Optimal PMU placement for full network observability

using Tabu search algorithm

Jiangnan Peng a, Yuanzhang Sun b, H.F. Wang c,*

a China Power Investment Corporation, Beijing, People’s Republic of Chinab Department of Electrical Engineering, Tsinghua University, Beijing, People’s Republic of China

c Department of Electronic and Electrical Engineering, University of Bath, 39 Burnt House Road, Bath BA2 2AQ, UK

Received 4 April 2003; received in revised form 17 May 2005; accepted 25 May 2005

Abstract

This paper presents a fast analysis method for power system topology observability. The method is based on the linearized power system state

estimator model and uses augmented incidence matrix. In the paper, the Optimal PMU (phasor measurement units) Placement (OPP) problem is

formulated as to minimize the number of PMU installation subjecting to full network observability and enough redundancy. A global optimization

algorithm, Tabu search, is proposed to solve the combinatorial optimization problem and a priority list based on heuristic rule is embedded to

accelerate optimization. The effectiveness and flexibility of the proposed algorithms are demonstrated by numerical results tested in IEEE 14,

IEEE 57 and NE 39 bus systems.

q 2005 Published by Elsevier Ltd.

Keywords: Network observability analysis; Optimal PMU placement (OPP); State estimation; Tabu search (TS)

1. Introduction

In recent years, applications of phasor measurement units

(PMUs) have been attracting more and more attentions in

power systems security monitoring and control. This is due to

the advantages that PMUs can offer real-time synchronized

phasor measurements (voltage, current, powers, frequency etc.)

contrary to the conventional SCADA measurement devices

[1,2]. The prerequisite for an efficient and accurate control is

the development of adequate meter placement scheme, which

can realize the network full observability.

As we know, observability analysis is a fundamental

component of real-time state estimation, which acts as the

back bone of EMS applications. The theory of network

observability can be divided into two main classes of

algorithms: numerical and topological methods. The topologi-

cal methods are based on whether a spanning tree of full rank

can be constructed. The numerical methods rely on whether the

measurement information gain or Jacobian matrix is of full

rank. In this area, a lot of interesting work has been reported

0142-0615/$ - see front matter q 2005 Published by Elsevier Ltd.

doi:10.1016/j.ijepes.2005.05.005

* Corresponding author.

E-mail address: wanghf60@hotmail.com (H.F. Wang).

[3–9] and each of them has its own advantages and limitations.

Conventionally, numerical methods involve huge matrix

manipulation, and are computationally expensive. Moreover

the accuracy of solution is apt to suffering from the

computation error. In theory, if all nodes in power system

have been installed with PMUs, then the whole system state is

fully observable. However, considering the cost of the

equipment together with communication links, optimal PMU

placement (OPP) problem demands to reduce the number of

PMU installation and concerns about where and how many

PMUs should be implemented to a power system to achieve

full-state observability at minimal cost. This problem began

being addressed recently and certain progress has been

achieved [10–13].

This paper introduces a novel topological method based

on the augment incidence matrix proposed in [13] and the

Tabu Search (TS) [14–18]. By doing so, solution of the

combinatorial OPP problem requires less computation and is

of higher robustness. The method is much faster and more

convenient than the conventional observability analysis

method using complicated matrix analysis. The paper is

organized as follows. Section 2 presents the proposed

observability analysis method based on incidence matrix for

PMU applications and derives a linear phasor estimation

model benefited from the definition of defining different

measurement layers. In Section 3, the OPP problem is

Electrical Power and Energy Systems 28 (2006) 223–231

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J. Peng et al. / Electrical Power and Energy Systems 28 (2006) 223–231224

formulated and TS algorithm is applied to solve the combined

optimization problem. Three IEEE test system results are

given in Section 4 and the work is concluded in Section 5.

