Electrical Power and Energy Systems 33 (2011) 28–34

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Electrical Power and Energy Systems

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Optimal placement of PMUs to maintain network observabilityusing a modified BPSO algorithm

Mahdi Hajian a,⇑, Ali Mohammad Ranjbar b, Turaj Amraee b, Babak Mozafari b

a Department of Electrical and Computer Engineering, University of Calgary, Alberta, Canada T2N 1N4b Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 23 November 2007Received in revised form 10 April 2010Accepted 13 August 2010

Keywords:Power system observabilityPhasor Measurement UnitOptimal placementBinary Particle Swarm Optimization

0142-0615/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijepes.2010.08.007

⇑ Corresponding author. Tel.: +1 403 210 5467; faxE-mail address: mhajian@ucalgary.ca (M. Hajian).

a b s t r a c t

This paper presents a novel approach to optimal placement of Phasor Measurement Units (PMUs) forstate estimation. At first, an optimal measurement set is determined to achieve full network observabilityduring normal conditions, i.e. no PMU failure or transmission line outage. Then, in order to consider con-tingency conditions, the derived scheme in normal conditions is modified to maintain network observ-ability after any PMU loss or a single transmission line outage. Observability analysis is carried outusing topological observability rules. A new rule is added that can decrease the number of required PMUsfor complete system observability. A modified Binary Particle Swarm Optimization (BPSO) algorithm isused as an optimization tool to obtain the minimal number of PMUs and their corresponding locationswhile satisfying associated constraint. Numerical results on different IEEE test systems are presentedto demonstrate the effectiveness of the proposed approach.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Phasor Measurement Units (PMUs) were introduced in early1990 as devices capable of measuring synchronous real-time volt-age and current phasors in power systems [1]. Synchronous mea-surements of PMUs located in different buses are achieved viasame-time sampling of voltage and current waveforms using acommon synchronizing signal. The most common used synchro-nizing signal is obtained from Global Positioning System (GPS)reference which is able to provide an accuracy down to 1 ls [2].Real-time phasor measurement at different nodes can improvethe performance of monitored control systems in different applica-tions such as state estimation, fault location, transient and smallsignal stability analysis [3–6].

One of the applications of phasor measurements in power sys-tems is state estimation which is performed in a control center toprovide a platform for monitoring, and security applications suchas contingency analysis and optimal power flow [1]. The first stepin state estimation is to gather measured data from different sub-stations in a power network. These measurements must be suffi-cient to make the system observable [7]. Assuming all nodes ofthe system are equipped with PMUs, all system state variablescan be directly monitored and there is no need to estimate othervariables. However, due to installation cost of PMUs or non-exis-tence of communication facilities, an ubiquitous placement of

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PMUs is rarely conceivable. Therefore, the problem is to find theminimum number of PMUs to achieve full observability of the net-work. In addition, a reliable PMU placement set should be robustenough to maintain power network observability anticipating pos-sible contingencies such as PMU failure or a transmission lineoutage.

The PMU placement problem has been addressed in several lit-eratures. The problem was first introduced in [8]. A modifiedbisecting search and simulated annealing optimization algorithmare used to optimally select locations of PMUs. Topological observ-ability rules are employed in order to examine observability of anetwork. In [9], an integer programming based formulation is usedto find an optimal PMU placement scheme considering conven-tional measurements. In [10], the PMU placement problem issolved via tree search method considering complete and incom-plete observability. In [11], the Tabu search algorithm is used tooptimize the number of required PMUs for full network observabil-ity. Augmented incidence matrix is proposed to analyze observ-ability of a candidate PMU set.

The possibility of contingencies such as measurement loss or atransmission line outage in the placement problem is also consid-ered in the literature. Optimal placement methods consideringcontingencies for conventional measurements are addressed in[12,13]. In [14], an optimal PMU placement set is found using anondominated sorting genetic algorithm and topological observ-ability analysis considering single line contingencies. In [15], aheuristic approach is proposed to obtain a PMU placement set ro-bust to a single measurement loss and branch outage. In [16], an

M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34 29

integer quadratic programming is used in order to optimally find aPMU placement set considering PMU failure or a branch outage.However, in this work the effect of zero-injection buses is ignored.In [17], integer linear programming is proposed for solving theoptimal PMU placement problem anticipating PMU loss or a lineoutage.

