0000 p - A New Algorithm to Find the Optimal Numbers

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Abstract-- This paper proposes monitor positioning algorithm

which determines the optimal number of monitors and their locations. The bus incidence matrix is modified using monitor locations. The principles of monitor positioning are suggested and the weighting factors of each component are defined following these principles. Ambiguity indices are newly suggested to evaluate the monitor locations in terms of topological ambiguity. Finally, the overall algorithm for monitor positioning is developed and applied to IEEE 37 node test feeder. It realizes the experts knowledge on monitor positioning into sophisticated automatic computing algorithm.

Index Terms-- ambiguity, event, event direction, monitor,

optimization, power quality

I. INTRODUCTION N distributed monitoring scheme [1], [2], selecting the

number and location of monitors is a critical problem because it is directly related to the efficiency of the

monitoring system. This problem will be called as monitor positioning. It is a matter of course that more monitors will result in more accurate identification results. Ideally, it is best to install monitors at all components but all components can not be monitored for economic reason. There is a tradeoff between the accuracy of results and the economical efficiency [3]. Thus appropriate locations for monitor installation must be carefully selected for maximum efficiency. The followings are arising important issues.

How many monitors are needed? Where are their locations? How can the adequacy of the locations be estimated?

Typically, the number of monitors required is determined or guided by the customer considering the size of their site and economic capability. Therefore it becomes important to decide the location of monitors under the given number of monitors. To decide the locations, some principles for monitor positioning are also needed. Through these principles, monitors can be positioned at proper locations. The last problem can be solved by introducing the concept of topologi-

Dong-Jun Won is with Advanced Power Technologies Center, Department

of Electrical Engineering, University of Washington, Seattle, WA 98195-2500 USA (e-mail: [email protected]).

Il-Yop Chung is with Department of Electrical Engineering, Korea University, Seoul 136-701, Korea (e-mail: [email protected]).

Seon-Ju Ahn and Seung-Il Moon are with the School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected], [email protected]).

cal ambiguity. Therefore, in this paper a new algorithm on monitor

positioning is developed. The coverage matrix is constructed and the principles of monitor positioning are suggested. Following these principles, the weighting factors of each component are defined. Finally, this method newly defined ambiguity indices. At the end of this paper, the overall algorithm for monitor positioning is developed. II. TREE REPRESENTATION AND MODIFIED INCIDENCE MATRIX

A. Graph Theory and Incidence Matrices In power system analysis, the graph theory and incidence

matrices have been frequently used for computer simulation. The power system network is represented by a simple graph and converted into the corresponding incidence matrix to get a network matrix e.g. impedance matrix Z or admittance matrix Y [4]. For monitor positioning, these are also useful methodology. Therefore, the graph and corresponding incidence matrices are introduced in this section. Let us first introduce the example power system. This example system will also be used in the proceeding chapters.

(1)1

2 3

4 5

6

7 8

9

10

11

12

(2)

(3) (4)(5) (6)

13 EquivalentsourceTransformer

Bus

Line

Load

(n)n

n

Fig. 1 One-line diagram of example power system

Fig. 1 shows the topology of typical distribution power system. This system consists of an equivalent source, a transformer representing distribution substation, 4 distribution lines and 7 loads. Each component gets its number separately so that total 13 components are numbered.

This one-line diagram can be converted into Graph, which represents the geometrical interconnection of power system network. In graph, there exist nodes and elements. The lines and other connecting segments are elements and their terminals are nodes [4]. Fig. 2 shows the graph representation of example system.

A New Algorithm to Find the Optimal Numbers and Locations of Power Quality Monitors

Dong-Jun Won, Member, IEEE, Seon-Ju Ahn, Student Member, IEEE, Il-Yop Chung, Member, IEEE, Seung-Il Moon, Member, IEEE

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Authorized licensed use limited to: National Sun Yat Sen University. Downloaded on July 20,2010 at 12:56:27 UTC from IEEE Xplore. Restrictions apply.

2

1

2

3

45

6

7

13

12

11

10

9

8

(1)

(2)

(3)(4)

(5) (6)

(0)

(n)

n

Node

Element

Fig. 2 Graph representation of example system

A node and an element are incident if the node is a terminal of the element. Following this relation, an incidence matrix can be constructed from the graph. Element-node incidence matrix represents the incidence of elements to nodes. The following matrix is the element-node matrix of Fig. 2. Rows are the elements of the graph and columns represent the nodes of the graph. Matrix dimension is 713 which is equal to the number of elements and nodes of the example system.

