Expert Systems with Applications 38 (2011) 7263–7269

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Expert Systems with Applications

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Optimal PMU placement for power system observability using binary particleswarm optimization and considering measurement redundancy

A. Ahmadi a, Y. Alinejad-Beromi a,⇑, M. Moradi b

a Faculty of Electrical and Computer Engineering, Semnan University, Semnan, Iranb Department of Technical and Engineering, Faculty of Electrical Engineering, Islamic Azad University, Nowshahr Branch, Iran

a r t i c l e i n f o

Keywords:Phasor measurement unit (PMU)Network observabilityBinary particle swarm optimizationOptimal placement

0957-4174/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.eswa.2010.12.025

⇑ Corresponding author. Tel.: +98 21 66735235; faxE-mail address: Mehrvarzan_dr@yahoo.com (Y. Al

a b s t r a c t

This study presents a binary particle swarm optimization (BPSO) based methodology for the optimalplacement of phasor measurement units (PMUs) when using a mixed measurement set. The optimalPMU placement problem is formulated to minimize the number of PMUs installation subject to full net-work observability and to maximize the measurement redundancy at the power system buses. In order toensure full network observability in an electric power network the topology-based algorithm is used andSeveral factors considered; such as the available data from existing conventional measurements, thenumber and location of zero injection buses, the number and location of installed PMUs and of course,the system topology. The efficiency of the proposed method is verified by the simulation results of IEEE14-bus, 30-bus, 57-bus-118 bus systems, respectively. The results show that the whole system can beobservable with installing PMUs on less than 25% of system buses. For verification of our proposedmethod, the results are compared with some newly reported methods which show the method as a novelsolution to obtain redundant measurement system with the least number of phasor measurement units.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Today with the present of the global positioning system (GPS),communication network and digital signal processing technique,it is quite possible to monitor the operation of power systems. Se-cure operation of any power system is closely related to measure-ment and monitoring of the system operating conditions. Phasormeasurement units (PMU) can gather real-time phasor of bus volt-ages and branch currents in a wide-area electric network. The pha-sors from different nodes, which refer to the same time–spacecoordinate, can improve performances of monitored control sys-tems in various field of modern power systems, such as flow calcu-lation, state estimation, transient stability analysis and frequencystability analysis (Phadke, 1993).

If we place the PMUs in all busses in a power system, it can becompletely observable and so do not need any more calculation.But it is neither economical nor necessary to install PMUs at allnode of a wide-area interconnected network. Since, due to the factthat each PMU can measure not only the bus voltage but also thecurrents along the lines incident to the buses, so selecting suitablebuses and placing PMUs on them can make the entire system ob-servable. A power system is considered completely observable

ll rights reserved.

: +98 21 66733274.inejad-Beromi).

when all the states in the system can be uniquely determined(Zhigang, Zhifan, & Zengqiang, 2000).

In recent years, many investigators presented different methodsfor finding the minimum number and optimal placement of PMUsfor making a power system completely observable. In Baldwin,Mili, Boisen, and Adapa (1993), a bisecting search method is imple-mented to find the minimum number of PMUs to make the systemobservable. The simulated annealing method is used to randomlychoose the placement sets to test for observability at each step ofthe bisecting search. The authors in Nuqui and Phadke (2005)use a simulated annealing technique in their graph-theoretic pro-cedure to find the optimal PMU locations. A genetic algorithm isused to find the optimal PMU locations by authors in (El-Zonkoly,2006; Milosevic & Begovic, 2003). The authors in Rakpenthai,Premrudeepreechacharn, Uatrongjit, and Watson (2005) use thecondition number of the normalized measurement matrix as a cri-terion for selecting the candidate solutions, along with binary inte-ger programming to select the PMU locations. In Xu and Abur(2005, 2004), the authors use integer programming to find theminimum number and locations of PMUs. However, the issue ofmeasurement redundancy was not addressed, and the problem oflocal minimum may affect the solution. In Chakrabarti and Kyriak-ides (2007, 2008) the authors propose an exhaustive search basedmethodology to determine the minimum number and optimallocations of PMUs for complete observability of the power system.The main problem of the methods described by different authors is

Fig. 1. PMU placement rule 1.