In this paper, the optimal PMU placement considering PMU lossor a line outage is addressed using a modified Binary ParticleSwarm Optimization (BPSO) technique. Topological observabilityanalysis is used to assess observability of each placement candi-date. In order to attain maximum utilization of existing data, suchas characteristics of zero-injection buses, a new rule is added to theprevious rules of topological analysis. Using the modified observ-ability analysis rules and the modified BPSO, the optimal PMUplacement is accomplished during normal conditions, i.e. assumingno contingency. In order to consider PMU loss or a line outage, thetopological observability analysis is adopted to examine networkobservability anticipating contingencies. Using the modified BPSOand the adopted topological observability rules, the derived place-ment scheme in normal conditions is updated to maintain systemobservability during contingency conditions.

The remainder of this paper is organized as follows. The observ-ability analysis using PMU data is presented in Section 2. An over-view of Particle Swarm Optimization technique is described inSection 3. The proposed PMU placement problem in normal andcontingency conditions is addressed in Section 4. Numerical resultsof the proposed algorithm are provided in Section 5. Finally, Sec-tion 6 concludes the paper.

Fig. 1. A group of zero-injection buses.

2. Observability analysis using PMU data

Given a N-bus network provided with m measurements of volt-age and current phasors, the linear equations relating measure-ments and the state vector are

z ¼ H � xþ e ð1Þ

where the vector z is linearly related to the n-dimensional state vectorx containing N-bus voltage phasors (i.e. n = 2N � 1 ). H is the (m � n)matrix, and e is the (m � n) additive measurement error vector.

The observability of a system can be examined considering net-work topology, types, and locations of measurements. Generally,two different observability concepts are defined for a linear systemmodel (Eq. (1)), namely numerical and topological analysis.Numerical observability is defined as the ability of a system modelto be solved for state estimation. The full rank of matrix H (i.e.2N � 1) is considered to be the criteria to declare full observabilityof a system [18]. However, due to high calculation burden of veri-fying the rank of matrix H, this approach would not be preferredfor practical applications. In addition, for any placement schemein which the corresponding matrix H is not of full rank, this meth-od is not able to specify locations of unobservable buses.

Topological observability analysis is defined as the existence of atleast one spanning measurement tree of full rank in a network [8].This tree connects all observable nodes and branches which can beobserved by direct measurements or calculations. The followingrules are commonly used to assess the existence of this tree.

(1) Buses with PMU are assigned with direct voltage phasormeasurement and direct current phasor measurement ofincident lines. The vector of direct measurement of voltageand current phasors are denoted as DV and DI, respectively.

(2) If voltage and current phasors at one end of a line are known,then the unknown voltage phasor at the other end of the linecan be calculated (assuming known impedance of transmis-sion lines).

(3) If voltage phasors of both ends of a line are known, then thecurrent phasor of this line can be calculated. Voltage andcurrent phasors of buses and lines observed by rules 2 and3 are defined as pseudo-measurements and indicated withPV and PI, respectively.

(4) If all line current phasors incident to a zero-injection bus areknown except one, the current phasor of the unknown onecan be calculated through KCL equations. A zero-injectionbus is a bus in which the net power injection is zero.

(5) If there is an unknown zero-injection bus and voltage phasorsof its adjacent buses are all known, then the voltage phasor ofthe zero-injection bus can be obtained by the node equations.

Topological observability analysis based on a network graph isnormally accomplished by the above rules. However, another rulecan be introduced which enhances the performance of observabil-ity analysis. This rule is a generalization of rule 4 which was firstintroduced in [19].

(6) Considering a group of unknown zero-injection buses, asshown in Fig. 1, with all known voltage of adjacent buses,the node equations for each of the buses in the group canbe written as follows:

XN

j¼1

YijV j ¼ 0; for i ¼ 1; . . . ; k ð2Þ

where k is the number of zero-injection buses in the group, Vj isthe voltage of the jth bus, and Yij is the ijth element of the admit-tance matrix. In Eq. (2), there is k complex equations and giventhat the voltage of the adjacent buses are all known, the num-ber of unknown complex variables is exactly equal to k. Hence,the zero-injection buses in the group are all observable.

Voltages and currents observed by the rules 4–6 are defined asextended measurements and represented by EV and EI,respectively.

