lescope-2003-a higher order interior point method to minimize active power loss

7/17/2019 LESCOPE-2003-A Higher Order Interior Point Method to Minimize Active Power Loss

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A Higher Order Interior Point M ethod to Minimize

Active P ow er Loss in Electric Energy Systems

Marcos J .

Rider . Student Meinber.

IEEE.

V. L eona r do P auca r .

Senior

Member.

IEEE.

A r i ova l do

V .

Garcia.

Member-.

IEEE

a n d M a n f r e d

F.

BedriAana.

Student

Member-.

IEEE

Absrrircr-In this paper the artive power loss minimization

problem is formulated as an optinial power f low (OFF), inrluding

equality and inequality nonlinear ronstraints which represent the

power system serurity conditions. The

OPF

has been solved using

the multiple predirto r--ror rerto r interior point

method. of t h e

family of higher order interior point methods, enhanced wi t h an

optimal computation of the step length. The optimal roniputation

of the primal a n d dual step sizes minimizes the primal and dual

objertive funrtion errors. respectively, assuring a rontinuous

decrease of the errors during the iterations of the interior point

method. The proposed methodology has heen applied to minimize

t h e artive power loss of I E E E - 30 , 1E E E - 57 , I E E E - I 18 a n d I E E E -

300

bus test systems.

1-es t

results indicated

t h a t

the ronvergenre

is farilitated a n d the numher of iterations may be small.

Index 7knns-A rt ive power I

oss

m in in iza1 on. rompet

it

ive

elertrir markets. intttrior po in t method, nonlinear programming,

. optimal power flom.

1. I N T R O D U C T I O N

CTlVE power loss minimization of electrical power

A

ystems

(EPS)

is considered a requirement of current

comp etitive electricity marke ts. In EPS planning and

operation, the security and reliability are assessed using a

number of computer programs which include the optimal

power flow (OPF). An objective of OPF problem solution

is

to

determine the optimal steady-state operation of

EPS.

The

OPF

problem may be modeled such as a nonlinear programming

(NLP) problem with technical and economic constraints. [I]

Classical methods to solve OPF include the sequential linear

and quadratic programming, gradient and Newton methods.

Recently it has been used the interior point method

(IPM).

Originally developed to solve linear programming problems

(LP), the IPM provides better computational performance for

large scale problems than classical approaches such as the

simplex method 123. IPM may be adapted to solve NLP

problems. The IPM has been applied

to

optimize the power

systems operation with peat success

[3]

solving problems

such as the optinial power flow 141, reactive power dispatch

[ 5 ] $state estimation 161, loadability maximization 173. voltage

Manuscript sent March 30. 2003 This work has been supported bv thr

Brazilian Institulions CNPq and CAPES

Marcos

J

Rider and Ariovaldo V Garcia are with the Depanment of

Electric Energ\ S\ stems State Universiw of Campinas (UNICAMP)

Campinas - SP Brazil (k-mails

{

mirider, arij@dsee fee unicamp br)

V Leonardo Paucar and Manfred

F

Bedriiiana are with the Depanment of

Electrical Enpineering Federal Universih, of Maranhao (UFMA) Slo

Luic

MA Brazil (E-mails { Lpaucar manfred bedrinana]@ieee orel

stability [SI, hydrothermal coordination 191 and secur ih

constrained eco nom ic dispatch [ I O ] . Most o f IPM applications

in power systems use the primal-dual interior point methods

(PD-IPM) despite of the large number o f iterations and some

divergence problems during the search directions computation

by Newton method

(NM).

Mehrotra proposed best search

directions by solving

two

linear equation systems

[ I I ]

which

define the predictor and corrector steps of the predictor-

corrector primal-dual interior point method (PCPD -IPM ).

Results with PCPD-IPM may be enhanced by adding anothei

corrector step to the optim izatio n process resulting in the

multiple predictor-corrector interior point method (M PC-IP M)

that belongs

to

the fanlily of higher order IPM [ 121. Empirical

evidence suggests that NM: when used in the IPM procedure.

needs an adequate computation of the primal and dual step

sizes in order to facilitate the NM iterations convergence [ 131.

In this paper is proposed a methodology for power loss

minimization using an OPF model solved with an enhanced

M P C - I P M

procedure that includes an optimal computation

of

primal and dual step sizes which minimize the primal and dual

objective function erro rs, respectively. T he methodology has

been applied to the IFEE-30, IEEE-57, IEEE-I 18 and IEEE-

300 test systems. Results have indicated that convergence is

facilitated and th e iterations number m ay be sm all.