7264 A. Ahmadi et al. / Expert Systems with Applications 38 (2011) 7263–7269

the time consuming for large systems as well as the measurementredundancy.

Recently the particle swarm optimization (PSO) technique hasbeen used successfully in a number of power system applications(Alinejad-Beromi, Sedigizadeh, & Sadighi, 2008; del Valle, Venay-agamoorthy, Mohagheghi, Hernandez, & Harley, 2008). We useda binary particle swarm optimization (BPSO) based method forfinding the minimum required number of PMUs when using amixed measurement set and also to maximize the measurementredundancy. The next sections of the paper are organized as fol-lows. A brief explanation of the proposed approach for observabil-ity analysis based on PMUs is given in Section 2. The PMUplacement problem formulation is discussed in Section 3. In Sec-tion 4, a brief discussion of the BPSO and its enhanced version ispresented. Case studies and analysis of the results are given in sec-tion 5. Section 6 concludes the paper.

Fig. 2. PMU placement rule 3.

2. Observability analysis based on PMUs

There are two kinds of algorithms used to determine networkobservability: numerical methods and topological methods.Numerical methods have great calculation quantity and their accu-racy is liable to the influence of cumulative errors. So, it is neces-sary to analyze power system observability through fast topologymethod (Zhao, Li, Mi, & Yu, 2005).

2.1. Linear model of system with PMUs

Consider an N-bus system provided with m-measurements ofvoltage and current phasors contained in vector Z. The vector Z islinearly related to the N-dimensional state vector X containing N-nodal voltage phasors, resulting in n = 2N � 1 state variables (Pha-dke, Thorp, & Karimi, 1986). This yields the linear model:

Z ¼ HX þ e ð1Þ

where Z is a vector of measurement values, X is a vector of statevariables to be estimated, H is a measurement matrix and e is ameasurement error vector. Since, the PMU measurement based onGPS synchronism technology is far more accurate than traditionalanalog measurements based on SCADA, we can neglect the errorand choose directly the PMU measurement values as the estimatevalue. This yields the simple linear observation model. The voltagephasors at all bus voltages are chosen as state variables. The mea-sured values are the bus voltage phasors and the injection currentphasors. By splitting the vector Z into the voltage and currentsub-vectors (Zv and ZI), and the vector X into the measured andnon-measured sub-vectors (VM and VC), Eq. (1) can be rewritten asfollows:

ZV

ZI

� �¼ HX ¼

I 0YIM YIC

� �VM

VC

� �ð2Þ

where I is the identity matrix, YIM and YIC are submatrices whose en-tries are series and shunt admittances of the network branches.Conventional observability analysis can check the satisfaction ofthe following condition:

rankðHÞ ¼ 2n� 1 ð3Þ

The Eq. (3) predicates that if the measurement matrix is of full markstate estimation can be done. But, the calculation quantity of verify-ing the high-dimension matrix H is great which is hard for applica-tion. Generally, for any placement scheme, the Eq. (3) is not alwaystrue and cannot provide the influence on power system observabil-ity of placing PMUs on different buses.

2.2. Observability topology analysis method

Topology methods use the decoupled measurement model andgraph theory. In these methods decision is based on logical opera-tions therefore, they require only information about network con-nectivity, measurements types and their locations. If a full rankspanning tree can be constructed with current measurement set,the system will be observable. In this paper we use observabilitytopology analysis method based on PMUs according to observabil-ity rules below:

1. For a PMU installed bus, voltage phasor of that bus and currentsphasors of all incident branches to that bus are known. Theseare called as direct measurements (Fig. 1).