2. Observability analysis using augment incidence matrix

for PMU application

Observability is defined as the ability to uniquely estimate

the states of a power system using given measurements. It is

well-known that the state estimation can not work even if

measurements are redundant. Observability analysis is required

to decide meter placement in order to maintain solvability of

the observation equations in various conditions. Implemen-

tation of PMU presents an opportunity for improving

observability analysis and state estimation. Linear phasor

estimation model

Considering a power system with n nodes, the output

of the state estimation is 2nK1 dimensional state vector

XZ[V1,.,Vn, q1,.,qnK1]T containing the voltage magni-

tudes and phasor angle. The measurement equation can be

expressed as [2]

ZZ hðxÞC3 (1)

where measurement vector ZZ[Z1,Z2,.,Zm]T, Z2Rm;

measurement function h(x), h:R2nK1/Rm; measurement

error vector 32Rm.

Since PMU can directly measure phasor, in order to

establish the fast phasor estimator model, we select the optimal

candidate measurement of essential measurements as the PMU

correlative measurement sets (CMS). CMS includes measure-

ments of the voltage phasor at a bus where a PMU is placed and

all the current phasors incident to the bus. CMS, denoted by

Zcor, can be expressed as

Zcor Z fZig; Zi Z ½ _Vi; _Ii1 ;.; _Iil�T iZ 1;.; np (2)

where np is the number of PMU placement and, il is the total

number of branches incident to bus i. Hence, np PMU can

directly provide 2np bus voltage and the corresponding ilcurrent phasor at all the measurement node i. Obviously, the

CMS can be represented as ZcorZ[ZvZI]T, where Zv, ZI are

bus voltage phasor and branch current phasor, whose

dimensions are mVZnp and

m1 ZXnpiZ1

il

respectively. Since, the PMU measurement based on GPS

synchronism technology is far more accurate than traditional

analog measurements based on SCADA, we can neglect the

error and choose directly the PMU measurement values as the

estimate value. This yields the simple linear observation

model. For the PMU direct measured voltage VM and the

adjacent non-measured voltage VC, Eq. (1) can be rewriten as

follows [2]

ZV

ZI

" #ZHxZ

I 0

MIBYBBATMB MIBYBBA

TCB

" #VM

VC

" #(3)

where, MIB is ml!b measure-branch incidence matrix

associated with the current phasor measurements, YBB is b!n

diagonal matrix of branch admittances, AMB and ACB are

measured and non-measured node-branch incidence subma-

trices, respectively.

Conventional observability analysis can check the satisfac-

tion of the following condition

rankðHÞZ 2nK1 (4)

The above condition means that the measurement matrix is

full rank. However, it is difficult to directly assess H due to its

massive dimension. Generally, an arbitrary PMU placement set

cannot guarantee the satisfaction of Eq. (4). Moreover, Eq. (4)

can not provide the quantitative index for analyzing different

PMU placement scheme. Hence, it is necessary to develop a

quantitative assessment technique to guide the selection of

PMU placement sets based on the following observability

conditions:

Condition 1. If one end bus voltage and branch current

phasor of a branch is known, then the other end bus voltage

phasor can be calculated via branch equation.

Condition 2. For a branch with known voltage phasor at its

both ends, its current phasor can be calculated.

Condition 3. For a zero-injection node without a PMU

placement, if just only one of the incidence branch current is

unknown, then the current can be calculated by KCL law.

Condition 4. When all the bus voltages of an unknown zero-

injection node are known, the voltage can be computed by

corresponding node voltage equation.

Based on these conditions above, it is convenient to classify

the measurements, from the topological point of view, into

three layers, the measurement, pseudo-measurement, and the

extension-measurement. The states variables connected

directly to PMUs buses (i.e. the CMS) can be defined as the

measurement layer variables. Similarly, pseudo-measurement

variables are voltages or currents those can be calculated by

CMS via Ohm law (conditions 1 and 2 above) and extension-

measurement variables are the voltages or currents that can be

inferred by using the KC circuit laws (conditions 3 and 4

above) via zero-injection node. Hence, Eq. (3) can be rewritten

according to measurement, pseudo-measurement and

extended-measurement respectively. The relation between the

PMU installation bus and its adjacent bus can be expressed as

follows:

ZV

ZI

" #ZHxZ

I 0

YIM YIN

" #VM

VM

" #(5)

where, VNC is pseudo-measurement voltage phasor of the

neighbor buses.

