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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007 101

An Optimal PMU Placement Method AgainstMeasurement Loss and Branch Outage

Chawasak Rakpenthai, Suttichai Premrudeepreechacharn, Member, IEEE, Sermsak Uatrongjit, Member, IEEE, andNeville R. Watson, Senior Member, IEEE

AbstractThis paper presents a new method for an optimalmeasurement placement of phasor measurement units (PMUs) forpower system stateestimation. The proposed method considerstwotypes of contingency conditions (i.e., single measurement loss andsingle-branch outage) in order to obtain a reliable measurementsystem. First, the minimum condition number of the normalizedmeasurement matrix is used as the criteria in conjunction withthe sequential elimination approach to obtain a completely deter-mined condition. Next, a sequential addition approach is used tosearch for necessary candidates for single measurement loss andsingle-branch outage conditions. These redundant measurements

are optimized by binary integer programming. Finally, in orderto minimize the number of PMU placement sites, a heuristictechnique to rearrange measurement positions is also proposed.Numerical results on the IEEE test systems are demonstrated.

Index TermsContingency, measurement placement, phasormeasurement units (PMUs), state estimation.

I. INTRODUCTION

THE phasor measurement units (PMUs) are measuring

devices synchronized via signals from global positioning

system (GPS) satellite transmission[1]. They are employed to

measure the positive sequence of voltage and current phasors.The PMUs are more accurate and can take measurements

synchronously. Consequently, the performance of state es-

timation is improved. Since the voltage and current phasors

are measured, the state estimation equations become linear

and it is easier to find the solution than the nonlinear system

state estimation[2]. The problem of PMU placement becomes

an important issue in the power system state estimation as

the devices are increasingly accepted. The PMU placement

method should be performed under three considerations: 1)

the accuracy of estimation, 2) the reliability of estimated state

under measurements failure and change of network topology,

and 3) the investment cost. Since a rigorous formulation ofthe optimal PMU placement becomes very difficult and it is

a time-consuming process to search for the global optimal

Manuscript received November 2, 2005. This work was supported by theThailand Research Fund (TRF) through the Royal Golden Jubilee Ph.D. Pro-gram under Grant PHD/0055/2547. Paper no. TPWRD-00640-2005.

C. Rakpenthai is with Department of Electrical Engineering, Faculty of En-gineering, North-Chiang Mai University, Chiang Mai 50230, Thailand (e-mail:[email protected]).

S. Premrudeepreechacharn and S. Uatrongjit are with Department of Elec-trical Engineering, Faculty of Engineering, Chiang Mai University, Chiang Mai50200, Thailand (e-mail: [email protected]; [email protected]).

N. R. Watson is with Department of Electrical and Computer Engi-neering, University of Canterbury, Christchurch 8020, New Zealand (e-mail:[email protected]).

Digital Object Identifier 10.1109/TPWRD.2006.881425

solution, a systematic procedure which presents nearly optimal

solutions is usually desired to design PMU placement.

In [3], a phasor measurement placement method based on

the topological observability theory using graph theorem anal-

ysis is proposed. A minimal number of buses with measure-

ments is found through both a modified bisecting search and

simulated annealing-based method. However, the possible con-

tingency in the power system is not considered, the measure-

ment set is not robust to loss of measurements and branch out-

ages. In[4], an optimal PMU placement method based on thenondominated sorting genetic algorithm (GA) is proposed. The

problem is to find the placement of minimum PMUs set so that

the system is still observable during its normal operation and

any single-branch contingency. Each optimal solution of ob-

jective functions is estimated by the graph theory and simple

GA. Then, the best tradeoff between competing objectives is

searched by using nondominated sorting GA. Since this method

requires more complexity computation, it is limited by the size

of the problem. In[5], the integer programming based on net-

work observability and the cost of PMUs has been applied to

find the PMU placement. This method can be applied to the case

of the mixed measurement set which PMUs and conventionalmeasurements are employed in the system. These papers find

the minimal buses where PMUs should be installed such that

the power network is observable. It is assumed that the installed

PMUs have enough channels to record the bus voltage phasor

at their associated buses and current phasors along all branches

that are incident to the buses. However, the topological observ-

ability is not guarantee that the state estimation can be solved

[6]. Furthermore, it usually gives large condition number of the

measurement matrix. As a result, the computed solution may not

be accurate due to roundoff error during numerical computation.

