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IEEE 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 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.
Index Terms—Integer linear programming (ILP), optimal
phasor placement (OPP), phasor measurement unit (PMU), zero
injection measurement.
I. INTRODUCTION
HASOR Measurement Units (PMUs) provide time synchronized phasor measurements in a power system [1].
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
due to nonexistence of communication facilities in some substations. Moreover, as a consequence of Ohm’s Law, when a PMU
is placed at a bus, neighboring busses also become observable
P
Manuscript received June 12, 2007; revised October 14, 2007. First published
April 15, 2008; current version published September 24, 2008. This work was
supported by PowerAnser Labs, IIT Bombay, India. Paper no. TPWRD-003542007.
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 programming (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 be
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:
(1)
subject to bus observability constraints defined as follows:
Bus
Bus
Bus
Bus
Bus
Bus
Bus
Bus
Bus
Bus
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(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
DUA et al.: OPTIMAL MULTISTAGE SCHEDULING OF PMU PLACEMENT: AN ILP APPROACH
Bus
Bus
Bus
Bus
(12)
(13)
(14)
(15)
The objective function in (1) is the total number of PMUs required for complete system observability, which has to be minimized. 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 formulation for minimum PMU placement problem, henceforth referred
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 observable, 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] consider the requirement that system should remain
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 resulting formulation, observability constraints arising out of zero
injection busses turn out to be nonlinear. This increases the complexity 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 communication 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 placement. 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.”
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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. If a bus is observed by number of PMUs, then system
observability redundancy index (SORI) is given by
.
If in a system, multiple optimal solutions exist, then it is worthwhile 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 formulation of phasing of PMUs. Section III deals with modeling
of zero injection busses, and Section IV explains maximizing redundancy in observability. In Section V, we present case studies
involving IEEE 14, 57, and 118 bus systems. Section VI concludes the paper.
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 .
for every bus belonging to set and
In other words,
for every bus not in . Now consider the problem
of phasing PMU placement into time horizon such that in
the th time horizon,
PMUs are installed.1 The problem of
PMU phasing involves finding nonintersecting subsets of ,
, such that their union generates the set
i.e.,
i.e.,
(16)
In other words, all PMUs identified in set are installed at the
end of time horizons. Also, the set of PMUs available for installation at the th time horizon is given by2
(17)
It is obvious that if PMUs obtained from minimum PMU placement formulation are phased over multiple time horizon, then
1
+
2Initially
+
S~
+
= S.
111
=
.
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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 the intermediate
period, we should choose in such a way so as to maximize the
number of observable busses in the system. Also, we notice that
constraint set given by (2)–(15) will be infeasible in intermediate phases. Hence, to model partial observability constraints,
let us introduce a variable known as bus observability confir, it implies that the busmation variable with bus . If
is observable while if
, then it implies that the bus is untime horizons, the optimal
observable. Now for the first
phasing can be obtained by solving
subsidiary ILP problems 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.4 Then
,
,
,
, and
.
Formulation for various phases is as follows.
Phase-I: We define objective function (18) to maximize the
number of observable busses
For example, inequality (20) implies that if
, then
.
Thus, bus-1 will remain unobservable if PMU is not installed at
, i.e., PMU is installed at bus 2, then
bus 2. However, if
can take a value of unity, i.e., bus-1 becomes observable.
Similarly, inequality (23) implies that bus-4 can be made observable by placement of one or more PMUs among the busses
2, 7, and 9. Thus, constraints (20)–(33) signify that all the busses
of the system can be made observable from PMU locations on
busses 2, 6, 7, and 9. Also, variables and can take only binary values, i.e.,
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 remaining busses, the objective function is as follows:
(34)
(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
The constraint that only two PMUs can be placed in this phase
among the three busses 6, 7, or 9 can be modeled by
the following equality constraint:
(19)
(35)
,
,
,
, and
All other ’s, i.e., , , , , ,
are set to zero. Therefore, constraints governing the observability for phase-I can be obtained by modifying inequalities
(2)–(15) as follows:
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 are placed in this
phase. We conclude that
. A generic ILP formulation,
henceforth referred as phasing for the subsidiary ILP formulation is given in Appendix B.
Bus
Bus
Bus
Bus
Bus
Bus
Bus
Bus
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
Bus
Bus
Bus
Bus
Bus
Bus
(28)
(29)
(30)
(31)
(32)
(33)
0
3PMUs in the optimal set which are not installed in the first t 1 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.