For zero-injection bus, when all its adjacent voltage phasors

are known, it can be regarded as an extended-measurement and

J. Peng et al. / Electrical Power and Energy Systems 28 (2006) 223–231 225

be calculated as follows

VMA

VZ

" #Z

I 0

KYZM YZC

" #K1ZZV

0

" #(6)

where, ZZV is the known voltage phasor around the zero

injection bus and VZ is the voltage phasor of zero injection bus.

YZM and YZC represent the admittance matrix of zero injection

bus and its neighbor buses respectively. For the PMU bus and

their neighbor bus, the state vector can be attained using Eq.

(5). And the extension states can be calculated via Eq. (6). If

the number of measurement, pseudo and extension states for a

specified PMU is larger; the PMU has more contribution to the

network observability. Also we can see that the value of branch

admittance matrix YBB has no impaction in topology

observability. Hence, we can assume it to be a unity matrix,

i.e. YBBZI. Thus the measurement matrix in Eq. (4) can be

written in a simple form as follows

HZI 0

MIBATMB MIBA

TCB

" #(7)

It is clear that the network observability mainly depends on

the measurements placement scheme (M) and the network

topological structure (A).

2.1. Observability analysis using augment incidence matrix

Network observability can be checked through the above

different measurement layer variables. We can define different

variables to express the bus voltage and branch current

according to their measurement layers, e.g. symbols W,V,U,

for extended-measurement, pseudo-measurement and

measurement layers respectively. Then by searching the three

layer measurements for every PMU in incidence matrix, how

many PMUs contribute to estimate the each bus voltage and

line current can be estimated. Finally, the network observa-

bility and measurement redundancy can be identified. The

whole procedure can be illustrated by IEEE 14 bus system as

follows.

The observability augment incidence matrix ~A of studied

system is defined in Figs. 1 and 2. The values of row WI, VI,

UI or the column WV, VV, UV stand for times of the

corresponding branch current or bus voltage which falls into

extension-measurement, pseudo-measurement and measure-

ment. It should be indicated that UV is the PMU placement

set. UVZ1 means that the node is placed with PMU and

UVZ0 that the node has no PMU placement. Column SV is

the node voltage observability state variable, whose value

stands for the total observable times of corresponding bus

voltage. The zero-injection node is marked by initial value 0.

Other columns are the incidence vector between branches m

and n with the formation MmnZ ½0/1mK1n/�T. Row SI is

branch current observability state variable, whose value

stands for the total observable times of corresponding line

current. Other rows are the ith node incidence vector Ti,

where non-zero elements mean the corresponding branch

incidence to the ith node.

When set a PMU on a node, the corresponding measure-

ment, pseudo-measurement and extension-measurement can be

searched from augment incidence matrix ~A according to the

following rules.

(1) Measurement assigning rule. The directly measure-

ments, CMS, can be found through the following search: first,

set the node voltage observability UV(i)Z1 for bus i installed

with a PMU; then assign a current phasor measurement to each

branch incident to bus i by searching the incidence vector Ti.

i.e. UI(k)Z1, k2Ti, ks0.

(2) Pseudo measurement assigning rule. Apply observa-

bility condition 1 and search every observable current (UIZ1)

branch incidence vectorMji, where there are only two non-zero

elements; Assign the non-zero element node j except the

measurement node i, a pseudo-voltage measurement, VV(j)Z1.