2. If voltage and current phasors at one end of a branch are knownthen voltage phasor at the other end of that branch can beobtained. These are called pseudo measurements.

3. If voltage phasors of both ends of a branch are known then thecurrent phasor of this branch can be obtained directly (Fig. 2).These measurements are also called pseudo measurements.

4. For a zero-injection bus i in a N-bus system we have:

XN

j¼1

YijV j ¼ 0

Therefore, if there is a zero-injection bus without PMU whose allincident branches current phasors are known but one, then thecurrent phasor of the unknown one can be obtained using KCLequations (See Fig. 3).

5. If there is a zero-injection bus with unknown voltage phasorand voltage phasors of its adjacent buses are all known thenthe voltage phasor of the zero-injection node can be found bynode equations.

6. If there exists a group of adjacent zero-injection buses whosevoltage phasors are unknown but the voltage phasors of alladjacent buses to the group are known then the voltage phasorsof zero-injection buses can be obtained through node equations.The measurements obtained from rules 4–6 are called extendedmeasurements.

Fig. 3. PMU placement rule 4.

Fig. 4. 7-Bus example system.

A. Ahmadi et al. / Expert Systems with Applications 38 (2011) 7263–7269 7265

3. PMU placement problem formulation

3.1. Topology based formulation

PMU placed at a given bus is capable of measuring the voltagephasor of the bus as well as the phasor currents for all lines inci-dent to that bus. Thus, the entire system can be made observableby placing PMUs at strategic buses in the system. The objectiveof the PMU placement problem in this paper is to minimize thenumber of PMUs that can make the system observable, and tomaximize the measurement redundancy in the system. The Objec-tive function therefore should evaluate, for the position vector ofeach particle, (1) whether the system is observable, (2) in case itis observable, what is the number of PMUs employed, and (3) themeasurement redundancy. The measurement redundancy is de-fined as in (London, Alberto, & Bretas, 2007). The redundancy levelof a measurement is equal to the number (p � 1) which corre-sponds to the smallest critical p-set to which the measurement be-longs. For instance, if the number of times a bus is observed by aPMU is increased by one, the measurement redundancy at thatbus is also increased by one. In this paper, the OPP is formulatedbased on topological observability method as follows:

MINXN

I¼1

wixi þ C � ðM � AXÞTðM � AXÞ ð4Þ

Subject to FðXÞP 1̂ðfiðxÞP 1 i ¼ 1;2; . . . ;NÞ.The parts of the objective function representing the measure-

ment cost with PMU such that the entire system becomes observa-ble and the measurement redundancy, respectively. Where N is thenumber of system total buses, wi is weighting factor accounting forthe cost of installed PMU at bus i, C is constant coefficient. X is abinary decision variable vector, whose entries are defined as:

xi ¼1; if a PMU is installed at bus i

0; otherwise

�ð5Þ

The elements of the binary connectivity matrix A for a power sys-tem are defined as,

Ai;j ¼1; if i ¼ j

1; if i and j are connected0; otherwise

8><>: ð6Þ

The entries of the product AX in (4) therefore represent the numberof times a bus is observed by the PMU placement set defined by x.The vector M can be chosen according to the desired level of mea-surement redundancy in the system. For example, if a measurementredundancy level of 2 is desired at all buses, all the elements of Mare set to 3. The vector ð ~M-AXÞ computes the difference betweenthe desired and actual number of times a bus is observed. Minimi-zation of this difference is therefore equivalent to maximizing themeasurement redundancy. The second term in objective functionis therefore a metric for the measurement redundancy offered bythe PMU placement set. Finally F(X) is a vector function whose en-tries are nonzero if the corresponding bus voltage is observableusing the given measurement set and according to observabilityrules that described later; otherwise its entries are zero.

