase. For

example,14bussystem achievescomplete systemobservability

after phase-II. However, the system observability is guaranteed

from at least two nodes, only in the last phase. It is to be noted

here that number of busses having

number of busses having BOI equal to 1 after every phase. This

indicates efficacy of formulation-Max Obs.

, total number of

is greater than

C. Computational Evaluation

The simulationshavebeen run on computerhaving following

configuration:

CPU—Pentium(R) IV 3.00 GHz;

Level L2 Cache—2 MB

System Memory—1-GB RAM.

Table IV gives the CPU time for various formulations on the

IEEE 118 bus system.

For the sake of comparative evaluation, the approach pre-

sented in [6], which requires nonlinear constraints, has been

implemented using Tomlab optimization toolbox. Both our

proposed method and method of [6] lead to same number of

PMUs for system observability in presence of zero injection

constraints, thereby validating the proposed method. For IEEE

118 bus system, the CPU time taken is 1.1250 s as against

0.0312 s (Table IV, s.no. 2, column 4) for the proposed formu-

lation. Thus, it is seen that modeling of zero injection as linear

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1818IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008

Fig. 7. IEEE 57 Bus System: Location of 14 PMUs for complete system ob-

servability (considering zero injection busses).

constraints (formulation-OPP-Z) has reduced computational

burden by 36 times.

Finally, Fig. 7 illustrates placement of PMUs on IEEE 57 bus

system. For the zero injection bus 4, currents

are knownas aconsequenceofPMUs onbusses76and 15.Con-

sequently, current

is calculated using Kirchhoff’s current

law, thereby making bus 18 observable. Note that neither bus

18 nor its adjacent busses 4 or 19 have a PMU. This shows how

zero injection busses can extend observability to a neighboring

bus.

, , and

VI. CONCLUSION

Salient contributions of this paper as follows.

1) Algorithm for optimal multistage scheduling of PMU

placement has been devised. It ascertains that final place-

ment obtained by phasing is identical to one obtained

without imposing phasing constraints.

2) Zero injectionconstraints hasa capabilitytofurtherreduce

PMU requirement. We develop a linear model for zero in-

jection constraints.

7Due to PMUs at busses 6 and 15, voltage phasors at busses 3, 4, 5, and 6 are

known. Hence, respective branch current can be computed.

3) Optimal PMU placement has multiple solutions. We pro-

pose BOI and SORI and show thatsolution with maximum

SORI outscores other optimal solutions.

Results on IEEE 14, 57, and 118 bus systems demonstrate

the claim made. The proposed model is free of unwarranted

complexities. This claim is vindicated by the excellent compu-

tational performance of the algorithm.

APPENDIX

A. Formulation: OPP

For an

elements

bus system, if the PMU placement vector

defines possibility of PMUs on a bus, i.e.,

having

if a pmu is installed at bus

otherwise

and if

of PMU at bus , then a weighted minimum PMU placement

problem can be defined as follows.

Formulation: OPP

is the weight (or cost) associated with placement

(53)

subject to the following constraints:

(54)

where

is a unit vector of length , i.e.,

and

and is the binary connectivity matrix of the system, i.e.,

if either

are adjacent nodes;

otherwise.

or if and

Formulation OPP is a classical ILP formulation. Minimum

PMU placement problem is obtained by setting all weights to

unity.

B. Formulation: Phasing

We now consider the problem of phasing the placement of

PMUs obtained from the solution of OPP. Phasing over time

horizons require solution of

lems. Asubsidiary optimization problem for PMU placement in

th stage can be formulated as follows.

Formulation: Phasing

subsidiary optimization prob-

(55)

such that

(56)

(57)

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DUA et al.: OPTIMAL MULTISTAGE SCHEDULING OF PMU PLACEMENT: AN ILP APPROACH 1819

and

(58)

(59)

where we set

nodes made observable by placement of

the th stage. Inequality (56) models the constraint of incom-

plete observability due to availability of fewer PMUs than the

minimum required. If a node

then (56) forces corresponding

node

is observable, i.e.,

to 1, which is consistent with (56). Constraint (57) imposes

condition that nodes which do not have PMUs in solution to the

original formulation OPP will also not have PMUs in the inter-

mediate steps. Constraint (58) limits the number of new PMUs

installed in stage

to. Constraint (59) honors the commit-

ment made on placement of PMUs in the previous stages.