Similarly, apply observability condition 2 and search all the

pseudo-voltage nodes (VVZ1); Assign the pseudo-current

measurement to the unobservable branch current incident to

these nodes, VI(l)Z1, l is the line between pseudo-voltage

nodes i and j; Then the system observability index can be

calculated as

SVðiÞZUVðiÞCVVðiÞ iZ 1; 2;.; n (8)

SIðjÞZUIðjÞCVIðjÞ jZ 1; 2;.; b (9)

(3) Extension-measurement assigning rule. For all the non-

measurement (UVZ0) and observable (SVs0) zero-injection

node Z, check whether observability condition 3 is satisfied; If

it is true, assign the branch current as extension-currentWIZ1;

When the new extension-current measurement is found, repeat

applying observability conditions 2 and 3 to find new

extension-measurement, all will be assigned as extension-

measurement WV, WI; Check if observability condition 4 is

satisfied; assign the corresponding voltage and current as

extension-measurements.

The corresponding three layers of measurements of node

voltage and their search routes are also labeled in the Figs. 1(b)

and 2(b).

After applying above assigning rules, the performance index

of network observability of the PMU set can be calculated as

follows

SVðiÞZUVðiÞCVVðiÞCWVðiÞ iZ 1; 2;.; n (10)

SIðjÞZUIðjÞCVIðjÞCWIðjÞ jZ 1; 2;.; b (11)

The above search algorithm can be summarized by the

following flow chart

The algorithm is a searching process of the augment

incidence matrix in the case of adding a PMU in network.

When removing a PMU from system, the same search

algorithm can be applied with the modification to subtract

the value of all found measurement to network observability

Fig. 1. Initial PMU placement results of IEEE-14 bus system. (a) Augment incidence matrix. (b) Node voltage measurement variables.

J. Peng et al. / Electrical Power and Energy Systems 28 (2006) 223–231226

index. In addition, the search algorithm can be extremely fast

by taking advantage of the sparsity of the incidence matrix ~A.When all the np PMUs complete placement, the whole

system node observability vector �SV and branch observability

vector �SI can be computed via adding all the PMU

measurement variables

�SV ZXm

SVðmÞ �SI ZXm

SIðmÞ mZ 1;.; np (12)

The network full observability can be checked through if all

the individual part of vector �SVR1. If true, the network full

Fig. 2. Optimal PMU placement results of IEEE-14 bus system. (a) Augment incidence matrix. (b) Node voltage measurements variables.

J. Peng et al. / Electrical Power and Energy Systems 28 (2006) 223–231 227

observability is guaranteed. The following logical function is

used to evaluate the checking procedure

Obsðnp; SðnpÞÞZ 15 �SVðiÞR1; iZ 1; 2;.; n (13)

or Obsðnp; SðnPÞÞZ05 at least exist one individual component

of �SVðiÞZ0; ci2n (14)

The number of individual value of vector �SV!1 is the

unobservable number of nn. In Eqs. (13) and (14), S(np) is the

PMU placement set and, also UV is the PMU allocation

vector.

The redundancy measurement index can be defined as

Rðnp; SðnpÞÞZ �SV C �SIKðnCbÞ (15)

The maximum redundancy is the case that with the same

number of PMU’s installation, the optimal placement scheme

will have the maximum redundancy. Compared with the

Start

Intialize PMU placement set

and the intial number np_inital

Initialize the upper limitnp_max = np_inital

and the lower limit np_min=0.1*n

Set current number

np=[0.3np_max+0.7np_min]

Execute TS to generatenew solution

Is full observable?

Is full observable?

End

YN

N

Is satisfied quitcondition

Y

N

Y

set the new upper limitnp_max = np

Set the new lower limitnp_min=np

np_max-np_min = 1?

Y

N

Select current placment set Uvusing heuristic rule

Fig. 3. Flow chart of OPP solution algorithm.

J. Peng et al. / Electrical Power and Energy Systems 28 (2006) 223–231228

traditional observability analysis method, the method

proposed above is simple, fast and quantitative, suitable for

the applications in large power systems and convenient in the

assessment of the optimal PMU placement scheme.