=

0000011100000110000011100000

010000101100000010001001010000010010001001000110000001010000110

13121110987654321

)6()5()4()3()2()1()0(\ ne

matrixincidence

nodeElement

(1)

The node (0) is the reference node which means common

ground. By deleting the column corresponding to the reference node, bus incidence matrix is obtained. Bus-incidence matrix shows the relation between the elements and real nodes. It also includes all the information about the topology of the system so that the graph can be constructed from this matrix conversely.

=

100001100000110000

010000011000001000001000000000000100000100000110000010000011

13121110987654321

)6()5()4()3()2()1(\ ne

matrixincidenceBus

(2)

B. Tree Representation

In data structure of computer engineering, a tree consists of a set of nodes and a set of directed edges that connect pairs of nodes [5]. A rooted tree is the tree which has only one root that does not receive information from other nodes. For this reason, rooted tree is suitable for representing the topology of radial distribution power system.

For power quality events happen in the lines or the elements connected to the lines, other components except line are merged with lines. Buses and loads are merged with nearby lines. The merged components become the nodes of rooted tree and are newly numbered in Fig. 3. Equivalent source and transformer are not included in this tree because they are part of external system.

L1

External system

Internal system

L2

L3

L4 L5

L6

L7 L8

L9

L10

L11

L12

L1

L2 L6L3

L12L11

L10L9

L8L7L4 L5

Children of L3

Parent of L4 & L5

Parent of L2 , L3 & L6

Descendants of L6

Child of L1 Child of L1

(a) (b)

Fig. 3 Rooted tree representation of example system (a) One-line diagram of example system for rooted tree (b) Rooted tree representation and parent-child relationship

In Fig. 3(a), L1 becomes the root of tree. For L1 is

connected with L2, L3 and L6 in Fig. 3(a), the node L1 is connected with the node L2, L3 and L6 in Fig. 3(b). Other nodes are connected with one another in this way. In rooted tree, there is parent-child relationship between nodes. For example, L1 is the parent of L2, L3 and L6. L3 is the parent of L4 and L5. L4 and L5 are the children of L3. In this way, L7 ~ L12 become the descendants of L6. These relationships will be used later to construct modified incidence matrix.

C. Up/Down Area Definition

If a monitor is installed in power system, the monitor can identify the relative location of event source [6], [7]. In other words, the monitor can determine whether the event has come from UP area or DOWN area. Determination of relative location of event source is called event direction identification.

The relative location of event source is classified into UP area and DOWN area. Fig. 4 shows the UP/DOWN area in example system. These areas can be defined in two different viewpoints. In electrical view, the power always flows from source to load through power quality monitor in radial distribution system. Therefore DOWN area can be defined as the area to which the power flows from the monitoring point.


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UP area is the other area which is not included in DOWN area, that is, the area from which the power flows to monitoring point and their neighborhood area. In topological view considering tree structure, DOWN area can be defined as the monitored component and its descendants in tree. UP area is the other area which is not included in DOWN area, that is, the ascendants of the monitored component and their neighborhood in tree.

Up Area

Down Area

Monitor

Direction ofPower flow

Fig. 4 Definition of UP/DOWN area considering monitor location

D. Modified Incidence Matrix

Once the monitors are located in power system, the UP/DOWN area according to the monitor location is uniquely determined. This means that the bus incidence matrix can be modified considering the monitor locations. If the monitor location replaces the bus locations in bus incidence matrix, the modified incidence matrix can be obtained, which is called coverage matrix.

Coverage matrix informs us of the relation between the monitor and the component. It indicates whether the component is in UP area or DOWN area of a monitor. In other words, it reveals the coverage of the monitors. The coverage matrix is ML matrix. L is the number of lines and M is the number of monitors. The elements of coverage matrix are determined as follows.

+=

monitorjofareaUPiniscomponentiifmonitorjofareaDOWNiniscomponentiif

athth

ththij 1

1

M1

M2 M3

M4 M5M6 M7

L1

L2

L3

L4 L5

L6

L7L8

L9

L10 L12

L11

Ln : LineMn : Monitor

External system

Internal system

Fig. 5 Example system with 7 monitors

In Fig. 5, total 7 monitors are located in the example system.

