Electrical Power and Energy Systems 42 (2012) 71–77

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Electrical Power and Energy Systems

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An optimal PMU placement technique for power system observability

B.K. Saha Roy ⇑, A.K. Sinha, A.K. PradhanIndian Institute of Technology, Kharagpur 721 302, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 August 2011Received in revised form 4 January 2012Accepted 21 March 2012

Keywords:Phasor measurement unit (PMU)Network observabilityNetwork connectivityOptimal placement

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.03.011

⇑ Corresponding author.E-mail address: bksr12@gmail.com (B.K. Saha Roy

Power system state estimation with exclusive utilization of synchronous phasor measurements demandsthat the system should be completely observable through PMUs only. To have minimum number ofPMUs, the PMU placement issue in any network is an optimization problem. A three stage optimalPMU placement method is presented in this paper using network connectivity information. The methodinitially considers PMU in all buses of the network. Stage I and Stage II of the algorithm iteratively deter-mine (i) less important bus locations from where PMUs are eliminated and (ii) strategically important buslocations where PMUs are retained. Stage III of the algorithm further minimizes the number of PMU usingpruning operation. The set of PMUs obtained after Stage III is an optimal set of PMU locations for networkobservability. Proposed method is further extended for assuring complete observability under single PMUoutage cases. Simulation results for IEEE 14-bus, 24-bus, 30-bus, 57-bus, 118-bus and New England 39-bus test systems are presented and compared with the existing techniques. Results show that the pro-posed method is simple to implement and accurate compared to other existing methods.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Power utilities are facing increasing number of threats of secu-rity of operation due to over stressed power network in competi-tive power market scenario. State estimation is a tool whichprovides the real time state of the system. It is an integral part ofenergy management system (EMS) for security analysis and otherpower system applications [1]. Pre-requisite of state estimationis that the system must be fully observable from the available mea-surements. Before introduction of synchro-phasor technology,state estimation was relying on SCADA systems. Superiority ofPMU measurements over SCADA measurements are that PMU pro-vides the phase angle measurement directly and all the measure-ments are time synchronized [2]. PMU provides voltage phasor ofthe bus where it is installed and current phasors of all the branchesincident to that bus. Direct measurement of all the system states ispossible by placing PMUs in all buses of a network without runningany state estimator. The PMU and its associated communicationsystem are costly and the voltage phasor of the incident buses toPMU installed bus can be obtained with the help of branch param-eter and branch current phasor. If the network is observablethrough optimally placed PMU, a linear state estimator providessystem states in a single iteration. The main objective of optimalPMU placement is to determine the minimal number of PMUs tobe installed at strategic locations so that the entire power systembecomes completely observable for state estimation.

ll rights reserved.

).

Some of the important contributions in PMU placement area arebisecting search and simulated annealing based method [3], nondominated sorting based genetic algorithm approach [4], simu-lated annealing based graph theoretic approach [5], integer pro-gramming (IP) based approach [6,7]. Random selection of PMUplacement sets makes bisecting search approach computationallyless efficient. The IP approach uses linear programming (LP) solverand branch and bound algorithm [8]. Branching involves develop-ment of subproblem of the original problem and bound involvesenumeration of the subproblem through linear programming.The process requires use of LP solver in every iteration which againdepends on the size of the system and thus the method is compu-tationally intensive. For benchmarking of global optimal solutionthe exhaustive binary search method [9] is suitable but computa-tional burden is heavy for large size networks. In [10] a phasedinstallation scheme of PMUs is proposed such that the final place-ment will be optimal. Tabu search metaheuristic method [11] isproposed to obtain minimum number of PMU for complete observ-ability of power system. A non-dominated sorting based differen-tial evolution algorithm [12] is proposed for multi-objectiveoptimization of PMU placement problem. In [13] a metaheuristicbased iterative local search method is proposed to find the optimalsolution where an initial PMU placement is considered whichmakes the system fully observable. A two stage PMU placementmethodology is proposed in [14]. The stage-I takes care of topolog-ical observability and stage-II checks for numerical observabilitybased on exhaustive search method which increases computa-tional burden of the method heavily. Binary particle swarm optimi-zation (BPSO) based approach is proposed in [15]. The optimization