Finally, Fig. 2 shows the flowchart of the observability analysisbased on aforementioned rules. In this flowchart, OV and OI repre-sent the vectors of observed buses and lines, respectively. The in-put of this algorithm is a candidate placement scheme. After theassessment of topological observability rules, the number and loca-tion of observed buses and lines (i.e. components of OV and OI vec-tors which are greater than 1) are deduced.

3. Particle swarm optimization algorithm

3.1. Continuous version of PSO

Particle Swarm Optimization (PSO) was first introduced in [20].It provides a population-based search procedure in which individ-

Fig. 2. Flowchart of topological observability analysis.

30 M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34

uals, called particles, fly around in a multidimensional search spaceand change their positions with time. Position changes of particlesis based on their individual experience, and experience of neigh-boring particles in finding the optimal point of an optimizationproblem.

3.2. Discrete binary version of PSO (BPSO)

In BPSO, the search space is discrete and variables can only takeon values of 0 and 1. The BPSO was first introduced by Kennedyand Eberhart [21]. That BPSO was achieved with a simple modifica-tion to the real-value particle swarm optimization. In [22], theauthors proposed a modified BPSO with a better performance thanthe previous BPSO in finding optimal solution of an optimizationproblem. The modification was performed through transforming

the continuous equations to the binary mode. However, the effectof previous velocity of particles in updating velocity vectors wasnot introduced.

In this paper, a modified BPSO is proposed in which velocitiesand positions of particles are updated based on followingequations:

dk1;i ¼ pbesti � xk

i ð3Þ

dk2;i ¼ gbest � xk

i ð4Þ

vkþ1i ¼ vk

i þ c1 � dk1;i þ c2 � dk

2;i ð5Þxkþ1

i ¼ xki þ vkþ1

i ð6Þ

In the above equations, �, �, and + are xor, and, and or operators,respectively. xk

i and vki are the binary position and velocity vectors

corresponding to particle i at iteration k, respectively. pbesti is thebest experience of particle i and gbest is the best experience inthe whole population in finding the optimal point of the problem.dk

1;i is the ‘‘cognition” part, which shows private thinking of a parti-cle itself. dk

2;i is the ‘‘social” part, which reflects cooperation amongparticles. c1 and c2 are random binary vectors generated at eachiteration of the algorithm.

Contrary to the real-value PSO, in BPSO a better performance isachieved by setting a single Vmax to the whole of a particle insteadof imposing the maximum velocity to each dimension of the parti-cle [22]. In BPSO, Vmax is defined as the maximum number of onesin a particle. This limit is checked after updating velocities of par-ticles using Eq. (5). In the case of violating the limit, the number ofones in the velocity vector is reduced random until it becomes lessthan Vmax.

3.3. BPSO approach for a discrete optimization problem

The PMU placement problem leads to a discrete binary optimi-zation problem that can be expressed as the problem described inEqs. (7) and (8).

Minimize JðxÞ ð7Þsubject to FðxÞ ¼ 0 ð8Þ

where JðxÞ and FðxÞ are the objective function and equalityconstraint and x is the vector of variables to be optimized. In orderto better demonstrate the BPSO algorithm, the steps required forsolving this discrete binary optimization problem are elaboratedas follows:

Step 1: A random population is created based on the dimensionsize of the problem. In the case of optimal PMU placement, eachindividual consists of a number of bits corresponding to exis-tence (1) or nonexistence (0) of PMU in each bus of the system.Step 2: The problem is transformed into an unconstrained oneby creating an augmented objective function incorporating pen-alty factors for any value violating the constraint:

Minimize J�ðxÞ ¼ JðxÞ þ aFðxÞ ð9Þ

where a is a penalty factor corresponding to the equality con-straint Eq. (8).Step 3: The augmented objective function is evaluated for eachindividual. The current position, xi, is stored in pbesti, ifJ�(xi) < J�(pbesti). The best experience among pbests is denotedas gbest.Step 4: The member velocity v of each particle is modifiedaccording to Eq. (5) considering the maximum velocity limit.Step 5: Positions of particles are updated according to Eq. (6).Step 6: If maximum number of iterations is not reached yet, thealgorithm is pursued from step 3, otherwise it is stopped andgbest is announced as the global optima of the problem.