] ] .THE

PTIMAL POWER

FLOW

MODEL

The

OPF

problem consists of minimizing the active power

loss subject to security constraints. Control variables are the

voltages at generalion buses, shunt reactive powers and

transformers tap positions. Th e optimization model is given as

rib

in l'n,, C ) n . /

COS 6n.y + 6, -

i1 T

,J

+

P,lnt

(

1

J I

subject to:

~ V ~ V , Y ~ c o s ( 0 , + d , - 6 , ) - P ~ , + P U i = O :

= l ,

...,nb (2)

/ = I

~ V I V l Y , , s i ~ i ( / 3 , ,d,,

-

d,)-Q,,

+

Q,, =

0

; i = 1

,__. ,b

(3)

n

J=l

QZY (IG,

: i = I, ... ng

(4)

5)

6)

(7)

5

VI

5 VIma ;

= I,

...,~ 7 b

QSh,""" QSh,

5

QSh,"'" :

i =

I,

...,nsh

Tap, "" 5 Tap, 5

Tap,?

; i =

I,

...,n f

where

ns

is the slack bus number. PG, nd Qc, are the real

0-7803-7863-6/03/ 17.002003

EEE80



and reactive power Fenerated at bus i PI,, nd ell,re the real

and reactive power loads at bus I

I/ ,

and

6,

are the voltage

magnitude and angle at bus i. and

B,,

are the magnitude and

angle of element

i

of admittance matrix.

nb, ng, nsh

and

nt

are

the number of buses, generators, shunt compensators and

transformers. respectively. Eqs.

( 2 ) : (3)

represent the power

flow

equations. Inequalities (4)-(7) represent the maximum

and minimum limits of the reactive power generations

(eG),

bus voltages

(V),

shunt conipensators

( Q s h ) .

and transformer

tap positions

( T up) ,

espectively.

111

TH E

HIGHER-ORDER

N T R I O R

POINTMETHOD

The OPF formulation shown in ( 1

)-(7)

can also be written

Min

f (x)

as

a

general nonlinear programming problem, given by

subject io :

g(x) = 0

h'

I

h(x)

5

h"

( 8 )

X I

I I t

where

x

represent the control variables,

(x)

is the objective

function; g(x) and h(x) are equality and inequality constraints,

2 and

I

are superscripts for upper and lower limits. The

inequality constraints

of

(8) are transform ed in equality

constraints by using non-negative slack variables (s,, s2,

s3,s,)

141,

[SI

hat are included inf(x) as logarith mic term s.

Min f (x)

subject to :

g(x) = 0

- s , - s , - h ' + h " = O

h(x)

-

S?

+

h" = 0

-s3 -s4 - x i

XI

= O

- X -Sq

+XI1

= o

SI? 2, sj 4

2

0

(9)

Including the slack variables in f(x):

= I

J I

SubJect to g x)

= 0

- s l - s 2 - h h l + h U O

10)

s3 - s 4

- x i + x u = o

-x-s4 + x u

= o

- h ( x ) - s ? + h "

= O

where

ph

> 0 is a barrier parameter.

ti

and

ndh

are the

The Lagrangean function L of 1 0 ) s

numbers

of

control variables and ineq uality constraints.

J = l J = l

- ? ' 7 g ( ~ ) - i : ( - ~ , - ~ ? - h h l + h U ) - r 9 ( - h ( x ) - s z

h " )

1:

( - S 3

- s q - x i +x )-z::

( -W-S4

+ X I ' )

...(

11)

where 1 : I z3 . and i are Lagrange multipliers vectors.

According to Fiacco and McCormick's theorem

[ 141, if p'

decreases monotonically to zero during the iterative process:

the solution of

I O )

approaches the local solution of

(9).

If

is local minimum of ( I O ) ,

x *

meets the first order necessarr;

optimality cond itions

of

Karush-Kuhn-Tucker (KKT): that is:

VS,L

=

-

p k S ; ' e

+

z l

= O

VS,L =

-

pkS z 'e+ z1+

z 2

= O

V s , L

=

-

p k S < ' e+

z1

= O

VS 4L= - p k S i 1 e z3 + z4 = O

V1,L = s3 + s4 + x' - x

= O

V i 4 L

=

Ix + s4 - x l ( = O

V2, L

=

s1

+ s2 + h' - h" = O

V i ? L = h(x)+ SI

-

h" = O

VYL = - g w = O

(12)

V xL

=

Vf(x) -

Jg(x)'.y

+

Jh(X)'i?