72 B.K. Saha Roy et al. / Electrical Power and Energy Systems 42 (2012) 71–77

process takes care of dual objective such as minimization of PMUnumber and maximization of measurement redundancy. A modi-fied BPSO based optimal PMU placement technique is availablein [16]. Contingency constrained optimal PMU placement tech-niques are reported in [17,18]. In [19] heuristics based optimalPMU placement method is proposed, where the author addressesissues like zero injection bus as pseudo measurements. Methodfor selection of optimal PMU with available SCADA measurementsconsidering single and multiple measurement loss is reported in[20].

In this paper a topological observability based three stage opti-mal PMU placement technique is proposed. For topological observ-ability a set of minimum PMUs is required to make the systemcompletely observable. It is assumed that there are strategic busesin every system using that the PMU placement becomes an easytask. This is the motivation of the proposed three stage optimalPMU placement method. The method assumes PMUs in all busesinitially. The first two stages eliminate PMUs sequentially from lessimportant buses and find the priority bus locations based on sim-ple network connectivity where the PMUs are retained. Optimalityof the set of PMUs obtained from Stage I and Stage II is furtherchecked by pruning operation in Stage III. The set of PMUs obtainedafter pruning is the optimal set for complete observability of thepower system. Proposed method considered that the PMU will per-fectly work. Like any other metering devices PMU may malfunc-tions due to problem in PMU itself or with the communicationsystem. To increase the reliability of measurements it is of interestto avoid such maloperating situations. It is desired that each busshould be observable at least by two PMUs to take care of singlePMU loss in the system. Probability of loss of two PMUs simulta-neously is considered to be very low or negligible here. The pro-posed methodology is extended to find an overall optimalsolution to make system completely observable under normaloperating conditions and for single PMU loss situation. The pro-posed method is found to be simple, accurate and fast incomputation.

The paper is organized as follows. Section 2 discusses the powersystem observability using PMU. Section 3 provides the problemformulation for optimal PMU placement. Solution methodology isdiscussed in Section 4. Case studies and discussions on simulationresults are presented in Section 5. Section 6 provides conclusion ofthe paper. Appendix A provides the explanation of the rules usedand an illustration of the algorithm is exemplified in Appendix B.

2. Power system observability analysis

A power system state estimator demands complete observabil-ity of the network from the available measurements. Power systemis observable by a set of measurements when the measurementscan build a spanning tree of full rank of the system [21]. Power sys-tem observability analysis is usually carried out in two differentways namely numerical and topological observability analysis.

2.1. Numerical observability analysis

The measurement model used in state estimation is

z ¼ hðxÞ þ e ð1Þ

where x is the system state vector i.e. the voltage phasor of all busesof the network, z is the measurement vector, h(x) is the non-linearfunction that relates the measurement vector to the state vector ofthe system and e represents the measurement error vector. SincePMUs provide accurate measurements (voltage and current pha-sors) the measurement error is small and can be neglected. Exclu-sive use of PMU measurements results in linear state estimator

formulation [22]. The linear state estimator model is givenbelow:

z ¼ hx ð2Þ

where h is the measurement function matrix. When the system iscompletely observable by PMUs the linear state estimator providessystem state by solving linear set of equations in (2). Numericalobservability analysis checks whether the rank of measurementmatrix is full or not. For an N bus network the measurement sets(voltage and current phasors) obtained from PMUs can make thesystem numerically observable if

RankðhÞ ¼ 2N � 1 ð3Þ

Optimal PMU placement for numerical observability of powersystem can be carried out in two different ways

Method I: Introduction of PMU in the network one by one toimprove the rank of the measurement matrix. No further PMUis introduced when (3) is satisfied.Method II: Considering PMU in all buses, PMUs are sequentiallyeliminated from different buses one by one. Elimination processis overruled at a bus which makes the rank of the measurementmatrix deficient.