3. Opp formulation and solution

3.1. Problem formulation

The OPP problem is to determine the minimum number npand the optimal location set S(np) of PMU’s to satisfy network

topology observability and preset redundancy criterion. It can

be formulated as follows

J Zminnp

fmaxRðnp; SðnpÞÞg (16)

s.t.

Obsðnp; SðnpÞÞZ 1 (17)

Mathematically, OPP is a large-scale combinatorial optimi-

zation problem and the system observability is dependant of two

factors, the number of PMU np and the placement set S(np). So far

there has been no proposed scheme to directly obtain the optimal

solution of number np_min. Usually, it is estimated by try and

error. The search space ofnp is [0, b], and S(np) is the combination

number Cnpb . Obviously, the OPP is a highly nonlinear,

discontinuous and multi-modal (i.e. there may exist multiple

local optimum) problem. Moreover, its objective function is

nonconvex, nonsmooth, and nodifferentiable. Hence, it is

essential to employs a derivative-free optimization technique in

solving OPP problem in order to keep computational cost low. In

this paper, TS algorithm is proposed to solve the OPP problem.

3.2. Set of initial PMU placement

The initial PMU setting is arranged in the following way

[10]

(i) Place a PMU at the bus with the highest number of

incident branches in system unobservable region;

(ii) Determine the measurement, pseudo and extension

measurements by the current PMU placement set using the

method proposed in the above section;

(iii) If system is not full observable, then continue step (i)

and until observed region covers the whole system.

It is cleat that this procedure will lead to an initial PMU set

which can realize whole system observable but not an optimal

scheme as outlined by Eqs. (16) and (17). The initial PMU

allocation of IEEE-14 bus system is shown in Fig. 1(b) and the

initial number np_initialZ5. The corresponding augment

incidence matrix result is given in Fig. 1(a). Apparently, it

satisfies the observability evaluation function Obs(np, S(np))Z1

and the redundancy value is 8.

3.3. Solution method

Fig. 3 shows the procedure of OPP solution approach

proposed, which separates the searching of the minimum PMU

number np from observability analysis. A modified bisecting

search is used to iteratively find the optimal PMUs number npand the updating of np employs the following equation

np Z INT½0:7np_min C0:3 np_max� (18)

According to experience, the minimum number of np is

typically around 1/4 to1/3 of the total number of system buses.

Hence, a good choice for initial lower and upper limits of npsearch range is suggested to be np_minZ2, np_maxZnp_initial.

For a given number of PMUs, the placement set variable S(np)

or UV solution is a combinatorial optimization problem,

the TS algorithm can be employed to identify the optimal

placement set.

Tabu search (TS) was early proposed by Glover and has

become a well established combinatorial optimization

approach which has been applied in many fields [15–18].

The theory of TS is well documented in [14]. In [13,15], TS

is regarded superior to other heuristic approaches such as SA

and GA. Unlike other heuristic techniques, the advantage of

TS algorithm is the fact that it uses a flexible memory of

search history to prevent cycling and to avoid entrapment in

local optima. It has been theoretically proved that TS

J. Peng et al. / Electrical Power and Energy Systems 28 (2006) 223–231 229

algorithm can yield global optimal solution. The basic

elements and general algorithm of TS can be found in [14].

TS algorithm in this particular application for the optimal

PMU setting can be introduced as follows:

(a) Step 1. This is to set the iteration counter kZ0 and the

initial solution S(np) to be the current solution as and the best

solution, i.e. SbestZSnowZS0. The Tabu list is empty, HZf,with length Tl. By using incidence matrix method, the network

observability of current placement scheme is assessed.

(b) Step 2. This is to generate a set of trial solution, Strial, in

the neighborhood of the current solution based on the specified

moving rule, i.e. to create the candidate set C(Snow). Base on

the evaluation function value, the best trial solution Sbesttrial in

C(Snow) is chosen and the Tabu status of Sbesttrial is checked; if it is

not in the Tabu list, then put it in the Tabu list, set SnowZSbesttrial

and go to Step 4; if it is in the Tabu list, go to Step 3.