The monitors are positioned arbitrarily just to show the elements of coverage matrix. The resulting coverage matrix is (3). In (3), for M1 is installed at L1, a11 is +1. Other elements

which are the descendants of L1 are all set to +1, that is, a21 ~ a12 1 are all +1. For M2 is installed at L3, a32 is +1. Because L4 and L5 are the descendants of L3, a42 and a52 are also +1. Similarly, other elements of matrix are determined.

++++++++++++++++++++++++++

=

111111111111111111111111111111111111111111111111111111111111111111111111111111111111

\

12

11

10

9

8

7

6

5

4

3

2

1

7654321

LLLLLLLLLLLL

MMMMMMMmc

matrixCoverage

(3)

By just looking at this matrix, the topology and UP/DOWN

area can be inferred. This matrix is fairly useful for monitor positioning and event source locating. In monitor positioning, this matrix is used for ambiguity checking.

III. WEIGHTING FACTORS

A. Four Principles of Monitor Positioning

For there is a tradeoff between the accuracy of results and the economical efficiency, the proper number of monitors and their locations must be determined. In the past, this procedure was performed manually by the power quality experts. The experts installed the monitors according to their own guidelines and knowledge on power quality and system topology. For automatic approach to power quality diagnosis, these guidelines must be clearly stated and formulated. Thus this paper suggests a few principles to select the appropriate monitor locations. The principles can be summarized as the following four categories. Principle 1: (N-1) criterion by KCL

If N lines diverge from a bus, there are N possible points to monitor. In this case, it is ineffective to monitor all N points because the information of one line can be computed by those of other (N-1) lines from KCL (Kirchhoffs Current Law). However, (N-2) monitors are not sufficient because ambiguity happens between the two lines that are not monitored. Therefore, (N-1) is the minimum number of monitor to avoid ambiguity. Principle 2: Higher weight on load branch

Typically the damages caused by power quality problems appear at the customer loads. Customer loads are selected as important monitoring points and monitored first in most cases. For this reason, the load branch and the branch near load must be considered first in selecting (N-1) lines to monitor. Main


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feeder or lateral line should be in the next order. In this manner, ambiguity usually happens along the main feeder or lateral line. It means that it becomes easy to use fault distance calculation along the lines. This principle is also consistent with the fact that monitors are typically installed at the service entrance of customer loads. Principle 3: Higher weight on the component with many descendants

The component which has many descendants can monitor more extensive areas than the component with few descendants. It means that it can produce more accurate results. Therefore, these components with many descendants must receive more attention in monitoring. This principle has the tendency to install monitors at the points near power source. This tendency may be thought to be somewhat opposite to the principle 2. However, it will be proven in proceeding section that principle 2 and 3 can cooperate with each other to get reasonable results. Principle 4: Installation at branch beginning point

Monitors have to be installed at beginning point of branch, that is, just after a branch diverges from a bus. In this manner, the branch where a monitor is installed can be included in the monitoring coverage of the monitor. The fact that the branch is not included in the monitoring coverage requires the complex modifications of tree and incidence matrix. This principle is a kind of technical approach to avoid these problems.

M1

M2 M3

M4 M5M6 M7

L1

L2

L3

L4 L5

L6

L7L8 L9

L10 L12

L11

Principle 1 & 3

Principle1 & 2

Principle 1

Fig. 6 Application of monitor positioning principles

Four principles are applied to the example system in Fig. 6. M1 is compulsory monitor location at service entrance. M2 and M3 follow principle 1 for two monitors should be positioned from 3 possible positions. M2 and M3 also follow principle 3 for L3 and L6 has several times of descendants compared to L2. M5 and M6 follow principle 1 and principle 2 as shown in the figure. M4 and M7 follow only principle 1 because two parallel loads are topologically same branches.

B. Definitions of Weighting Factors

According to the principles suggested in section III.A, each component of system has its own degree of importance. The degree of importance should be formulated for the application to computing algorithm. It can be formulated by weighting

factors which properly estimates the degree. In this section, these weighting factors are defined according to the principles.

To formulate the weighting factors, it is necessary to give each component a property and a number. The property of component is classified as main feeder, lateral line and load. The component is numbered considering bus number and the number of branch connected to the bus. For example, cij is the jth component connected to the bus i. The results of component numbering and component properties are shown in Fig. 7.