In both approaches to reach the optimal (minimum) PMU place-ment solution, large number of combinations has to be checked.Each combination involves rank testing which increases the com-putational burden of numerical observability analysis.

2.2. Topological observability analysis

In topological observability analysis power system is repre-sented by a topological graph. The graph has ‘N’ number ofnodes representing the network bus bar and ‘E’ number ofedges representing the branches of the network connectingthe bus bars. In topological approach the optimal PMUplacement set is searched such that each bus of the networkis observable by at least one PMU. In this paper a simple topo-logical observability analysis method is proposed which isexclusively based on PMU measurements. Loss of single PMUor communication link failure is also considered in the presentpaper.

3. PMU placement problem formulation

For an N-bus system the optimal PMU placement problem isformulated as follows:

minXN

i¼1

wixi

s:t: FðXÞP b

ð4Þ

where X is the binary decision variable vector for PMU placement,whose entries are defined as

xi ¼ 1 if PMU is placed at ith bus

0 otherwise:for i ¼ 1; . . . ;N:

b is an unit vector of length N, i.e. b = [111. . .. . .]T. wi is the cost ofPMU installed at ith bus. F(X) is the observability constraint vectorfunction, whose entries are nonzero if the corresponding individualbuses are observable w.r.t a given measurement set and zero other-wise. If wi is assumed unity, the optimal PMU placement problem isrepresented as

I13

I43I23

ZIB

3

2

1

4

Fig. 1. Modeling of ZIB.

B.K. Saha Roy et al. / Electrical Power and Energy Systems 42 (2012) 71–77 73

minXN

i¼1

xi

s:t: FðXÞP b

ð5Þ

Constraint vector function ensures full network observability. Asolution i.e. a set of minimum xi is to be found out which will sat-isfy (5). The constraint vector function is formed using the binaryconnectivity matrix (A) of power system. The binary connectivitymatrix (A) represents the bus connectivity information of a powersystem. The elements of matrix A is defined as

am;n ¼ 1 if m ¼ n

1 if bus m is connected to bus n

0 otherwise

The constraint vector function for the test system and any par-ticular ith bus is given in (6) and (7).

FðXÞ ¼ AX P b ð6Þfi ¼ ai;1x1 þ . . .þ ai;;ixi þ . . .þ ai;NxN ð7Þ

If ai,n is zero, the product ai,nxi not appear in (7). If any xi appear-ing in fi is nonzero, fi is observable. If all fi in F are nonzero the sys-tem is completely observable.

4. Proposed technique to PMU placement

The aim of the proposed method is to obtain minimum numberof strategic bus locations where PMUs are to be placed to satisfysystem observability (6). Strategic buses are obtained based onnetwork connectivity information.

4.1. Description of the proposed method

The proposed three stage optimal PMU placement method con-siders PMU at each bus initially. If zero injection bus (ZIB) is usedas pseudo measurement, PMUs are not considered in ZIBs. ThePMUs are eliminated sequentially from the less important buslocations and retained at potentially important bus locations. Theterms used in the proposed method are described below.

Bus valency: Valency of ith bus of a network is the total numberof buses connected to that bus including the ith bus itself.MaxV: Maximum bus valency of the network.MinV: Minimum bus valency of the network.Radial bus (RB): Bus which is connected to only one other bus ofthe network. Bus valency of radial bus is minimum and it is 2.Equal valency (EV) bus: The set EV stores buses which have equalbus valency at any particular valency of the test network.Allocated PMU bus (APB): In the process of PMU eliminationfrom different buses there are some important bus locationswhere PMUs are retained. Those PMU bus locations are calledallocated PMU buses (APB).SB: Buses of EV those are connected to any bus in APB is storedin set SB.Higher valency (HV) bus: Some of the EV buses connected to acommon bus having valency higher than EV buses, forms agroup with group centre at that common bus. HV store thosehigher valency buses which have maximum number of groupelement. At least two EV bus is needed to form a group.APB observable buses (APBOBs): Buses which are connected toany APBs.Candidate bus (CB): CB consist candidate’s bus locations fromwhere PMUs are to be eliminated. At starting CB consists allbuses except the ZIBs.