The optimal placement methods for conventional measure-

ments against contingency have already been addressed in

[7][9]. A sequential selection process based on measurementsensitivities and measurement failures performance indices has

been presented in [7]. A topology method considering only

single-branch outage is presented in[8]. In addition, a numer-

ical algorithm to optimally upgrade the existing measurements

in order to make the network observable under loss of single

measurement and any single-branch outage is proposed in[9].

This paper presents a PMU placement method for power

system state estimation based on the minimum condition

number of the normalized measurement matrix. The proposed

method finds an optimal measurement set necessary for com-

pletely system numerical observability and single measurement

loss and single-branch outage contingency. Then, the positions

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102 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007

of these measurements are rearranged by a heuristic algorithm

in order to minimize the number of PMU placement sites.

This paper is organized as follows. The PMU placement

problem is formulated inSection II. Next, the proposed PMU

placement algorithm is described in Section III. Then, nu-

merical results of the proposed algorithm are provided in

Section IV.The conclusion is given inSection V.

II. PROBLEMFORMULATION

Let the voltage phasors at all bus voltages be chosen as state

variables. The measurement values are the bus voltage phasors,

the injection current phasors, and the branch current phasors.

Then, a linear measurement model used in power system state

estimation is represented by

(1)

where

vector of measurement values;

vector of state variables to be estimated;

measurement matrix;

measurement error vector.

Note that if the measurement values are obtained without

error, the estimated problem(1)can be considered as a linear

algebraic problem. Each row of the measurement matrix should

be linearly independent to minimize the number of measuring

devices in the power system network. Moreover, the condition

number of is the one of the essential factors for solving pro-

cedure. A large condition number may lead to unsolvable or in-accurate solutions[10].Therefore, PMU placement problem is

tofind the minimal measurements with small condition number

of .

Some failure on a measurement may happen in the power

system or a network topology may be changed due to CB

operation when a fault occurs. Both conditions may make

the system become unobservable. Thus, good measurement

placement should also consider the case of measurements loss

and branch outages.

III. PROPOSEDPMU PLACEMENTALGORITHM

As shown in Fig. 1, the proposed placement algorithm

consists of four stages. In the first stage, a measurement set

for a completely determined condition is searched. It is called

an essential measurement set. Next, a redundant measurement

set is selected from the candidate measurements under each

contingency condition. Both stages use the minimum condi-

tion number of normalized measurement matrix as criteria in

selecting the measurement positions. Then, from these mea-

surements, the optimal redundant set is selected by using the

binary integer programming technique. The essential measure-

ments and the optimal redundant measurements are rearranged

by the proposed heuristic method in order to minimize the

number of PMU placement sites in the final stage. These stagesare explained in the following subsections.

Fig. 1. Flowchart of the proposed PMU placement algorithm.

A. Finding Essential Measurement Set

The algorithm starts by searching the essential measurementset, in another words, the positions and types of measurements

under the completely determined condition (i.e., the number of

the measurements is equal to the number of estimated states).

The proposed methods are different from the placement method

in[11]and[12]since the normalized measurement matrix (i.e.,

the norm of each row vector of the measurement matrix is nor-

malized to unity), is used in the sequential elimination proce-

dure. In [13], the authors show that this technique yields the

better condition number than the method in[11],especially for

large power systems. The normalized measurement matrix can

be employed as the measurement matrix of the linear measure-

ment model for state estimation. Usually, the calculation of a

power system is performed using the per-unit system [14]. Sincethe power system has high power rated, the base impedance

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RAKPENTHAI et al.: OPTIMAL PMU PLACEMENT METHOD 103

of per-unit system is also high and always larger than the real

impedance of transmission lines. This makes the per-unit ad-

mittance of transmission parameters larger than one. It implies

that the norm of each row vector of measurement matrix is not

less than one. Therefore, the measurement and error values of

(1)may be unchanged or reduced.

Let the number of candidate instruments be and the numberof state variables be . The sequential elimination can be ex-

plained by the following pseudo code.

Sequential Elimination Pseudo Code

Input:

, 2

matrix

;

Output:

, 2 matrix;

Normalize each row of

to unit row vector;

;

for

to 0

,

;

for

to nrow,

;

Eliminate ith row from

;

;

end

Find imin such that ;

Eliminate iminth row from

;

end

return

;

B. Selection of Redundant Measurements

In this paper, as stated above, single measurements loss and

single-branch outage are considered contingency conditions. It

should be noted that the voltage measurement at the slack bus

is the critical measurement which cannot be loss. The sequen-

tial addition method is used to search the redundant measure-

ments for each contingency condition. This step is to find neces-

sary measurements from candidate measurements of each con-

tingency condition such that the state estimation is still solvable

under these conditions.