The expressions for the nonlinear constraints can be formedbased on the knowledge about the locations and types of existingmeasurements. Given a PMU at a bus, it is assumed that the busvoltage phasor and all current phasors along lines connected tothat bus will be available. This also implies that this bus voltage,along with all adjacent bus voltages will also be available (solv-able). The procedure for building the constraint equations will bedescribed for three possible cases where there are (1) only PMUmeasurements, (2) PMU measurements and injections (they may

be zero-injections or measured injections) or (3) PMU measure-ments, injections and flows measurements.

3.2. Considering injection measurement and flow measurements

A system, which contains injections as well as flow measure-ments considers the most general situation where both injectionand flow measurements may be present, So, the objective is toplace PMUs in order to merge the observable islands formed bythe existing conventional measurements and render a fully obser-vable system.

Injection measurements whether they are actual measurementsor zero-injections, are treated the same way. Two different meth-ods have been proposed to take into account the buses with injec-tion measurements. These are topology transformation andnonlinear constraint functions methods. In this paper, we usedtopology transformation for considering these buses. This observa-tion allows a topology transformation where the bus, which hasthe injection measurement can be merged with any one of itsneighbors. A word of caution needs to be added here in that, ifthe optimal solution chooses the newly formed fictitious bus (mer-ger of two actual buses) as a candidate bus, it may place one PMUon one of these two buses or two PMUs on both. In this paper, atopology analysis is applied to check the observability of the sys-tem once this happens. This also assures that the minimum num-ber of PMUs will be placed.

This method is illustrated below by an example. A small 7-bustutorial example is shown in Fig. 4. In this example, the injectionmeasurement at bus 3 will be taken into account when determin-ing the PMU locations. Note that if the phasor voltages at any threeout of four buses 2, 3, 4 and 6 are known, then the fourth one canbe solved using the Kirchhoff’s Current Law applied at bus 3 wherethe net injected current is known.

Note that if the phasor voltages at any three out of four buses 2,3, 4 and 6 are known, then the fourth one can be solved using theKirchhoff’s Current Law applied at bus 3 where the net injectedcurrent is known. Fig. 5, shows the updated system diagram afterthe merger of buses 3 and 6 into a new bus 60. The newly createdbranch 60-4 reflects the original connection between buses 3 and 4.

Fig. 5. System diagram after the merger of buses 3 and 6.

7266 A. Ahmadi et al. / Expert Systems with Applications 38 (2011) 7263–7269

Flow measurement on branch 1-2 in the 7-bus example systemwill be used to illustrate the approach when using this device inthe system.

Existence of this flow measurement will lead to the modifica-tion of the constraints for buses 1 and 2 accordingly.

Modification follows the observation that having a flow mea-surement along a given branch allows the calculation of one ofthe terminal bus voltage phasors when the other one is known.Hence, the constraint equations associated with the terminal busesof the measured branch can be merged into a single constraint. Inthe case of the example system, the constraints for buses 1 and 2are merged into a joint constraint as follows:

f 1 ¼ x1 þ x2 P 1f 2 ¼ x1 þ x2 þ x60 þ x7 P 1

�ð7Þ

f 1new ¼ f 1þ f 2 ¼ x1 þ x2 þ x60 þ x7 P 1

Note that this new constraint ensures that if either one of the volt-age phasors at buses 1 or 2 is observable, then the other one willalso be observable. Applying this modification to the constraintsthe following set of final constraints will be obtained.

f ðxÞ ¼

f 1new ¼ x1 þ x2 þ x60 þ x7 P 1

f 4 ¼ x4 þ x5 þ x60 þ x7 P 1

f 5 ¼ x4 þ x5 P 1

f 60 � x2 þ x6 þ x60 P 1

f 7 ¼ x2 þ x4 þ x7 P 1

8>>>>>><>>>>>>:

ð8Þ

Note that, the constraints corresponding to buses 1 and 2 aremerged into a single constraint. The constraint associated withbus 3 where there is an injection measurement, is eliminated asexplained.