M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34 31

4. Optimal PMU placement considering normal andcontingency conditions

The proposed approach for the optimal PMU placement isimplemented in two sequences. At first, the optimal placement iscarried out with the goal of minimizing the total number of re-quired PMUs for complete system observability during normalconditions, i.e. no PMU failure or line outage. Then, the derivedplacement set is modified so that the network regains its observ-ability during the two types of contingency, (i.e. PMU loss or abranch outage). Both of the problems are solved using the de-scribed modified BPSO algorithm.

4.1. Optimal PMU placement in normal conditions

The objective of this stage is to minimize the total number ofPMUs required for complete system observability assuming nor-mal conditions. The optimization problem can be described math-ematically as:

MinimizeXN

i¼1

PMUi ð10Þ

subject to Nnobs ¼ 0 ð11Þ

where PMUi indicates the installation of PMU at bus i, it is 1, if aPMU is installed on bus i and 0 vice versa. Nnobs represents the num-ber of unobserved buses in the network.

Before implementation of the modified BPSO to solve the prob-lem, several definitions are pointed out as follows:

4.1.1. Initial placementAs an initial point of the optimization problem, a PMU place-

ment is created by graph theoretic search procedure. A reasonablestarting point can greatly accelerate the convergence speed of thesolution and also increase the possibility of finding the global opti-ma in complex optimization problems [8].

The initial placement is generated based on the following steps.

Step 1: A PMU is placed at a bus located in an unobservableregion which has the maximum number of incident lines.Step 2: Topological observability rules (Fig. 2) are verified, andnumbers and locations of unobserved buses, if any, aredetermined.Step 3: If there is an unobservable region remaining, step 1 isrepeated, otherwise the procedure is stopped.

This placement scheme results in a PMU placement set, whichsatisfies complete system observability. However, as shown in sim-ulation results, this scheme is not necessarily the optimal one, andan optimization tool is required to derive the minimum number ofrequired PMUs.

4.1.2. Reducing search spaceIn order to decrease the number of PMU candidate locations, a

set of buses is identified to be eliminated from search space. Thesebuses are considered to be those that have only one incident line(radial buses), and buses with zero-injection power (zero-injectionbuses). This elimination is attributed to the fact that placing a PMUon a radial bus provides minimum advantage in obtaining phasorvoltages of adjacent buses. Also, placing a PMU on a zero-injectionbus leads to covering of data from these buses and consequentlythe loss of their benefit in observability analysis (i.e. rules 3–5).

4.1.3. PMU placement procedureThe described BPSO is used as the optimization tool and the aug-

mented objective function described in Section 3.3 is formed based

on the objective function and the equality constraint presented inEqs. (10) and (11). An initial population of BPSO is created basedon the graph theory described in Section 4.1.1. The augmentedobjective function for each particle is calculated based on the num-ber of PMUs and the number of unobservable buses corresponding tothat particle. The number of unobservable buses is determined byassessing topological observability rules described in Section 2.

4.2. Optimal PMU placement in contingency conditions

In this stage, an algorithm is developed in which the placementconfiguration derived in normal conditions is modified to maintainnetwork observability during a single PMU loss or line outage. Theobjective function of this part can be expressed mathematically asfollows:

MinimizeXN

i¼1

PMUi ð12Þ

subject to Nnobs ¼ 0jsingle PMU loss or branch outage ð13Þ

The flowchart of topological observability analysis described inSection 2 is adopted to consider the effect of PMU loss or topologychange and identify the total number of unobserved buses duringcontingency conditions. An initial starting point is also generatedto improve the performance of the optimization algorithm in find-ing the global optima of the problem.

The details of PMU placement in contingency conditions are de-scribed in the following sections.

4.2.1. Loss of a single PMUIn order to consider loss of a single PMU, the observability anal-

ysis of a placement scheme is performed as follows. For each PMUlocated in a placement scheme, the topological observability anal-ysis of the network (Fig. 2) is carried out omitting the selected PMUfrom the placement scheme. The total number of unobservablebuses is declared as the sum of unobserved buses during loss ofeach PMU in the placement scheme.

4.2.2. Loss of a single branchIn this part, the observability analysis is performed to consider

the impact of a branch outage on network observability. In order tomaintain network observability during a line outage, each bus ofthe system must be observable from two paths. It is clear that ifone of the paths is lost (single line outage), that bus is still obser-vable through the other path. Note that observability of buses withone incident line is not of interest during single line outage, be-cause those buses become isolated from the network after the lineoutage. In summary, the observability analysis algorithm in contin-gency conditions is described by the flowchart depicted in Fig. 3.