Modeling of ZIB: ZIBs are the buses which have neither any gen-eration nor any load. At ZIBs no current is injected into the sys-tem. This is used as pseudo information to make systemobservable with less number of PMUs compared to the casewhen information of ZIBs is not considered. Let us consider atypical 4-bus example as in Fig. 1 for ZIB modeling. In the 4-bus example, buses are numbered as 1, 2, 3 and 4. Bus 3 is aZIB. Application of KCL at bus 3 provides

I13 þ I23 þ I43 ¼ 0 ð8Þ

Among the four buses if any three bus voltages (V) are knownthe 4th bus voltage can be calculated using (8). When ‘K’ numberof connected ZIBs forms a super node with all known adjacentbus voltages, the ZIBs can be solved using (9). M is the numberof bus connected to ith bus and Yij is the transfer admittance be-tween ith bus and jth bus.

XM

j¼1

YijVj ¼ 0 for i ¼ 1; . . . ;K ð9Þ

PMU elimination starts from minimum valency buses i.e. fromthe radial buses. To observe a radial bus PMU is essential at radialbuses or the bus to which a radial bus is connected. If PMU isplaced at radial bus it makes two buses observable but if PMU isplaced at bus where radial bus is connected it makes more thantwo buses observable. Hence PMUs are eliminated from radialbuses and retained at buses where radial buses are connected.PMU elimination process continues for the higher valency busesup to maximum valency. In the process of elimination PMUs are re-tained at important bus locations such as (1) a higher valency busconnected to more than one running valency buses of EV, and (2) abus from where elimination of PMU leads to the system unobserv-ability. At every iteration the tested buses (PMU eliminated or re-tained) are removed from the CB.

The three stages of the proposed algorithm are enumeratedbelow.

Stage-I: It finds the important bus locations where PMUs areretained among different valency buses.Stage-II: Some of the buses neither formed any group nor con-nected to any APB bus remain untested in stage-I. Among thosebuses lower valency buses with higher connectivity to APBOBsand connected to ZIBs are considered first. PMUs are eliminatedor retained one by one from these buses subject to satisfyingsystem observability. On testing all lower valency buses,untested higher valency buses are tested up to maximumvalency.Stage-III: This is a pruning stage. Pruning checks the possibleways for further reduction of PMUs from the set of PMU buslocations obtained in stage-I and stage-II. It searches a nonPMU bus which is connected to more than one PMU bus.Pruning checks whether placement of one PMU at non PMUbus can make two or more PMU buses redundant in satisfying

Yes YesNo

No

Set: V=2;

EV nil ? V<maxV ?

c

Order the EV buses connected to maximum APBOBs and connected to ZIBs 1

st followed by the rest.

Eliminate one PMU from ordered EV buses. Retain PMU when system observability violated. Update EV, CB and APB.

b

Find EV with valency V=V+1 from updated CB excluding ZIBs. Find APBOB buses.

Fig. 2b. Flow chart of Stage II.

No

Do pruning and check observability

Observable ? Don’t prune

c

74 B.K. Saha Roy et al. / Electrical Power and Energy Systems 42 (2012) 71–77

observability constraints. It also checks redundancy of the indi-vidual PMUs to satisfy system observability constraints.

Elimination and retention of PMU obey the following rule.

Rule-1: Eliminate PMU from SB buses one by one subject to sat-isfaction of (6).Rule-2: Retain PMU at any highest valency bus among HV buses.

4.2. Algorithm for optimal PMU placement under normal operatingcondition

Flow charts of three stages of the proposed algorithm are de-tailed in Figs. 2a–2c.

4.3. Algorithm for optimal PMU placement considering single PMU loss

The optimal set of PMUs obtained for normal operating condi-tion is not sufficient to observe all buses under single PMU lossor any communication failure. Extra PMUs are needed along withthe optimal set obtained for normal operating condition to takecare single PMU loss. Additional PMUs are selected in such a waythat finally an overall optimal set of PMUs obtained that makethe system completely observable under normal operating condi-tion as well as under single PMU loss condition. To obtain the over-all optimal set the optimization problem is restated as

No

Yes

Yes

No

Yes

Yes

No

Eliminate PMU from RB and retain PMU at bus connected to RB. Retained PMU buses are stored in APB. Remove tested buses from CB.