4. Particle swarm optimization

Particle Swarm Optimization is an algorithm developed by Ken-nedy and Eberhart (1995) that simulates the social behaviors ofbird flocking or fish schooling and the methods by which they findroosting places, foods sources or other suitable habitat. In the basicPSO technique, suppose that the search space is d-dimensional,

� Each member is called particle, and each particle (ith particle) isrepresented by d-dimensional vector and described asXi = [xi1,xi2, . . . ,xid].� The set of n particle in the swarm are called population and

described as pop = [X1,X2, . . . ,Xn].� The best previous position for each particle (the position giving

the best fitness value) is called particle best and described asPBi = [pbi1,pbi2, . . . ,pbid].

� The best position among all of the particle best positionachieved so far is called global best and described asGB = [gb1,gb2, . . . ,gbd].� The rate of position change for each particle is called the parti-

cle velocity and described as

Vi ¼ ½v i1; v i2; . . . ;v id�

At iteration k the velocity for d-dimension of i-particle is updatedby:

k� �

k� �

Vkþ1id ¼ wvk

id þ c1r1 pbid � xkid þ c2r2 gbd � xk

id ð9Þ

where i = 1,2, . . . ,n and n is the size of population, w is the inertiaweight, c1 and c2 are the acceleration constants, and are two ran-dom values in range [0,1]. The optimal selection of the previousparameters found in (Shi & Eberhart, 1998; Zhang, Yu, & -X Hu,2005).� The i-particle position is updated by

xkþ1id ¼ xk

id þ vkþ1id ð10Þ

For binary discrete search space, Kennedy and Eberhart (1997)have adapted the PSO to search in binary spaces, by applying a sig-moid transformation to the velocity component Eq. (11) to squashthe velocities into a range [0,1], and force the component values ofthe locations of particles to be 0’s or 1’s. The equation for updatingpositions Eq. (10) is then replaced by Eq. (12).

sigmoid vkid

� �¼ 1

1þ e�vkid

ð11Þ

xkid ¼

1; if rand < sigmoid vkid

� �0; otherwise

(ð12Þ

The PSO technique can be expressed as follow:

(Step 1) (Initialization): Set the iteration number k = 0. Generaterandomly n particles, fx0

i ; i ¼ 1;2; . . . ;ng where:

X0i ¼ x0

i1; x0i2; . . . ; x0

id

; and their initial velocities :

V0i ¼ v0

i1;v0i2; . . . ;v0

id

:

� �

Evaluate the objective function for each particle f X0i . If

the constraints are satisfied, then set the particle bestPB0

I ¼ X0i i, and set the particle best which give the best

objective function among all the particle bests to globalbest. Else, repeat the initialization.

(Step 2) Update iteration counter k = k + 1.(Step 3) Update velocity using Eq. (9).(Step 4) Update position using the sigmoid function Eqs. (11) and

(12).(Step 5) Update particle best: If fi Xk

i

� �< fi PBk�1

i

� �then PBk

i ¼ Xki

else: PBki ¼ PBk�1

i .(Step 6) Update global best: f ðGBkÞ ¼min fi PBk

i

� �n oIf f(GBk) < f(GBk�1) then GBk = GBk else GBk = GBk�1.

(Step 7) Stopping criterion: If the number of iteration exceeds themaximum number iteration, then stop, otherwise go tostep 2.

In this paper we used BPSO for optimal PMU placement basedon observability topology analysis method. In this method, The restof the buses are made observable by placing a minimum number ofadditional PMUs. The position vectors of the particles represent thepotential solutions for the PMU placement problem. As mentionedbefore, a fitness function needs to be defined to evaluate thesuitability of the solutions found by the particles at each stage of

Table 1Results with and without considering zero-injections.

Systems Location of zero-injection buses Number of PMUs

Ignoring zero-injection Using zero-injection

IEEE14-BUS 7 4 3IEEE30-BUS 6-9-11-25-28 10 7IEEE57-BUS 4-7-11-21-22-24-26-34-36-37-39-40-45-46-48 17 13IEEE118-BUS 5-9-30-37-38-63-64-68-71-81 32 29

Table 2PMU location with and without considering zero-injections.