L1(C11 , Mf)

L2(C21 , Lo)

L3(C22 , La)

L6(C23 , Mf)

L4(C31 , Lo)

L5(C32 , Lo)

L7(C41 , Lo)

L8(C42 , Mf)

L9(C51 , Lo)

L10(C52 , Mf)

L11(C61 , Lo)

L12(C62 , Lo)

Component Property Mf : Main feeder La : Lateral line Lo : Load

Component number : Cij

Main feeder

Lateral line

Load

Fig. 7 Tree representation with component number and property

With these properties and notations, the weighting factors can be formulated as follows considering principle 1, 2 and 3. Principle 3 results in the weighting factor i , principle 2 corresponds to the weighting factor

ij and principle 1 is related to the weighting factor ij .

ABi

i = (4)

i

ijij C

D= (5)

==

11,1

)(

,1

cciforothers

ibusofcallamongweighting

nimummihasandloadnotiscifB

ij

ijiji

iji

ij (6)

=

=n

iiNA

1

(7)

=

=busiofbus

descendantikki

th

NB&

(8)

iLoLo

iLaLa

iMfMfi NwNwNwC ++= (9)

=loadiscifw

linelateraliscifwfeedermainiscifw

D

ijLo

ijLa

ijMf

ij

,,,

(10)

where,

i : bus number j : child number at a bus n : total number of bus

iN : total number of child at bus i iMfN : number of main feeder at bus i


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iLaN : number of lateral line at bus i iLoN : number of load at bus i

Mfw : predetermined weighting for main feeder

Law : predetermined weighting for lateral line

Low : predetermined weighting for load

ijc : component number

i is the normalized value representing the number of descendants of a bus i compared to the total number of components. Therefore

i informs us of how extensive area can be monitored at bus i.

i holds for all the branches connected to bus i.

ij is the normalized value representing the weight of a component j compared to the total weights at a bus i. Main feeder receives the half value of weighting factor of lateral line and load has twice of the weight of lateral line, that is,

MfLo

La www 22 == . Therefore the load branch receives

higher weight than main feeder or lateral line. ij reduces the

weighing factor of a component which has minimum weighting factor at a bus i to be as small as that of a component at the end of the system. It means that the component will have less possibility to be monitored by (N-1) criterion because the component may be selected at last order.

i , ij and ij will be used together in optimization routine which will be formulated in chapter V.

IV. AMBIGUITY INDEX

A. Definitions of Ambiguity Indices Ambiguity happens when (N-1) criterion is not satisfied.

When 5 children are connected to a bus, if only 2 children are monitored, the remaining 3 children cannot be discriminated by the monitors. It is because the 3 children are in the same position topologically. In other words, they are topologically ambiguous. Fig. 8 shows the topological ambiguity problem in example system.

M1

M3M2

M4

L1

L2

L3

L4 L5

L6

L7L8 L9

L10 L12

L11

Ambiguity happens! :Violate (N-1) Criterion

Fig. 8 Illustration of topological ambiguity

For all the components can not be monitored, ambiguity problem happens. If the number of monitors is not sufficient, there exists topological ambiguity in distinguishing the

components. This topological ambiguity can be expressed in the form of ambiguity index. There can be two kinds of ambiguity. One is System Total Ambiguity (STA) and the other is Section Maximum Ambiguity (SMA)

=

= ni

i

ctionsseallkfor

kND

N

N

STA

1

, (11)

=

= ni

i

kND

ctionsseallkfor

N

N

SMA

1

,max (12)

where, kNDN : the number of components which cannot be

distinguished at section k

STA represents the ability to discriminate the location of event source under the assumption that the event can happen at all components. SMA represents the degree of how equally the section is divided. Two indices are normalized by the total number of components of the system. Using these two indices, the System Average Ambiguity (SAA) can be calculated. SAA has adopted the geometric mean to give smaller value when the ambiguity is equally distributed all over the sections.

SMASTANSAAn

ii =

=1 (13)

B. Ambiguity Checking with Coverage Matrix

Ambiguity is inspected using coverage matrix in computing algorithm. The following flowchart briefly explains the algorithm to get ambiguity index using coverage matrix.

No

For every component in each section

YesIt is Monitor-installed component ?

Its parent is ambiguous ?

Satisfy (N-1) criterion?