Find buses with valency V=V+1 from CB and store in EV.

Form system A matrix, compute valency of all CB; find MaxV, MinV. Set V=MinV=2 ;

Find SB buses from EV buses.

Eliminate PMUs as per rule 1. Retain PMU at bus violating observability. Update EV, APB and CB. n = no’s of bus in EV.

n =0 ?

Start

Find HV from CB ; nh = no’s of bus in HV

nh=0 ?

Apply rule 2. Update APB, CB. Reset EV, HV

PMU retained ?

V=2V< MaxV?

b

Consider PMU in all buses except ZIBs when ZIBS are considered as pseudo measurements

Fig. 2a. Flow chart of Stage I.

No

Yes

Yes

Optimal solution obtained

Pruning complete ?

Stop

Fig. 2c. Flow chart of Stage III.

minXN

i¼1

xi

s:t: FðXÞP b

ð10Þ

where b = [222. . .. . .]T. Length of b is N. Doubly observable buses areeither observable from two PMUs or from a single PMUs and alsoobservable using KCL at ZIBs. Singly observable buses are either ob-servable from a PMU or using KCL at ZIB. When all but one bus in-volved with any ZIB are doubly observable, KCL at ZIB makes theunknown bus doubly observable. The following steps successfullyprovide the overall optimal set of PMUs required for making thesystem observable under single PMU loss contingency.

Step 1: Place PMU at all the radial buses those are not connectedto any ZIBs along with the optimal set of PMU buses obtainedfor complete observability under normal operating condition.Step 2: Find the singly observable buses with the already placedPMUs, store these buses in set {SOB}.Step 3: If the number of buses in SOB is zero, go to step 6 else goto step 4.Step 4: Find the maximum valency bus from SOB excluding thePMU buses.Step 5: Select any bus from the list of bus obtained in step 4 asPMU bus. Remove the buses which become doubly observabledue to the additional PMU bus from SOB and go to step 2.Step 6: Perform pruning to discard any redundant PMU for mak-ing the system buses double observable.

Table 1Optimal PMU placement results for normal operating condition without considering ZIB.

System Optimal PMU locations Optimal number

IEEE 14-bus 2,67,9 4IEEE 24-bus 2,3,8,10,16,21,23 7IEEE 30-bus 2,3,6,9,10,12,15,19,25,27 10NE 39-bus 2,6,9,10,12,14,17,19,20,22,23,25,29 13IEEE 57-bus 1,4,9,20,24,27,29,30,32,36,38,39,41,45,46,51,54 17IEEE 118-bus 1, 5,9, 12,13,17,21,23,26,28,34,37,41,45,49,53,56,62,63,68,71,75,77,80,85,86,90,94,101,105,110,114 32

Table 2Comparison of optimal PMU placement results with available techniques for normal operating condition without considering ZIB.

Method IEEE 14-bus (Nos.) IEEE 24-bus (Nos.) IEEE 30-bus (Nos.) NE-39 (Nos.) IEEE 57-bus (Nos.) IEEE 118-bus (Nos.)

Proposed 4 7 10 13 17 32Xu and Abur [7] 4 N/A 10 N/A 17 32Chakrabarti and Kyriakides [9] 4 7 10 13 N/A N/AHurtgen and Maun [13] 4 N/A N/A N/A 17 32

N/A: Not available.

Table 3Optimal PMU placement results for normal operating condition considering ZIB.