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 injection busses while Fig. 2(b) shows the bus 2 as a zero injection
bus. For system in Fig. 2(a), it can easily be seen that a minimum
of two PMUs are required to make the system completely observable. These can be placed on any two of the four busses.
For example, if a PMU is placed on bus 1, another PMU is required 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 known as the bus 2 is a zero injection bus, i.e.,
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DUA et al.: OPTIMAL MULTISTAGE SCHEDULING OF PMU PLACEMENT: AN ILP APPROACH
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Fig. 3. Modeling of zero injection busses.
Fig. 2. Optimal PMU placement for a four-bus system (a) neglecting zero injection constraints and (b) considering bus-2 as zero injection bus.
Hence, knowing the line parameters, the voltage at bus 4 can be
calculated5 as:
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 requiring 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
is the line admittance between bus 1 and bus . Consequently, bus can also be made observable by calculating bus
voltage as follows:
where
is the line impedance between busses 1 and . Every
zero injection node leads to one additional constraint. Hence, in
the best case, the minimum number of PMUs required to observe 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 framework is to selectively allow for existence of some unobservable busses even in the traditional single-stage formulation (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.
5For simplicity, we have assumed series branch model. However, the description applies equally for model of transmission line.
indicate the set of busses
2) For a zero injection bus- , let
adjacent to bus- . Let
. Then, number of
unobservable busses in each cluster defined by set (i.e.,
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
. Thus, additional constraint
to be modeled in the ILP formulation becomes
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 incorporating zero injection constraints is given by
(36)
subject to bus observability and zero injection constraints as follows:
Bus
Bus
Bus
Bus
Bus
(37)
(38)
(39)
(40)
(41)
Bus
Bus
Bus
Bus
Bus
Bus
Bus
Bus
Bus
(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
zero injection constraints are modeled, complete system observability 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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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 formulation-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 Observability Index (BOI) as a performance indicator on quality of the
as the number
optimization. Let us define BOI for busof PMUs which are able to observe a given bus. Consequently,
maximum bus observability index is limited to maximum conof a bus plus one, i.e.,
nectivity
Fig. 5. Four cases resulting from the formulated problems.
A. PMU Outage
Now we define SORI as the sum of bus observability for all the
busses of a system. Then
where
represents SORI. Consider a six-bus system shown
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
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
. Alternatively, for PMU placement in Fig. 4(b),
SORI,
. Hence, the PMU
BOI for busses 1 to 6 are unity, 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 subject to
constraints of OPP and additional linear equality constraint that
number of PMUs in the solution should be restricted to number
, where
is the minimum number of PMUs obtained for
complete observability as per master problem-OPP. This formulation, referred as Max Obs is detailed in Appendix D. We
now consider the problem of modeling 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 the inequalities in (2)–(15) by 2. For example,
in case of the IEEE 14 bus system, bus-1 will be made observable from at least two busses if we replace inequality (2) in formulation 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 solver6 has
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 APPROACH
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TABLE I
RESULTS-OPP: MINIMUM NO OF PMUS
TABLE II
RESULTS-Phasing: NUMBER OF OBSERVABLE BUSSES (
u
)
Fig. 6. Results-Phasing: considering single PMU outage.
TABLE III
RESULTS-Max Obs: SYSTEM OBSERVABILITY REDUNDANCY INDEX (INITIAL
FINAL SORI)
SORI
!
Table I brings out results of formulation-OPP under the backdrop 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 injections with no PMU outage scenario, the number of PMUs required for system observability reduces, at most, by the number
of zero injection busses (Table I, s.no-1, column-3 and 4). Similarly, considering zero injections while maintaining system observability for a single PMU outage, the number of PMUs required 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.
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
, total number of
three bars correspond to SORI busses
, and total number of busses that are
observable busses
observable from at least two nodes. It is observed that complete
system observability is possible even before the last phase. For
example, 14 bus system achieves complete system observability
after phase-II. However, the system observability is guaranteed
from at least two nodes, only in the last phase. It is to be noted
is greater than
here that number of busses having
number of busses having BOI equal to 1 after every phase. This
indicates efficacy of formulation-Max Obs.
C. Computational Evaluation
The simulations have been run on computer having 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 presented 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 formulation. Thus, it is seen that modeling of zero injection as linear
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3) Optimal PMU placement has multiple solutions. We propose BOI and SORI and show that solution 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 computational performance of the algorithm.