(c) Step 3. This is to check the aspiration criterion of Sitrial. If

it is satisfied, update Tabu restrictions and aspiration level and

set SnowZSitrial.

(d) Step 4. This is to check the stop criteria. If any of them is

satisfied, then stop; else set kZkC1 and go back to step 2.

In this paper, an improved technique has been developed to

make TS more efficient in solving OPP problem. A heuristic

rule is suggested for solution effective moves. Whenever a

PMU is removed from a node, the bus with the least branches is

considered firstly. Whenever a PMU is added to a bus, the bus

with the most branches is considered firstly. It has been proved

by simulation that these two heuristic rules can speed up the

TS convergence and improve solution quality. Main operations

of TS include moves, Tabu list and aspiration etc., which are

explained further as follows.

3.3.1. Moves and set of candidate moves

Moves are the local search process in the neighborhood of

current solution Snow to generate trial solutions. The candidate

move set, C(Snow), is operated simply by just changing one

PMU placement at a time, e.g. solution S(np) moves as:

(01011)/(00111), (10011), etc.

3.3.2. Control of Tabu list

Tabu list length Tl plays an important role in searching time

and solution quality. If Tl is too short, solution quality can

deteriorate. However, if it is too long, it may pose massive

computation burden. Hence, a varying-length Tabu list is

suggested and used here. The length is set in the range

Tl2[Tlmin, Tlmax]. The limits are changed according to the

solutions in Tabu list and the neighboring structure of current

solution.

Table 1

Simulation results

Test systems n np_ini Rini np_min np_min/n Rmin

IEEE_14 14 5 8 3 0.214 2

NE_39 39 15 23 10 0.25 17

IEEE_57 57 16 37 13 0.228 14

3.3.3. Aspiration criteria

Aspiration criteria can override Tabu restrictions. The

criteria considered here are to override the Tabu status of a

move if this move yields a better solution than the one obtained

earlier with the same move.

3.3.4. Evaluation function

The evaluation function is devised as a penalty function as

follows

min Jðnp; SðnpÞÞZ snuðnp; SðnpÞÞCnpKl$Rnðnp; SðnpÞÞ (19)

where s is a large positive number, nu(np, S(np)) is the

unobservable node number corresponding to the solution S(np),

l is a relatively small positive number. This function can

integrally evaluate system observability, number of PMUs and

measurement redundancy performances. It is clear that the

minimum value of evaluation function is obtained only if

network full observability (nuZ0), the minimum number of

PMUs and the maximum R(np, S(np)) are achieved.

3.3.5. Stopping criterion

In our study, the search is terminated if one of the following

criteria is satisfied: (a) the number of iterations is greater than a

pre-specified number before the network full observability is

satisfied. This needs to increase the number of PMUs, np; (b)

for a feasibility solution, the number of PMUs can not decrease

further and at the same time the redundancy cannot improved.

3.3.6. Initialization

The heuristic search rule used in TS moving operation is

also employed to initialize the solution for every updated

number of PMUs, np. It is shown that this rule can greatly

reduce computational time compared with the random

generation of the initial solutions.

4. Simulation and test results

The proposed algorithms have been tested for the OPP

problem of IEEE 14 bus, New England 39 bus and IEEE 57 bus

systems. The simulation results are given in Table 1.

4.1. IEEE 14 bus system

IEEE-14 bus system only has one zero injection bus. There

are 5 PMUs in initial placement scheme and the optimal

scheme needs 3 PMUs to assure the observability conditions.

The result is shown in Figs. 1 and 2.

4.2. New England 39 bus system

There are 12 zero-injection buses and the PMU placement

scheme is shown in Fig. 4. Initially, the number of PMUs is 15, and

the redundancy value is 23. After optimization, the PMU number

reduces to 10 and the corresponding redundancy value is 17.