4.2.3. Initializing a starting pointA starting point is generated based on a modification to the

placement scheme derived in normal conditions. The procedureof creating the initial point is described as follows:

Step 1: The base scheme is considered and the first PMU iseliminated.Step 2: By adding the minimum number of PMUs to the adja-cent buses of the eliminated PMU, the observability of the net-work is recovered.Step 3: If all the PMUs in the base scheme are considered go tothe next step, otherwise the next PMU is chosen and step 2 isrepeated.Step 4: Topological observability analysis is carried out andbuses which are not observable through two paths are identi-fied. A PMU is then added to each of these buses.

Fig. 3. Observability analysis considering contingency conditions.

Table 1The numbers of zero-injection and radial buses in the IEEE test systems.

Test system No. of zero-injection buses No. of radial buses

IEEE 14-bus 1 1IEEE 30-bus 6 3IEEE 39-bus 12 9IEEE 57-bus 15 1IEEE 118-bus 10 7

32 M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34

Step 5: The base placement scheme plus the added PMUs areconsidered as the initial placement scheme.

This placement scheme leads to a PMU configuration, whichmaintains complete system observability after loss of a PMUs orbranch outage. However, as mentioned in normal placement, thisscheme is not necessarily the optimal one and the minimum num-ber of required PMUs should be obtained through the optimizationprocedure.

4.2.4. PMU placement procedureThe described BPSO is used as the optimization algorithm to de-

rive the minimum number of required PMUs for system observ-ability during contingency conditions. The augmented function isformed based on Eqs. (12) and (13). In order to calculate theamount of objective function for each particle the modified topo-logical observability analysis (Fig. 3) is employed.

Table 2A comparison among the results of different BPSO algorithms for the case of IEEE-39bus test system in normal-condition placement.

Algorithm Number of required PMUs

Best Average Worst

The modified BPSO 8 8.26 9The proposed BPSO in [22] 8 8.37 9The original BPSO in [21] 8 8.93 10

5. Simulation results

Simulations are carried out on IEEE 14-, 30-, 39-, 57-, and 118-bus test systems [23]. The numbers of zero-injection and radialbuses in the test systems are shown in Table 1. The simulationsare carried out in MATLAB environment on an Intel Pentium III(1.6 GHz) with 512 MB RAM. The following parameters are usedin the simulations: The value of a in Eq. (9) is 2. This is to placeemphasis on the equality constraint to ensure full network observ-ability. The maximum velocity of particles, Vmax, is 1/4 of dimen-sion size of each particle in normal-condition placement and 2/3in contingency-condition placement. The maximum iteration num-

ber in BPSO is selected to be 1000 with the population size of 100.Since heuristic algorithms, such as PSO, are based on a randomsearch in the search space of the problem, the result of each execu-tion of these algorithms might be different from another one.Therefore, they must be run several times to ensure that the opti-mal point of the problem is found. In the simulations presented inthis paper, the best solution of the modified BPSO is found after 50runs of the algorithm.

The performance analysis of the modified BPSO, and the PMUplacement results in normal and contingency conditions are pre-sented in the following sections.

5.1. Performance analysis of the modified BPSO

In order to demonstrate the effectiveness of the modified BPSOin finding the optimal solution of the problem, the performance ofthe algorithm is examined for the case of IEEE 39-bus test system.A comparison of the results in normal-condition placement be-tween the modified BPSO, BPSO presented in [22], and the originalBPSO in [21] is shown in Table 2 in terms of the best, average, andworst results after 50 runs of each algorithm with the same popu-lation size and iteration number as described before. As it can beobserved from this table, a better performance is achieved by usingthe modified BPSO algorithm. The average run time of the modifiedBPSO in this case is 13 min which is almost the same as the othertwo methods. The objective values of the best solutions throughout the iterations for the three algorithms are shown in Fig. 4.The modified BPSO can reach the optimal point of the problem inrelatively lower number of iterations. The modified BPSO findsthe optimal solution after 26 generations, while the proposed BPSOin [22] and the original BPSO in [21] reach the optimal point after38 and 55 generations, respectively.