System ZIBs locations Optimal PMU locations Optimal number

IEEE 14-bus 7 2,6,9 3IEEE 24-bus 11,12,17,24 1,2,8,16,21,23 6IEEE 30-bus 6,9,22,25,27,28 2,3,10,12,18,24,30 7NE 39-bus 1,2,5,6,9,10,11,13,14,17,19,22 3,8,12,16,20,23,25,29 8IEEE 57-bus 4,7,11,21,22,24,26,34,36,37,39,40,45,46,48 1,6,13,19,25,29,32,38,51,54,56 11IEEE 118-bus 5,9,30,37,38,63,64,68,71,81 1,6,8,12,15,17,21,25,29,34,40,45,49,53,56,62, 72,75,77,80,85,86,

90,94,101,105,110,11428

Table 4Comparison of optimal PMU placement results with available techniques for normal operating condition considering ZIB.

Method IEEE 14-bus (Nos.) IEEE 24-bus (Nos.) IEEE 30-bus (Nos.) NE 39-bus (Nos.) IEEE 57-bus (Nos.) IEEE 118-bus (Nos.)

Proposed 3 6 7 8 11 28Xu and Abur [7] 3 N/A 7 N/A 12 28Chakrabarti and Kyriakides [9] 3 6 7 8 N/A N/AHajian [16] 3 N/A 7 8 11 28

Table 5Optimal PMU placement results for single PMU loss condition considering ZIB.

System Optimal PMU locations Optimal number

IEEE 14-bus 1,2,4,6, 9,10,13 7IEEE 24-bus 1,2,7,8,9,10,11,15,16,17,20,21,23 13IEEE 30-bus 1,2,3,5,6,10,12,13,15,16,18,19,24,27,30 15NE 39-bus 2,3,5,6,8,13,16,17,20,22,23,25,26,29,34,36,37,38 18IEEE 57-bus 1,2,4,6,9,12,15,18,19,22,24,25,27, 29,30, 32,33, 36,38,41,47,50, 51,53,54,56 26IEEE 118-bus 1,2,5,6,8,9,11,12,15,17,19,20,21,23,25,27,28,29,32,34,35,37, 40,41,43,45,46,49,50,51,52,53,56,59,62,66,68,70,71,72,75,76,

77,78,80,83,85,86,87,89,90,92,94,96,100,101,105,106,108, 110, 111,112,114,11764

Table 6Comparison of optimal PMU placement results with topological observability based approach for single PMU loss condition considering ZIBs.

Method IEEE 14-bus (Nos.) IEEE 24-bus (Nos.) IEEE 30-bus (Nos.) NE-39 (Nos.) IEEE 57-bus (Nos.) IEEE 118-bus (Nos.)

Proposed 7 13 15 18 26 64Xu and Abur [7] 7 N/A 17 N/A 26 65

Table 7Comparison of computation time for obtaining optimal solution by the proposed method with two other simulated methods.

Method IEEE 14-bus IEEE 24-bus IEEE 30-bus NE-39 IEEE 57-bus IEEE 118-bus

Proposed 0.66s 0.76s 0.83s 0.84s 0.87s 1.34sBIP 1.16s 1.34s 1.24s 1.27s 1.94s 1.55sExhaustive search 1.3s 7.4 m 2.6hr

B.K. Saha Roy et al. / Electrical Power and Energy Systems 42 (2012) 71–77 75

8

6

2

5

109

4

1

3

121124

15 14

16

13

232019

17

18 21 22

7

Fig. B.1. Single line diagram of IEEE 24-bus test system.

Table B.1Bus valencies of different buses of IEEE 24-bus system.

Valency (V) Bus locations

2 73 4,5,6,14,18,19,20,22,244 1,2,3,8,13,15,17,21,235 11,12,166 9,10

76 B.K. Saha Roy et al. / Electrical Power and Energy Systems 42 (2012) 71–77

Step 7: Optimal set of PMUs obtained for complete observabilityof power system under normal operating condition as well asfor single PMU loss contingency.

5. Case studies and discussion on simulation results

The proposed three stage algorithm determines minimum num-ber of strategic bus locations where PMU must be placed for com-plete observability of the power system. First and second stage ofthe algorithm determines the important bus locations for allocat-ing PMUs. The third or Pruning stage checks the possible ways tofurther reduce any PMUs from the set obtained after Stage I andStage II. The set of PMUs obtained after pruning stage providesan optimal (minimum) solution for PMU placement for completeobservability of the system. The proposed method considered onlyPMU measurements for complete observability of the system.