In case of measurement loss, the measurement matrix will

be modified by removing the row corresponding to the lostmeasurement. However, when the network topology changes, it

is necessary to rebuild the measurement matrix according to the

outage condition. Candidate measurements which yield nor-

malized measurement matrices with condition number below a

predefined threshold are selected as redundant measurements.

Since bus voltage phasors are chosen as state variables, the

number of voltage measurements should be kept at minimum

and the branch current measurements are used as candidate

whenever possible. However, in case that no branch current

measurement can make the system observable under the given

contingency condition, bus voltage measurements are used as

candidates.

Define as the measurement matrix of the existing mea-surements,

as the measurement matrix of the candi-

date measurements, and

as the position vector of the

candidate measurements. The sequential addition procedure can

be described by the following pseudo code.

Sequential Addition Pseudo Code

Input:

, 0 2 matrix;

, 2

matrix;

, 2 matrix;

Threshold;

Output: , 2 matrix;

;

Normalize each row of

and

tounit row vector;

for

to P,

;

Append the ith row of

to

;

;

if ,

;

end

end

return ;

The output from the sequential addition (i.e., ), contains the

list of redundant measurements to be added into the essential

measurements to ensure the completely observable condition of

the power network when the contingency occurs. This list is then

used to construct a contingencycandidate matrix . The rowsand columns of correspond to the contingency and the redun-

dant measurements, respectively, where is one if the th

measurement is selected as a candidate for the th contingency;

otherwise, it is zero.

C. Finding Optimal Redundant Measurements

A binary integer programming has been applied to solve the

placement problem of the conventional measurements (i.e.,

voltage magnitude, power injection, and power-flow measure-

ments [8], [9]). In this paper, it is used to find the optimal

redundant measurements for the contingency. For simplicity,

the cost of each candidate redundant measurement is assumed

to be the same. The optimal redundant PMUs problem can be

formulated as a binary integer programming as

Minimize (2)

Subject to (3)

where , , and is a bi-

nary solution vector whose elements are 0 or 1. The redundant

measurements according to the nonzero elements of are con-

sidered as the optimal redundant measurements.

D. Minimization of PMU Placement Sites

Although one may install a PMU at each measurement posi-tion obtained from the above algorithms, practically, the PMUs

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and transducers are installed at the buses representing power

system substations. The measured data are exchanged and syn-

chronized to the central control center through GPS satellite or

optical-fiber channels. This process requires extensive commu-

nication equipment. Consequently, the number of PMU place-

ment sites should be minimized in order to reduce the communi-

cation costs. For the placement sites that require many measure-ments, multichannels PMU will be installed at these sites. Al-

though the bus voltage and the injection current measuring de-

vices cannot be moved from their original positions, it is noticed

that if the branch current measuring device installed close to one

site is moved to the other end, the condition number of the mea-

surement matrix is not changed. Therefore, the measurement

positions obtained from the minimum condition number criteria

can be rearranged to minimize the number of PMU placement

sites by the following heuristic algorithm.

Step 1) Based on the placement position list obtained from

the measurement placement algorithm, determine

the bus where either an injection current or a bus

voltage measuring device is installed. These busesare called major buses. Other buses are called minor

buses.

Step 2) If there is a branch current measuring device on the

branch connected to the major buses, the device is

moved close to the major buses.

Step 3) From the branches with branch current measuring

devices, which are not connected to major buses, de-

termine the minor bus with the maximum connec-

tion number of those branches with branch current

measurements. Then, the branch current measuring

device on the connected branches moves close to the

selected bus. Note that this minor bus will not beconsidered again in the next iteration.

Step 4) Repeat Step 3) until the maximum connection

number of branches with branch current measuring

devices is equal to one.

The pseudomeasurements are not considered in this heuristic

algorithm since there is no actual PMU installed.

Example: The six-bus power system is shown in Fig. 2(a)in

which there arefive placement sites.

Step 1) Determine the major buses and minor buses

Step 2) Move devices close to major buses

Step 3) Considering only bold numbers, it can be seen that

the bus no. 2 has the maximum connection number

of three since it is connected to bus nos. 3, 4, and 6.