1) Reduced Formulation: From a computational perspec-

tive, formulation-Phasing can be improvised by reducing the

number of constraints and variables, as follows.

1) We can drop all columns of

in (56). Simultaneously, all such

from the list of variables.

2) Forthevariableswhichhavebeensetto“1”intheprevious

phase(s), we can drop the corresponding columns as well

astherows(equations)inwhichthesevariablesparticipate.

This can be reasoned out as follows. Assuming that

been set to “1” in the previous phase, let us consider th

row of (56), i.e.,

. Objective (55) maximizes the number of

PMU from setin

is unobservable, i.e.,

to zero. On the other hand, if

, then objective (55) drives

,

corresponding to

’s will also be dropped

has

Now if

cance,i.e.,

contrast,if

implies that

th equation from (54) and also variable from subsidiary

formulations. In essence, th column of

3) Thelaststageofformulationneednotbesolvedbecauseall

PMUs from

, not installed until the last stage, will have

to be installed in the last stage.

, then information

irrespectiveofwhether

,itimpliesthat

had been set to “1.” Hence, we can remove

is of no signifi-

is1or0.In

,whichinturn

is dropped.

C. Formulation OPP-Z

Zero injection constraints can be modeled in the ILP frame-

work as follows.

Model: OPP-Z

(60)

subject to:

(61)

and

(62)

and

(63)

or

(64)

where

the set

sents the

busses of binary connectivity matrix

that busses which are not adjacent to zero injection busses are

definitely made observable.

, represents cardinality of setand

contains indices for zero injection busses.

row, i.e., row corresponding to the zero injection

repre-

. Constraint (62) ensures

D. Formulation: Max Obs

Index SORI which measures the redundancy in system ob-

servability can be expressed by a linear equation as follows:

To solve the problem of maximizing SORI, while guaran-

teeing system observability, with minimum number of PMUs,

we solve the following slave problem.

Formulation: Max Obs

(65)

subject to the following constraints:

(66)

(67)

where

plete observability as per master problem-OPP.

is the minimum number of PMUs obtained for com-

E. PMU Outage

Outage of a PMU should not lead to partial loss of observ-

ability. This can be modeled by modifying the constraints given

by (54) to

(68)

1) Line Outage: If two PMUs are observing a bus, then

a related line outage will not affect the node observability.

Hence, the problem of ascertaining observability under single

line outage is a subset of the problem of single PMU loss

considered above. However, if a user wants to model different

topologies arising out of contingencies, but does not want to

guarantee observability in the event of PMU outage, then we

write constraint equations as follows:

for bus which is not affected by contingency, else

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Page 9

1820IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008

ACKNOWLEDGMENT

The authors would like to thank Prof. A. G. Phadke for

bringing the importance of the phasing problem to our notice.

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101–107, Jan. 2007.

DeveshDuaiscurrentlypursuingtheM.Tech.degreeintheDepartmentofElec-

trical Engineering, Indian Institute of Technology, Bombay, India.

Sanjay Dambhare is currently pursuing the Ph.D. degree in the Department of

Electrical Engineering, Indian Institute of Technology, Bombay, India.

His research interests include power system protection, power system anal-

ysis, and FACTS.

Rajeev Kumar Gajbhiye (S’07) is currently pursuing the Ph.D. degree in the

Department of Electrical Engineering, Indian Institute of Technology, Bombay,

India. His research interests include large-scale power system analysis, power

system protection, and deregulation.

S. A. Soman (M’07) is a Professor in the Department of Electrical Engineering,

Indian Institute of Technology Bombay, India. He has authored a book on Com-

putational Methods for Large Sparse Power System Analysis: An Object Ori-

ented Approach (Kluwer, 2001). His research interests and activities include

power system analysis, deregulation, and power system protection.

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