G G

GG

G

GG GG G

30

39

1

2

25

37

29

17

26

9

3

38

16

5

4

18

27

28

3624

35

22

21

20

34

23

19

33

10

11

13

14

15

831

126

32

7

PMU CurrentMeasurement

Non-Measurement

G G

GG

G

GG GG G

30

39

1

2

25

37

29

17

26

9

3

38

16

5

4

18

27

28

3624

35

22

21

20

34

23

19

33

10

11

13

14

15

831

126

32

7

PMU CurrentMeasurement

Non-Measurement

(a) Initial PMU placement (b) Optimal PMU Placement

Fig. 4. PMU allocation results in NE 39 bus system. (a) Initial PMU placement; (b) optimal PMU placement.

J. Peng et al. / Electrical Power and Energy Systems 28 (2006) 223–231230

4.3. IEEE 57 bus system

There are 15 zero-injection buses and PMU placement

scheme is shown in Fig. 5. Initially 16 PMUs are needed, which

is reduced to minimum 13 PMUs. The redundancy value is 37

and 14, respectively.

G GG

G

G

G G

5

17

30

25

5429 5352

27

28

26 24

21

23 22

201918

5110

7

8 9

1234

6

35

34

33

3231

38

37

36

14 13 12

15

16

46

44

45

49

48

47

50

40

5739

55

41

42

56 11

43

PMUCurrent

MeasurementNon-

measurement

(a) Initial PMU placement

Fig. 5. PMU allocation results in IEEE 57 bus system. (a)

Fig. 6 shows the changes of average objective function values

of Tuba search applied to IEEE 14 bus and NE 39 bus power

system. From Fig. 6 it can be seen that the average objective

function values decrease along with the PMUs placement

number and yet the redundancy performance also decreases.

However, when the number of PMUs is less than theminimal np,

G GG

G

G

G G

5

17

30

25

5429 5352

27

28

26 24

21

23 22

201918

5110

7

8 9

1234

6

35

34

33

3231

38

37

36

14 13 12

15

16

46

44

45

49

48

47

50

40

5739

55

41

42

56 11

43

PMU CurrentMeasurement

Non-Measurement

(b) Optimal PMU Placement

Initial PMU placement; (b) optimal PMU placement.

1 1.5 2 2.5 3 3.5 4 4.5 53.6

3.8

4

4.2

4.4

4.6

4.8

5IEEE-14 Bus

PMU Number

Ave

rage

Obj

ectiv

e fu

nctio

n V

alue

s = 5l = 0.1

JAve

np

IEEE 14 bus system

9 10 11 12 13 14 154

5

6

7

8

9

10 NE-39 Bus

PMU Number

Ave

rage

Obj

ectiv

e fu

nctio

n V

alue

s = 10l = 0.1

JAve

np

NE 39 bus system

Fig. 6. Tuba search results of average objective function values for IEEE 14 bus

and NE 39 bus system. IEEE 14 bus system. NE 39 bus system.

J. Peng et al. / Electrical Power and Energy Systems 28 (2006) 223–231 231

the objective function value increases abruptly since the

observability condition is not satisfied at that situation.

5. Conclusions

In this paper, the OPP is solved by the Tabu search

algorithm and the full network observability is checked by

proposed fast observability analysis method. TS provides

solutions with high accuracy and less computational effort.

The new observability analysis method based on incidence

matrix only manipulates integer numbers and can fast,

conveniently and quantitatively assess the network observa-

bility as far as PMU placement scheme is concerned. The

effectiveness of the proposed scheme is demonstrated by

simulation results in IEEE 14, NE 39 and IEEE 57 bus power

system.

Acknowledgements

The authors acknowledge the support by the National

Natural Science Foundation of China for Outstanding Young

Investigators (Grant N.59825104) and National Key Basic

Research Special Fund of China (Grant N. G1998020314).

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