The component is ambiguous

Yes

No

No

For every section

Find sections from coverage matrix

End of component ?

End of section ?

Calculate STA, SMA and SAA

Yes

No

NoYes

Yes

Fig. 9 Ambiguity checking algorithm


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For ambiguity checking, each section must be found from coverage matrix. The algorithm is repeated for all sections and all the components of each section are investigated whether they are ambiguous or not. Ambiguity checking has three steps, that is, monitor checking, parent checking and (N-1) criterion checking. After all the components are investigated, STA, SMA and SAA are calculated.

This algorithm is applied to the example system shown in Fig. 8. (14) is the coverage matrix of Fig 8.

+++++++++++++++++

=

111111111111111111111111111111111111111111111111

\

8.3.

12

11

10

9

8

7

6

5

4

3

2

1

4321

LLLLLLLLLLLL

MMMMmc

FigofmatrixCoverage

(14)

Using (14), each section is classified by counting the

number of +1. The resulting sections are (L1, L6, L8, L9, L10, L11, L12), (L2), (L3, L4, L5), (L7). Among these components, L1, L2, L3 and L7 are monitor-installed components. Therefore, they are not included in calculating ambiguity index. L6 and L8 follow the (N-1) criterion. They are not also included. Therefore, the remaining components (L4, L5), (L9, L10, L11, L12) are included in calculating ambiguity index. The resulting ambiguity indices are

126=STA , 124=SMA and 899.4=SAA .

V. FORMULATION AND APPLICATION

A. Formulation of Optimization Problem

With the results in chapter III and IV, the optimization routine to find the optimal number and locations of monitors can be formulated. Objective function and its constraint condition are formulated in (15).

( )

=

= =

n

i

N

jijijijix

X

i

ij

x

XF

1 1min

)(min

(15)

subject to

= ==

+ 1

No

Equal WeightComponent?

k = k + 1

Store the optimum number of monitor (NM opt ),their locations (Xopt )

and system ambiguity index (SAAopt)

Minimize the objective function F(X))(min XF

X

Modify Xk

Yes

NMk = NMmax

NMk+1=NMk+1

Update SAAopt = min{ SAAopt, SAAk } andcorresponding NMopt , Xopt

Yes

NoNo

Yes

Construct tree from power system network(Topology matrix)

Set k = 1 and initial number of monitor to install(NMk = NMmin )

Calculate weighting factors ( , , )from the tree

i ij

Construct coverage matrix

NMmin = NMmax

Set SAAopt ,NMopt , Xopt

Yes

No

ij

Fig. 10 Overall monitor positioning algorithm

B. Application to IEEE 37 node test feeder

Fig. 11 shows the configuration of IEEE 37 node test feeder in the form of one-line diagram [8].


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SAA and optimal number of monitor

0

10

20

30

40

50

60

1 6 11 16 21 26NMmax

SAA

and

NM

opt

SAA NM_opt

Optimum(10~20) NMopt = 20 SAAopt = 8.7187

Optimum(20~61) NMopt = 25 SAAopt = 0.0

799

701

702

708

709

730

703

727

729

728 744

706 725

724

722

707

720

718714

704

713

742

712

705

738

737736

735

710

775732

734

731

733

741740

711

L1

L61L60L59 L58

L57L56

L55L54

L53

L52L51

L50L49

L48L47

L46

L43

L42

L45

L44

L40

L39L38 L41

L37L36

L35

L34

L33

L32L31

L30

L29

L28

L27

L26

L25L24

L23

L22

L21

L20L19L18

L17

L16

L15

L14

L13L12L11

L10

L9L8 L7

L6

L5

L4

L3L2

IEEE 37 node Test Feeder - Bus: 37 - Load: 25 - Line: 36 - Component: 61

Fig. 11 One-line diagram of IEEE 37 node test feeder

The total number of component is 61. Therefore, it is fairly difficult to select the appropriate monitoring point manually. Therefore, the monitor positioning algorithm is applied to this test feeder. The results are shown in Fig. 12 relating SAA with the number of monitor.

Fig. 12 Optimal number of monitor and SAA according to NMmax

As NMmax increases, the SAA decreases gradually. If NMmax25, SAA reaches to zero value. It means that no ambiguity exists and it is unnecessary to install more monitors. The solution at NMmax=25 is Xopt = {L1, L2, L27, L4, L9, L36,L39, L45, L42, L10, L47, L16, L54,L56, L28, L17, L48, L30, L21, L5,L22, L49, L58, L15, L31}.