The algorithm has been tested on IEEE 14-bus, 24-bus, 30-bus,New England 39-bus, IEEE 57-bus and 118-bus test system to val-idate the effectiveness of the proposed method. Detail systeminformation and single line diagram for each of the above networksis available in [23,24]. Table 1 shows the optimal number of PMUsrequired for different systems and corresponding bus locations. ForIEEE 14-bus system 4 PMUs are essential and the bus numbers forplacement are 2, 6, 7 and 9 without considering ZIBs. For IEEE 118-bus system 32 PMUs are required for complete observability. Theresults of the proposed method have been compared with the re-sults of three other topological observability based PMU placementmethods reported by Xu and Abur [7], Chakrabarti and Kyriakides[9] and Hurtgen and Maun [13]. Table 2 shows the comparative re-sults of minimum number of PMUs required for complete observ-ability of different systems. All the three methods show thatminimum 4 PMUs are required for complete observability of IEEE14-bus system and so does the proposed method.

Simulation results for optimal number and corresponding buslocations for system observability considering ZIBs informationare given in Table 3. For IEEE 14-bus system 3 PMUs are sufficientand the bus numbers for placement are 2, 6, and 9. Comparative re-sults of the proposed method considering ZIB with available PMUplacement methods are given in Table 4. Comparative results showthat the proposed method successfully finds the global optimalsolution for different systems.

The methodology is extended for an overall optimal solution totake care of single PMU loss contingencies. Table 5 shows the opti-mal number of PMUs required for different systems and corre-sponding bus locations for complete observability under singlePMU loss case considering ZIBs. For IEEE 14-bus system 7 PMUsare essential and the bus numbers for placement are 1, 2, 4, 6, 9,10 and 13. The result of the proposed method for single PMU losscase is compared with the results reported by Xu and Abur [7] andprovided in Table 6. Results show that proposed method is capableof producing optimal set of PMUs for single PMU loss condition.

The CPU time taken by proposed method under normal operat-ing condition is shown in Table 7 for different test systems. Com-putation time of the proposed method is compared with the timetaken in matlab binary integer programming (BIP) and exhaustivesearch approach without considering ZIBs. Performance compari-son shows the computational efficiency of the proposed methodcompared to the other methods. The simulation platform usedfor implementation of the methods is Intel Pentium 4, 3.0-GHzCPU with 512 MB of RAM.

6. Conclusion

A new methodology for optimal placement of PMU is presentedin this paper. The new iterative method makes the test systemstopologically observable by placing a set of minimum PMUs. Thethree stage algorithm is simple, fast and easy to implement. Thepresent method obtains optimal solution using simple networkconnectivity information. The overall optimal solution obtained issufficient to take care of system observability under normal oper-ating condition as well as for single PMU loss cases. Simulation re-sults for different networks show the effectiveness of the proposedmethod in obtaining the minimum number of PMUs required forcomplete observability of power systems and also its advantageof computational efficiency.

Appendix A

Explanation of rule 1: SB buses are connected to an already allo-cated PMU bus. Elimination of PMU from any of SB bus does notviolate its observability. Hence this is the choice of elimination.However this elimination may violate observability of otherbus. In that case PMU is retained to that particular SB bus.Explanation of rule 2: When maximum number of EV bus is con-nected to a higher valency (HV) bus, PMU is allocated to thatbus. In case large number of higher valency buses has connec-tion to similar number of EV bus, highest valency bus amongthe HV buses is the choice for PMU retention. If more thanone bus has highest valency, any of them is the choice. Alloca-tion of such PMU can make more than one PMUs of EV busesredundant.

In stage-II, a lower valency bus with higher connectivity to AP-BOBs and connected to ZIB is the first choice for PMU elimination.This is supposed to be a redundant PMU bus in satisfying systemobservability.

B.K. Saha Roy et al. / Electrical Power and Energy Systems 42 (2012) 71–77 77

Appendix B

The proposed algorithm is illustrated on IEEE 24-bus test sys-tem. Single line bus connection diagram of IEEE 24-bus system isgiven in Fig. B.1. ZIBs are bus number 11, 12, 17 and 24.