Systems Location of PMUs

Ignoring zero-injection Using zero-injection

IEEE14-Bus 2-6-7-9 2-6-9IEEE30-Bus 2-4-6-9-10-12-15-18-25-27 1-7-10-12-19-24-27IEEE57-Bus 1-4-7-9-15-20-24-25-27-32-36-38-39-41-46-50-53 1-4-9-14-19-22-25-29-32-38-51-54-56IEEE118-Bus 3-5-9-12-15-17-21-23-28-30-36-40-44-46-51-54-57-62-64-68-71-75-80-

85-86- 91-94-101-105-110-1142-8-11-12-15-19-21-27-31-32-34-40-45-49-52-56-62-65-72-75-77-80-85-86-90- 94-101-105-110

A. Ahmadi et al. / Expert Systems with Applications 38 (2011) 7263–7269 7267

iteration. In this paper fitness function based on Eq. (4) for usingBPSO is formulated as follows:

JðxÞ ¼ w1 �XN

i¼1

fi þw2 � NPMU þ C � j1 ð13Þ

w1, w2 and C are three weights with values such thatPNb

i¼1fi andNPMU and j1 are comparable in magnitude.

PNbi¼1fi and NPMU and j1

are the parts of the fitness function representing the number of ob-servable buses and the total number of PMUs and the measurementredundancy, respectively. NPMU, j1 are defined as follows:

NPMU ¼ XT X ð14ÞJ1 ¼ ðM � AXÞTðM � AXÞ ð15Þ

Other variables in this fitness function explained later. The individ-ual best position vector of a particle,pbesti, and the global best posi-tion vector gbest are evaluated based on this fitness function.

5. Case studies and simulation results

The proposed PMU placement method is applied for completeobservability of power systems to the IEEE14-BUS, IEEE30-BUS,IEEE57-bus and IEEE118-bus systems.

Case 1.In case 1, two groups of simulations are carried out on the fourtest systems, which initially have no flow measurements. In thefirst group of simulations, zero-injections are simply ignored

0 100 200 300 400 500 600 700 800 900 1000-44

-42

-40

-38

-36

-34

-32

Iteration

Fitn

ess

Func

tion

Valu

e J(

X)

Fig. 6. Converging trend of BPSO for case 1.

while in the second group, they are used as existing measure-ments. Comparative simulation results are shown in Table 1.Having zero-injections will reduce the number of requiredPMUs as can be seen from these results.Table 2 shows the optimal location of PMUs using binary parti-cle swarm optimization algorithm with and without consider-ing zero-injection buses information.Fig. 6 shows a converging trend of BPSO for the IEEE-57 bus sys-tem. To make the details clear, the vertical axis shows the dif-ference of the Minimum fitness function value between everygeneration and the first generation.In this case the BPSO parameters used in the first step to run thesearch for the optimal set of measurements to make the systemobservable and increase the measurement redundancy are setas follows: (see Table 3)Fig. 7 shows the results of optimal PMU placement for the IEEE-57 bus system, which has 15 zero-injection and no other con-ventional power flow or injection measurements. There are 13PMUs installed at buses 1, 4, 9, 14, 19, 22, 25, 29, 32, 38, 51,54, 56, which can make the whole system observable.Table 4 shows the effect of the maximization of PMU measure-ment redundancy on the 30-bus test system. The first set ofPMU locations in Table 4 is obtained by minimizing the numberof PMUs only, while ensuring the complete observability of thesystem. The second set of PMU locations in Table 4 is obtainedby minimizing the number of PMUs, as well as maximizing themeasurement redundancy at the buses. The target value for themeasurement redundancy is taken as 2, i.e., all the elements ofthe vector M in (15) are set to 3. This results shows theimprovement in the distribution of measurement redundancyat the buses in the case of the second solution described above.The third column in Table 4 shows the number of times thebuses 1 to 30 in the 30-bus system are observed by the two dif-ferent PMU placement sets. The number of times the buses areobserved is more in the second case, compared to the first case.To verify our results, in Table 5 we compared the results ofBPSO algorithm with that of genetic algorithm (GA) (Milosevic& Begovic, 2003), integer programming (IP) (Xu & Abur, 2004)and tabu search (TS) (Peng, Sun, & Wang, 2006) as shown inTable 5. Unlike the other algorithms, we considered the mea-surement redundancy in BPSO method.Case 2.In this case simulations are carried out using IEEE57-bus sys-tem. Three sets of flow measurements (P and Q) containingthree flow measurements each, are added in the system one