Careful investigation of Fig. 12 will also give the customer a good understanding on the relation between system efficiency and economic efficiency. Until 25max == NMNM opt , the SAA decreases as more monitors are installed. However, the SAA drops slowly at a certain regions and drops rapidly in some regions. By noticing this tendency, the customer can decide the number of monitor to install more efficiently in economic view. For example, if the customer has the economic capability of 15min =NM and 20max =NM , the customer may install 20 monitors. However, for SAA drops rapidly at 17=optNM or 19=optNM compared to SAA of

previous optNM , the customer can select 17 or 19 monitor

considering economic efficiency. If 6min =NM and 10max =NM , it is rather economic to choose 9 monitors than

10 monitors because the difference of SAA is fairly slight between 9=optNM and 10=optNM .

VI. CONCLUSIONS In this paper, monitor positioning algorithm has determined

the optimal number of monitors and their locations. It has constructed the coverage matrix and adopted weighting factors and ambiguity indices. The monitors have been positioned at appropriate locations using the weighting factors and the results have been proven to be adequate in the sense of ambiguity. The proposed algorithm has realized the experts knowledge into sophisticated automatic computing algorithm. It gives a guideline for monitor positioning and also suggests an answer for the number of monitor and locations.

VII. REFERENCES [1] L. Cristaldi, A. Ferrero, S. Salicone, A Distributed System for Electric

Power Quality Measurement, IEEE Transactions on Instrumentation and Measurement, vol. 51, no. 4, pp. 776 -781, Aug. 2002.

[2] Ben Byman, Tom Yarborough, Rudolf Schnorr von Carolsfeld, John Van Gorp, Using Distributed Power Quality Monitoring for Better Electrical System Management, IEEE Transactions on Industry Applications, vol. 36, no. 5, pp. 1481-1485, Sep.-Oct. 2000.

[3] G. Olguin, M.H.J. Bollen, Optimal Dips Monitoring Program for Characterization of Transmission System, Proceedings of IEEE/PES General Meeting, 13-17, July 2003.

[4] Glenn W. stagg, Ahmed H. El-Abiad, Computer methods in power systems analysis, New York: Mcgraw-Hill, Inc., 1968.

[5] Mark Allen Weiss, Data Structures & Problem Solving using JAVA, 2nd edition, Addison Wesley, Inc., 2002, pp. 570.

[6] Anthony C. Parsons, W. Mack Grady, Edward J. Powers, John C. Soward, A Direction Finder for Power Quality Disturbances Based Upon disturbance Power and Energy, IEEE Trans. Power Delivery, vol. 15, no. 3, pp. 1081-1086, July 2000

[7] Anthony C. Parsons, W. Mack Grady, Edward J. Powers, John C. Soward, Rules for Locating the Sources of Capacitor Switching Disturbances, IEEE Power Engineering Society Summer Meeting, vol. 2, pp. 794-799, 1999

[8] W. H. Kersting, Radial distribution test feeders, IEEE/PES Winter Meeting, vol. 2, pp. 908 -912, 28 Jan. - 1 Feb. 2001.

VIII. BIOGRAPHIES Dong-Jun Won (M 2005) received the B.S., M.S. and Ph.D. degrees in

electrical engineering from Seoul National University, Seoul, Korea in 1998, 2000 and 2004 respectively. He is currently a post-doctoral fellow at APT center, University of Washington..

Seon-Ju Ahn (S 2002) received the B.S. and M.S. degrees in Electrical Engineering from Seoul National University, Seoul, Korea in 2002 and 2004, respectively. He is currently Ph.D. student at Seoul National University.

Il-Yop Chung (S 2001) received his B.S., M.S. and Ph.D. degrees in electrical engineering from Seoul National University, Seoul, Korea in 1999, 2001 and 2005 respectively. He is currently a post-doctoral fellow at Korea University.

Seung-Il Moon (M 1993) received the B.S. degree in electrical engineering from Seoul National University, Seoul, Korea in 1985. He received the M.S. and Ph. D. degrees in electrical engineering from The Ohio State University in 1989 and 1993, respectively. Currently, he is an associate professor of school of electrical engineering and computer science at Seoul National University.


0000 p - A New Algorithm to Find the Optimal Numbers

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