A sensitivity analysis is also performed on the results to exam-ine the effect of changing the parameters of the modified BPSO. Theresults showed that the average execution time of the algorithmincreases almost linearly with the number of population size whilethe average number of required PMUs decreases as the populationsize increases. For example, changing the population size of thealgorithm to 50, and 150 results in the average execution time of7.2 and 22 min and the average number of required PMUs of 8.5and 8.15. Therefore, selection of the population size of the problemis a compromise between execution time and the quality of thesolution. The same trend is also observed regarding the effect ofmaximum iteration number of algorithm on the results.

10 20 30 40 50 60 70 80 90 1000

7

8

9

10

11

12

13

14

Iteration number

Obj

ectiv

e fu

nctio

n

The modified BPSOThe proposed BPSO in [22]The original BPSO in [21]

6

Fig. 4. A comparison among the best solutions of the modified BPSO, the proposedBPSO in [22], and the original BPSO in [21] for the case of IEEE-39 bus test system.

M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34 33

5.2. PMU placement in normal conditions

Assuming that there is no PMU failure or line outage, the place-ment problem is solved using the modified BPSO algorithm. Theinitial placement using graph theoretic search procedure leads toplacing 5, 11, 12, 19, and 38 PMUs in IEEE 14-bus, 30-bus, 39-bus, 57-bus, and 118-bus test systems, respectively. To decreasethe search space of the BPSO, radial and zero-injection buses areeliminated from the search space of the problem. Therefore, refer-ring to Table 1, the dimension size of the BPSO is 12, 21, 18, 41, 101in IEEE 14-bus, 30-bus, 39-bus, 57-bus, and 118-bus systems,respectively. In order to highlight the significance of the newobservability rules, i.e. rule 6, the PMU placement problem issolved in the following sections excluding and including this rule.In both cases, the average execution times of the algorithm are 1, 6,15, 43, and 85 min in IEEE 14-bus, 30-bus, 39-bus, 57-bus and 118-bus, respectively. Note that, in order to compute the average exe-cution time, the criteria of reaching the maximum iteration num-ber is changed to the one in which the algorithm is stopped afterreaching the predefined objective value.

Table 4The number and locations of the required PMUs obtained in IEEE 57-bus, and IEEE118-bus system including the new observability rule.

Test system Number ofrequired PMUs

Locations ofrequired PMUs

IEEE 57-bus 11 1, 5, 13, 19, 25, 29, 32, 38, 41, 51, 54IEEE 118-bus 28 2, 8, 11, 12, 17, 21, 25, 28, 33, 34, 40,

45, 49, 52, 56, 62, 72, 75, 77, 80, 85,86, 90, 94, 101, 105, 110, 114

Table 5

5.2.1. PMU placement excluding the new observability ruleIn this case, the new topological observability rule is not consid-

ered in observability analysis. Table 3 shows the number and loca-tions of the required PMUs for full network observability in each ofIEEE test cases. As can be seen from Table 3, by using the modifiedBPSO the number of the required PMUs for full system observabil-ity is decreased significantly compared with the number derived ininitial placement. Note that, PMU locations corresponding to theminimum number of the required PMUs might not be a uniquescheme. For example, in IEEE 39-bus system, the other placement

Table 3The number and locations of the required PMUs obtained excluding the newobservability rule.

Testsystem

Number ofrequired PMUs

Locations of required PMUs

IEEE 14-bus 3 2, 6, 9IEEE 30-bus 7 2, 3, 10, 12, 18, 24, 27IEEE 39-bus 8 3, 8, 12, 16, 20, 23, 25, 29IEEE 57-bus 12 1, 6, 9, 15, 19, 25, 27, 32, 38, 50, 53, 56IEEE 118-bus 29 2, 8, 11, 12, 15, 19, 21, 27, 31, 32, 34,

40, 45, 49, 52, 56, 62, 65, 72, 75, 77,80, 85, 86, 90, 94, 101, 105, 110

Cocon

* N

scheme which contains 8 PMUS installed is: 3, 8, 11, 16, 20, 23, 25,and 29, and in IEEE 57-bus test system the other placementscheme, corresponding to 12 PMUs, is: 1, 6, 9, 15, 19, 25, 27, 32,38, 41, 50, and 53. Since from the point of view of full networkobservability, these schemes are similar, only one of them is pre-sented in this section and the following sections.

5.2.2. PMU placement including the new observability ruleThe new topological observability rule is considered in observ-

ability analysis in this case. The number of the required PMUs isdecreased to 11, and 28 in IEEE 57-bus, and 118-bus systems,respectively. The new rule does not affect the number of the re-quired PMUS for the other cases. Table 4 presents the numberand locations of the required PMUs for full network observabilityin IEEE 57-bus, and IEEE 118-bus systems.