Thus, bus no. 2 is chosen in this iteration

Then, move the device close to bus no. 2

Step 4) After moving the device, bus nos. 5 and 6 have the

connection number of one. Thus, the iteration proce-

dure is stopped and the measurement positions of the

six-bus power system become as shown inFig. 2(b).

Notice that the number of placement sites is reduced

fromfive to three.

IV. NUMERICALRESULTS ANDDISCUSSIONS

In this section, the proposed algorithm has been applied to

some IEEE test systems. The bus voltage measurement is in-

stalled only on the reference bus of the entire system for thecompletely determined condition. In the following numerical

experiments, the threshold in sequential addition is set to double

of the condition number of the normalized

.

A. IEEE 14-Bus Test System

There are 55 possible measurement positions for the IEEE

14-bus test system. It consists of a bus voltage, 14 injections

current, and 40 branches current measurements. The condition

number of measurement matrix before and after normalization

is 258.55 and 12.83, respectively. The essential measurements

for the completely determined condition are obtained by se-

quential elimination. These measurements are bus voltage at no.1, injection current at bus nos. 8, 10, and 14, the branch current

on branch 1-2, 1-5, 2-3, 2-4, 4-7, 4-9, 5-6, 6-11, 6-12, and 6-13.

The condition number obtained from the proposed method is

10.08. The contingency conditions consist of 13 measurements

loss and 20 branch outages. Since the outage of branch 7-8 iso-

lates the corresponding synchronous condenser installed at bus

no. 8, it is unnecessary to observe the state variables of bus no.

8. Thus, this outage branch is not considered as a contingency

condition since the outage of branch 2-5, 3-4, 4-5, 7-9, 9-10,

9-14, 10-11, 12-13, and 13-14 are still giving a smaller condition

number than threshold, after sequential addition is performed

under the contingency conditions. These outage branches are

excluded from . Thus, only 23 contingency conditions (13measurement losses and 10 branch outages) and 21 candidate

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Fig. 3. Contingency conditions versus redundant measurements for the IEEE 14-bus test system.

Fig. 2. Six-bus power system (a) before and (b) after rearrangement.

redundancy measurements are considered in the optimal redun-

dant selection. Contingency conditions versus redundant mea-

surements can be shown as matrix (23 rows and 21 columns)

inFig. 3.

Define as the injection current measurement at bus no.

, and represents the branch current measurement on the

branch connecting bus no. and (near bus no. ). The rowsof matrix correspond to

The columns of matrix correspond to

After performing binary integer programming, the optimal

redundant measurements , , , and are obtained.

Then, both essential measurements and these optimal redundant

measurements are rearranged by the proposed heuristic tech-

nique. Finally, the optimal PMU placements are obtained as

shown inFig. 4.It can be seen that for the zero-injection current

at bus no. 7 which is a pseudomeasurement, the measurement is

not installed on this bus. The number of PMU placement sites

is 8. The condition number of normalized measurement matrix

is 10.67.

B. IEEE 30-Bus Test System

The measurement positions for the IEEE 30-bus test system

consist of bus voltage, 30 injections current, and 82 branches

current measurements, in total there are 113 possible mea-

surement positions. The condition number of measurement

matrices before and after normalization are 637.66 and 22.64,

respectively. The essential measurements for the completely

determined condition are obtained by sequential elimination.

These measurements are bus voltage at no. 1, injection current

at buses no. 11, 17, 22, 26, and 28, the branch current on branch

1-2, 1-3, 2-5, 2-6, 3-4, 4-12, 5-7, 6-8, 6-9, 6-10, 10-20, 10-21,

12-13, 12-14, 12-15, 12-16, 15-18, 15-23, 19-20, 23-24, 25-27,27-28, 27-29, and 27-30. The condition number obtained from

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Fig. 4. IEEE 14-bus test system with optimal PMU placement.

proposed methods is 18.06. With these essential measurements,

the contingency conditions consist of 29 measurements loss

and 41 branch outages. For IEEE 30-bus test system, the outage

of branch 9-11 and branch 12-13 isolate the corresponding

synchronous condensers while the outage of branch 25-26

isolates the corresponding load. Therefore, bus nos. 11, 13,

and 26 are unnecessary to bus observable. It implies that the

outages of these branches should be excluded in the considerate

contingency since the outage of branch 2-4, 4-6, 6-7, 6-28, 8-28,

9-10, 10-17, 10-22, 14-15, 16-17, 18-19, 21-22, 22-24, 24-25,

and 29-30 still give a condition number that is smaller than

threshold, after the sequential addition is performed. Therefore,there are only 52 contingency conditions (29 measurement

losses and 23 branch outages) and 42 redundant measurements

are considered by the binary integer programming. Conse-

quently, the size of in this case is 52 rows and 42 columns.

