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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 bus not 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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1814 IEEE 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.
and can 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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1816 IEEE 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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1818 IEEE 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 APPROACH1819
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 set in
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 set and
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
1820 IEEE 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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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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