Optimal Multistage Scheduling of PMU Placement: An ILP Approach

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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: [email protected]; [email protected]; rajeev81@ee. iitb.ac.in; [email protected]). 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

+ + + = S~ = S . 111

.

2Initially

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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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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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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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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.


(56) (57)


and

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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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ACKNOWLEDGMENT The authors would like to thank Prof. A. G. Phadke for bringing the importance of the phasing problem to our notice. REFERENCES [1] A. G. Phadke, “Synchronised phasor measurements in power systems,” IEEE Comput. Applicat. Power, vol. 6, no. 2, pp. 10–15, Apr. 1993. [2] X. Dongjie, H. Renmu, W. Peng, and X. Tao, “Comparision of several PMU placement algorithms for state estimation,” in Proc. Inst. Elect. Eng.E Int. Conf. Develop. Power Syst. Protection, Apr. 2004, pp. 32–35. [3] G. B. Denegri, M. Invernizzi, and F. Milano, “A security oriented approach to PMU positioning for advanced monitoring of a transmission grid,” in Proc. IEEE Int. Conf. Power Syst. Technol. , Oct. 2002, vol. 2, pp. 798–803. [4] D. J. Brueni and L. S. Heath, “The PMU placement problem,” SIAM J. Discrete Math., vol. 19, no. 3, pp. 744–761, Dec. 2005. [5] A. Abur and F. H. Magnago, “Optimal meter placement for maintaining observability during single branch outages,” IEEE Trans. Power Syst., vol. 14, no. 4, pp. 1273–1278, Nov. 1999. [6] B. Xu and A. Abur, “Observability analysis and measurement placement for systems with PMUs,” in IEEE Power Syst. Conf. Expo., Oct. 2004, vol. 2, pp. 943–946. [7] R. F. Nuqui and A. G. Phadke, “Phasor measurement unit placement techniques for complete and incomplete observability,” IEEE Trans. Power Del., vol. 20, no. 4, pp. 2381–2388, Oct. 2005. [8] T. L. Baldwin, L. Mili, M. B. Boisen, and R. Adapa, “Power system observability with minimal phasor measurement placement,” IEEE Trans. Power Syst., vol. 8, no. 2, pp. 707–715, May 1993. [9] K. S. Cho, J. R. Shin, and S. H. Hyun, “Optimal placement of phasor measurement units with GPS receiver,” in IEEE Power Eng. Soc. Winter Meeting, vol. 1, pp. 258–262.

[10] B. Milosevic and M. Begovic, “Nondominated sorting genetic algorithm for optimal phasor measurement placement,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 69–75, Feb. 2003. [11] X. Bian and J. Qiu, “Adaptive clonal algorithm and its application for optimal PMU placement,” in Proc. IEEE Int. Conf. Commun. , Circuits, Syst., 2006, vol. 3, pp. 2102–2106. [12] C. Rakpenthai, S. Premrudeepreechacharn, S. Uatrongjit, and N. R. Watson, “An optimal PMU placement method against measurement loss and branch outage,” IEEE Trans. Power Del., vol. 22, no. 1, pp. 101–107, Jan. 2007. 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.