After performing the binary integer programming, the op-

timal redundant measurements to be added to maintain system

observability under the contingency are found to be , , ,

, , , , , and . Both essential mea-

surements and these optimal redundant measurements are re-

arranged by the proposed heuristic method. The optimal PMU

placements are obtained as shown inFig. 5.It can be seen that

there are no measurements installed at bus nos. 22, 25, and 28since these are the zero-injection current measurements or pseu-

domeasurements. The number of PMU placement sites is 16.

The condition number of the normalized measurement matrix

is 19.95.

In addition, other IEEE test systems have been used to verify

the proposed method. From the results, it is found that the outage

of branch without the essential measurement installed still pro-

vides the small condition number of a normalized measurement

matrix. The numerical results of the proposed PMU placement

algorithm are summarized inTable I. Notice that more than half

of the system buses are required as PMU placement sites in

order to obtain the reliable measurement system and accurate

state estimation. Some zero-injection currents considered as thepseudomeasurements may be used since they also help decrease

Fig. 5. IEEE 30-bus test system with optimal PMU placement.

TABLE INUMERICALRESULTS OF THEPROPOSEDPMU PLACEMENTALGORITHM

For example 1, 6, 11 indicate that the number of bus voltage, injection current,

and branch current measurement types are 1, 6, and 11, respectively.

the placement sites. A minimal number of bus voltage measure-

ments is three for IEEE 39-bus system.

C. Comparison With Conventional Method

In order to demonstrate the effectiveness of the proposed

PMU placement algorithm, the method is compared with

the method in [4] in which the PMU placement against any

single-branch outage based on the graph theory is presented.

Here, as an example, the optimal PMU placement sites for

the IEEE 39-bus system from Table VI in[4] are chosen, they

are bus nos. 4, 6, 8, 12, 16, 18, 20, 22, 23, 25, 26, 29, and

39. In general, the measurement matrix is used for the state

estimation solution. However, the proposed method considers

the normalized measurement matrix and uses this matrix tosolve state estimation. Comparison results obtained by both

methods are summarized in Table II. Note that although the

method in [4] provides a smaller number of PMU placement

sites, the condition number is much higher than the proposed

method. Therefore, the precision of the estimated states tends

to be less accurate. On the other hand, the number of voltage

phasor measurements and the number of total measurements are

greater than the proposed method. Moreover, the measurement

system obtained from the proposed method is robust to single

measurement loss contingency.

V. CONCLUSION

In this paper, a new PMU placement algorithm for powersystem state estimation under single measurement loss and

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TABLE II

COMPARISON OFRESULTS FOR THE CASE OF THE IEEE 39-BUSSYSTEM

For example 13, 10, 39 indicate that the number of bus voltage, injection current, and branch current measurement types are 13, 10, and 39, respectively.

any single-branch outage has been presented. The proposed

method is numerical observability by using the minimum

condition number of the normalized measurement matrix as

criteria. The sequential elimination is used to find the essential

measurements for the completely determined condition. The

sequential addition is used to select the redundancy measure-

ments under the contingency. The binary integer programming

is also applied to select the optimal redundant measurements.

In addition, a minimal number of PMU placement sites is

obtained from the proposed heuristic approach.

Numerical results on the IEEE test systems indicate that theproposed placement method satisfactorily provides the reliable

measurement system that ensures the state estimation to be solv-

able under the given contingency conditions. Furthermore, due

to the well-conditioned measurement matrix, state estimation

accuracy is also improved.

REFERENCES

[1] A.G. Phadke, Synchronized phasormeasurements in power systems,IEEE Comput. Appl. Power, vol. 6, no. 2, pp. 1015, Apr. 1993.

[2] R. Zivanovic and C. Cairns,Implementation of PMU technology in

state estimation: an overview, in Proc. IEEE AFRICON, Sep. 1996,vol. 2, pp. 10061011.