A comparison between the number of the required PMUs in theproposed approach and the other approaches used in optimal PMUplacement is performed in Table 5. It can be observed that the pro-posed approach outperforms the most of the other techniques,especially in IEEE 57 and IEEE 118 bus system.

5.3. PMU placement in contingency conditions

Considering the possibility of PMU failure or a single line out-age, a placement scheme is modified to be robust enough to main-tain network observability following those contingencies. Theinitial placement as described in Section 4.2.3 leads to placing 8,20, 30, 42, and 95 PMUs in IEEE 14-bus, 30-bus, 39-bus, 57-bus,and 118-bus test systems, respectively. However, these configura-tions are not the optimal one, and the optimization routine as de-scribed in Section 4.2 is applied to obtain the minimum number ofthe required PMUs. The number and locations of the requiredPMUs for complete system observability is presented in Table 6for different IEEE test cases. The average simulation times of themodified BPSO are 4, 14, 35, 80, and 225 min in IEEE 14-bus, 30-bus, 39-bus, 57-bus and 118-bus, respectively.

Table 7 compares the number of the required PMUs in the pro-posed approach and the other relevant approaches used in optimalPMU placement. From this table, it can be observed that the pro-posed approach maintains the network observability with a lowernumber of PMUs compared to the other techniques.

mparison of results among different methods in PMU placement problemsidering normal conditions.

Method/system 14bus

30bus

39bus

57bus

118bus

Proposed method 3 7 8 11 28Dual search [8] 3 N/A* 8 N/A 29Integer programming [9] 3 N/A N/A 12 29Tree search [10] 3 7 N/A 11 N/ATabu search [11] 3 N/A 10 13 N/AInteger quadratic programming

[16]N/A 10 N/A 17 32

Integer linear programming [17] 3 7 8 11 28

/A is due to the unavailability of the result.

Table 6The number and locations of the required PMUs obtained considering a PMU failureor line outage.

Testsystem

Number ofrequired PMUs

Locations of required PMUs

IEEE 14-bus 7 1, 2, 4, 6, 9, 10, 13IEEE 30-bus 15 2, 3, 4, 8, 10, 12, 13, 15, 16, 18,

20, 22, 24, 27, 30

IEEE 39-bus 17 3, 7, 8, 12, 13,16, 20, 21, 23, 25,26, 29, 30, 34, 36, 37, 38

IEEE 57-bus 22 1, 2, 4, 9, 12, 15, 18, 19, 25, 28,29, 30, 32, 33, 38, 41, 47, 50, 51, 53, 54, 56

IEEE 118-bus 62 1, 3, 7, 8, 10, 11, 12, 15, 17, 19, 21, 22,24, 25, 27, 28, 29, 32, 34, 35, 40, 41, 44,45, 46, 49, 50, 51, 52, 54, 56, 59, 62, 66,68, 72, 73, 74, 75, 76, 77, 78, 80, 83, 85,86, 87, 89, 90, 92, 94, 96, 100, 101, 105,107, 109, 110, 111, 112, 115, 117

Table 7Comparison of results between different methods with regard to placement in normalstate.

Method/system 14 bus 30 bus 39 bus 57 bus 118 bus

Proposed method 7 15 17 22 62Integer quadratic

programming [16]N/A 21 N/A 33 68

Integer linearprogramming [17]

8 17 22 26 65

N/A is due to the unavailability of the result.

34 M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34

6. Conclusion

This paper presented a new approach for the optimal placementof phasor measurement units. At first, the PMU placement wassolved with the goal of minimizing the total number of requiredPMUs for the complete system observability. Then, the effect ofPMU loss or a branch outage was taken into consideration, and aplacement scheme was obtained which maintains complete sys-tem observability during the contingency conditions. A modifieddiscrete binary version of particle swarm algorithm was used asan optimization tool in finding the minimal number of the requiredPMUs for the complete system observability in both cases. Also, animproved topological observability analysis was proposed using anew rule based on observability analysis of zero-injection buses.Numerical results on the IEEE test systems indicated that the pro-posed placement method is capable of providing a placement

scheme in normal and contingency conditions that can competewith the other techniques used in this problem.

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