APPENDIX
A. Formulation: OPP
For an
elements
bus system, if the PMU placement vector having
defines possibility of PMUs on a bus, i.e.,
if a pmu is installed at bus
otherwise
is the weight (or cost) associated with placement
and if
of PMU at bus , then a weighted minimum PMU placement
problem can be defined as follows.
Formulation: OPP
(53)
subject to the following constraints:
(54)
where
is a unit vector of length , i.e.,
Fig. 7. IEEE 57 Bus System: Location of 14 PMUs for complete system observability (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
,
, and
system. For the zero injection bus 4, currents
are known as a consequence of PMUs on busses7 6 and 15. Conis calculated using Kirchhoff’s current
sequently, 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
and
is the binary connectivity matrix of the system, i.e.,
if either
or if and
are adjacent nodes;
otherwise.
Formulation OPP is a classical ILP formulation. Minimum
PMU placement problem is obtained by setting all weights to
unity.
B. Formulation: Phasing
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 placement obtained by phasing is identical to one obtained
without imposing phasing constraints.
2) Zero injection constraints has a capability to further reduce
PMU requirement. We develop a linear model for zero injection constraints.
We now consider the problem of phasing the placement of
PMUs obtained from the solution of OPP. Phasing over time
subsidiary optimization probhorizons require solution of
lems. A subsidiary optimization problem for PMU placement in
th stage can be formulated as follows.
Formulation: Phasing
(55)
such that
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.
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(56)
(57)
DUA et al.: OPTIMAL MULTISTAGE SCHEDULING OF PMU PLACEMENT: AN ILP APPROACH
and
1819
and
(58)
(63)
(59)
where we set
. Objective (55) maximizes the number of
nodes made observable by placement of
PMU from set in
the th stage. Inequality (56) models the constraint of incomplete observability due to availability of fewer PMUs than the
,
minimum required. If a node is unobservable, i.e.,
then (56) forces corresponding to zero. On the other hand, if
, then objective (55) drives
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 intermediate steps. Constraint (58) limits the number of new PMUs
installed in stage to . Constraint (59) honors the commitment made on placement of PMUs in the previous stages.
1) Reduced Formulation: From a computational perspective, formulation-Phasing can be improvised by reducing the
number of constraints and variables, as follows.
corresponding to
1) We can drop all columns of
in (56). Simultaneously, all such ’s will also be dropped
from the list of variables.
2) For the variables which have been set to “1” in the previous
phase(s), we can drop the corresponding columns as well
as the rows (equations) in which these variables participate.
This can be reasoned out as follows. Assuming that has
been set to “1” in the previous phase, let us consider th
row of (56), i.e.,
or
(64)
,
represents cardinality of set
and
where
reprethe set contains indices for zero injection busses.
row, i.e., row corresponding to the zero injection
sents the
busses of binary connectivity matrix . Constraint (62) ensures
that busses which are not adjacent to zero injection busses are
definitely made observable.
D. Formulation: Max Obs
Index SORI which measures the redundancy in system observability can be expressed by a linear equation as follows:
To solve the problem of maximizing SORI, while guaranteeing system observability, with minimum number of PMUs,
we solve the following slave problem.
Formulation: Max Obs
(65)
subject to the following constraints:
(66)
(67)
Now if
, then information
is of no signifiirrespective of whether is 1 or 0. In
cance, i.e.,
, it implies that
, which in turn
contrast, if
implies that had been set to “1.” Hence, we can remove
th equation from (54) and also variable from subsidiary
formulations. In essence, th column of is dropped.
3) The last stage of formulation need not be solved because all
PMUs from , not installed until the last stage, will have
to be installed in the last stage.
is the minimum number of PMUs obtained for comwhere
plete observability as per master problem-OPP.
E. PMU Outage
Outage of a PMU should not lead to partial loss of observability. This can be modeled by modifying the constraints given
by (54) to
(68)
C. Formulation OPP-Z
Zero injection constraints can be modeled in the ILP framework as follows.
Model: OPP-Z
(60)
subject to:
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:
(61)
for bus which is not affected by contingency, else
and
(62)
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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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Devesh Dua is currently pursuing the M.Tech. degree in the Department of Electrical 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 analysis, 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 Computational Methods for Large Sparse Power System Analysis: An Object Oriented Approach (Kluwer, 2001). His research interests and activities include
power system analysis, deregulation, and power system protection.
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