[3] T. L. Baldwin, L. Mili, M. B. Boisen, Jr, and R. Adapa,Power systemobservability with minimal phasor measurement placement, IEEETrans. Power Syst., vol. 8, no. 2, pp. 707715, May 1993.

[4] B. Milosevic and M. Begovic, Nondominated sorting genetic algo-rithm for optimal phasor measurement placement,IEEE Trans. PowerSyst., vol. 18, no. 1, pp. 6975, Feb. 2003.

[5] B. Xu and A. Abur,Observability analysis and measurement place-ment for systemwith PMUs, in Proc.IEEE PowerSystem Conf. Expo.,Oct. 2004, vol. 2, pp. 943946.

[6] A. Monticelli, State Estimation in Electric Power Systems: A General-ized Approach. Norwell, MA: Kluwer, 1999.

[7] Y. M. Park, Y. H. Moon, J. B. Choo, and T. W. Kwon, Design ofreliable measurement system for state estimation,IEEE Trans. PowerSyst., vol. 3, no. 3, pp. 830836, Aug. 1988.

[8] A.Abur and F.H. Magnago, Optimal meterplacement for maintainingobservability during single branch outage, IEEE Trans. Power Syst.,vol. 14, no. 4, pp. 12731278, Nov. 1999.

[9] F.H. Magnago and A.Abur, Aunified approach to robustmeterplace-ment against loss of measurements and branch outages,IEEE Trans.Power Syst., vol. 15, no. 3, pp. 945949, Aug. 2000.

[10] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Bal-timore, MD: Johns Hopkins Univ. Press, 1996.

[11] C. Madtharad, S. Premrudeepreechacharn, N. R. Watson, and D.Saenrak, Measurement placement method for power system stateestimation: Part I, in Proc. IEEE Power Eng. Soc. General Meeting,Jul. 2003, vol. 3, pp. 16321635.

[12] C. Madtharad,S. Premrudeepreechacharn,N. R. Watson, andR. Saeng-Udom,An optimal measurement placement method for power systemharmonic state estimation,IEEE Trans. Power Del., vol. 20, no. 2, pt.2, pp. 15141521, Apr. 2005.

[13] C. Rakpenthai, S. Premrudeepreechacharn, S. Uatrongjit, and N. R.Watson, An improved PMUs placement method for power systemstate estimation, presented at the 7th Int. Power Engineering Conf.,Singapore, Nov. 2005.

[14] J. J. Grainger and W. D. Stevenson, Jr., Power System Analysis. NewYork: McGraw-Hill, 1994.

Chawasak Rakpenthai received the B.Eng. andM.Eng. degrees in electrical engineering fromChiang Mai University, Chiang Mai, Thailand, in

1999 and 2003, respectively, where he is currentlypursuing the Ph.D. degree.

Currently, he is a Lecturer with the Faculty of

Engineering, North-Chiang Mai University. Hisresearch interests include applications of artificialintelligence in power system, power electronics,power system state estimation, and flexible actransmission systems (FACTS) devices.

Suttichai Premrudeepreechacharn (S91M97)received the B.Eng. degree in electrical engineeringfrom Chiang Mai University, Chiang Mai, Thailand,in 1988 and the M.S. and Ph.D. degrees in electricpower engineering from Rensselaer Polytechnic

Institute, Troy, NY, in 1992 and 1997, respectively.Currently, he is an Associate Professor with the

Department of Electrical Engineering, Chiang Mai

University. His research interests include powerquality, high-quality utility interfaces, power elec-tronics, and artificial-intelligence-applied power

system.

Sermsak Uatrongjit (M98) received the B.Eng.(Hons.) in electrical engineering from Chiang MaiUniversity, Chiang Mai, Thailand, in 1991, and theM.Eng. and Ph.D. degrees in physical electronicsfrom the Tokyo Institute of Technology, Tokyo,Japan, in 1995 and 1998, respectively.

Currently, he is an Assistant Professorwith theDe-partment of Electrical Engineering, Chiang Mai Uni-versity, Chiang Mai, Thailand. His research interests

include numerical methods for analog circuit simula-tion and optimization.

Neville R. Watson (SM99) received the B.E.(Hons.) and Ph.D. degrees in electrical and computerengineering from the University of Canterbury,Christchurch, New Zealand, in 1984 and 1988,respectively.

Currently, he is an Associate Professor with theUniversity of Canterbury. His main interests are inpower system analysis, transient analysis, and har-monic and power quality.

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