Stage IBus valeny for IEEE-24 bus system is computed and given in Ta-

ble B.1.Set V = MinV = 2 and MaxV = 6;Bus 7 is the radial or minimum valency bus and connected to

bus 8. PMU is eliminated from 7 and retained at bus 8 to makebus 7 observable. Bus 7 and 8 is removed from CB. APB = {8}.

Iteration 1V = V + 1 = 3. EV = {4,5,6,14,18,19,20,22,24}. None of EV is

connected to APB i.e. SB = {nil}.Higher valency buses i.e. HV = {2,10,16,17,21} are connected to

more than one bus of EV. Bus 17 is a ZIB and there is no PMU hencenot considered. PMU is retained at highest valency bus 10.APB = {8,10}. Bus 10 is removed from CB.

PMU is retained in bus 10. Set V = 2.Iteration 2V = V + 1 = 3. EV = {4,5,6,14,18,19,20,22,24}.SB = {5,6} of EV is connected to APB bus 10. PMU is eliminated

from SB one by one and system found observable. Bus 5 and 6 isremoved from CB. EV = {4,14,18,19,20,22,24}

HV = {16,21}. PMU is retained at highest valency bus 16. Bus 16is removed from CB; APB = {8,10,16}. PMU is retained in bus 16.Set V = 2.

Iteration 3V = V + 1 = 3. EV = {4,14,18,19,20,22,24}.SB = {14,19} connected to APB bus 16. CB is updated;

EV = {4,18,20,22,24}.HV = {21}. PMU is retained at bus 21. Bus 21 is removed from

CB; APB = {8,10,16,21}. Set V = 2.Iteration 4V = V + 1 = 3. EV = {4,18,20,22,24}.SB = {18,22} connected to APB bus 21. EV = {4,20,24}.

HV = {nil}. V < maxV.Iteration 5V = V + 1 = 4. EV = {1,2,3,13,15,17,23}.SB = {15,17} is connected to APB bus 16. EV = {1,2,3,13,23}.

HV = {nil}. V < maxV.Iteration 6V = V + 1 = 5. EV = {11,12}.SB = {11,12} connected to APB bus 10. EV = {nil} HV = {nil}.

V < maxV.Iteration 7V = V + 1 = 6. EV = {9}.SB = {9} is connected to APB bus 8. EV = {nil}. HV = {nil}.

V = maxV.Stage IIV = 2;Iteration 1:V = V + 1 = 3; buses EV = {4,20,24}. Bus 24 is a ZIB. Hence

EV = {4,20}. Both the buses have same connectivity to APBOB busesand none of them is connected to ZIBs. PMUs are eliminated frombus 4 and bus 20, system found observable. CB is updated. EV = {-nil} and V < maxV.

Iteration 2:V = V + 1 = 4; buses EV = {1,2,3,13,23}. Bus 13 has maximum

connection to OBs. Bus 1, 2, 3 and 23 have similar connection toOBs. Bus 3 and 23 are connected to ZIBs. OrderedEV = {13,3,23,1,2}. PMUs are eliminated one by one from EV andfound system unobservable at bus 23, 1 and at 2. Hence PMUs

are retained at bus 23, 1 and at 2. CB is updated.APB = {1,2,8,10,16,21,23}. EV = {nil} and V < maxV.

Iteration 3:V = V + 1 = 5; buses EV = {nil}. and V < maxV.Iteration 4:V = V + 1 = 6; buses EV = {nil};V = maxV.Stage III. Pruning stage.Set of PMU bus obtained are {1, 2, 8, 10, 16, 21 and 23}. There

are different possible PMU bus groups such as {2,10} and{16,21}. which are connected to a non-PMU bus 6, 15, etc. All pos-sible groups are checked by pruning operation. None of the non-PMU bus makes any group of PMU bus redundant. Pruning for indi-vidual PMU find bus 10 as redundant in satisfying observability.Hence optimal set is {1, 2, 8, 16, 21 and 23}.

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