Table 3BPSO parameters.

Maximum iteration 1000Population size 100Inertia weight 0.4w1 �2w2 1C 0.01

Fig. 7. Optimal PMU locations for the IEE-57 bus system.

Table 4Effect of the maximization of PMU measurement redundancy on the 30-bus testsystem.

Systemconfiguration

Optimal PMUlocations

Number of times eachbus is observed

Normal operating conditions,without maximizingmeasurement redundancy

1-5-10-12-18-23-27

1-2-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1

Normal operating conditions,maximizing measurementredundancy

1-7-10-12-19-24-27

1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-2-1-1-2-1-1-1-1-1

Table 5Number of PMUs resulting from BPSO and other optimization algorithm.

Algorithm 14-Bus 30-Bus 57-Bus 118-Bus

BPSO 3 7 13 29GA (Milosevic & Begovic, 2003) 3 7 12 29IP (Xu & Abur, 2004) 3 7 12 29TS (Peng et al., 2006) 3 – 13 –

Table 6Locations of flow measurements used for case 2.

Meas. set no. Flow measurements in the set

1 1–2 14–15 12–172 29–52 52–53 53–543 41–43 50–51 54–55

Table 7Simulation results for case 2.

System Flow measurementsset

Number ofPMUs

Location of PMUs

IEEE57-Bus

None 13 1-4-9-14-19-22-25-29-32-38-51-54-56

1 12 1-4-9-19-25-29-32-38-46-50-53-56

1 and 2 11 1-4-9-19-25-29-32-38-47-50-56

1, 2 and 3 10 6-14-16-19-25-29-32-38-51-56

0 100 200 300 400 500 600 700 800 900 1000-46

-45

-44

-43

-42

-41

-40

-39

-38

Iteration

Fitn

ess

Func

tion

Valu

e J(

X)

Fig. 8. Converging trend of BPSO for case 2.

7268 A. Ahmadi et al. / Expert Systems with Applications 38 (2011) 7263–7269

A. Ahmadi et al. / Expert Systems with Applications 38 (2011) 7263–7269 7269

at a time. The locations of these flow measurements in the sys-tem are given in Table 6.Simulation results are shown in Table 7. The required numberof PMUs is reduced from 13 to 10. Hence, as expected, havingconventional measurements reduces the number of requiredPMUs to make the entire system observable.Fig. 8 shows a converging trend of BPSO for optimal PMU place-ment in IEEE_57-bus system, which has 15 zero-injection and 9conventional power flow measurements.

6. Conclusions

This paper presents a new methodology for the optimal place-ment of PMUs for making a power system topologically observable.The problem formulation has some attractive properties whereconventional measurements such as injections and flows can alsobe taken into account if they already exist in the system. A binaryparticle swarm optimization (BPSO) based approach is used todetermine the optimal location of PMUs. The optimization processtries to attain dual objectives: (a) to minimize the number of PMUsneeded to maintain complete observability of the system, and (b)to maximize the measurement redundancy at all buses in the sys-tem. The method was successfully applied on IEEE test systems.The main contribution of this work lies in investigating the feasi-bility of using BPSO for the PMU placement problem.

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