+

j 7

q = 0

where 7 j ( x ) is gradient of f (x);

Jg(x)

and

Jh(x)

are the

Jacobians of equality and inequ ality constraints;

SI ,

S2,

S3.

and

S, are diagonal matrices containing SI, s2 s3. and

s,;

e is a

vector

of

ones.

Eq. (1 2 )

may be written in compact fo rm as

F ( w ) = 0

where,

F ( w ) =

-

kS , - ] e

+

il

-

p'S2'e

+ z 2 +

zl

-

k S ; ' e +

z1

-

pkS, 'e

+ z3 +

i

S?

+ S + X I - XI(

l x

+

s4 - I'

sl+

s2

+ hi

-

h"

Vf(x)

-

J ~ ( x ) ~ yJh(x)' z 2 + j " i 4

h(x)+ ~

-

h"

-

g(x)

Eq.

1

3)

may be solved

by

Newton method that implies in

solving a system of linear equations at iteration k :

J , ( d

) A d = -F(\d )

(14)

where Jf

is

the Jacobian

of

F 14,) and A141 is the vector of

corrections in the independent vector w.Afier solving

( 1 4 )

at

each iteration k an estimate of the variables is obtained by

a',, and aid are respectively the primal and dual step sizes.

The maximum primal and dual step sizes at iteration

k

are

190



k

2

k

y m i n ( L ) , y m in(-),

A . 4

As, As@

As2

min (

scalar

y

E

0,l)

is a safety factor to assure the non negativity

conditions of the next point, a typical value is

y =

0.99995.

The errors of the primal and dual fu nctions at iteration

k

are:

maxb, - l(xk )]m axb l(xk ) h,"

I(Vf(x')- Jg(xk)'yk + Jh(xk)'z,k + zi11,

k k k k

dinf(x , y

.z2.z4)=

lxkll*

...

(17b)

For Eq. (15) the primal and dual step sizes are defined as

akp [0 akpma]nd a k d E

[0

akdma] , respectively. Typically,

the effect of a; and

akd

n the primal and dual functions

errors, respectively, is shown in Fig. 1.

the one-dimensional search problem shown in Eq. (1 8).

min pinf ( x k +a +)

0 -< ai

5 aimax

(18)

subject to

After the computation of optimal primal step size a;, the

primal variables (x, sb

s2,

s3 e s4) are updated using the Eq.

(]Sa), then the step

akd

s

computed by solving the one-

dimensional search problem shown in Eq. 19).

min d in f (xk+ ' ,y k+ a $ A y , z , "+a$Az , , z ,k +aiAzz,)

subject to

With the optimal dual step size

akd ,

the dual variables

(y, zI

z2 z3 e z q) are updated through Eq. (15b). The two line search

problems for the computation of the optimal steps a: and akd,

see Eqs. (18) and (1 9), are solved using the bisection m ethod.

The so-called complementarity gap (residual value of the

complementarity condition)

is

computed at iteration

k

by

k k (19)

0

6 ad

6 admaw

pk

=($)'s, +( +:;)'st

+(z, )'s,

+(

+ z $ ) ' s $

(20)

In iteration

k

the value of

pk

is computed based on the

decrement of the complementari ty gap

where p E ( 0 , l )

is

the expected decrement of p k and is

The convergence criteria of the NM are

called the centering parameter.

A

typical value

is p

=

0.2.

p i n z ( x k )

I

dinf ( x k ,

k ,

t

z,k

)

o-'

I

p k

Computation of d w of

(14)

involves the factorization of JF

and the solution of

two

triangular systems. Factorization of JF

is a computationally intensive task of IPM iterations. Mehrotra

[ l

13

obtained best search directions d w by solving two linear

equation systems in each iteration k. These two systems define

the predictor and corrector steps, requiring the same matrix

JF

and different independent vectors in

14).

Thus, only one

factorization of

JF

is needed for both steps. Matrix

JF

is

Fig.

1.

Effect o f the primal and dual step siz es in the functions errors.

For NLP applications that adopt the IPM, the primal and

dual

step sizes are computed as the maximum primal and dual

Empirical evidence suggests that NM, when used in IPM, will

need an adequate computation of the primal and dual step sizes

in order to facilitate the convergence. In this work is proposed

an optimal computation of the primal and dual step sizes

which minimizes the primal and dual objective function errors,

respectively, assuring a continuous decrease during iterations

of IPM procedure (See Fig.

1).

The step akps the solution of

step sizes by using Eq.(16)

a;

=

akpma*,

kd

=

a

d m m ) .

D , O O O O O I O

0

O D , O

0 0 0 1 1 0

O O D O I O O O 0

O O O D 4 1 1 0 0 0

0 0 I I 0 0 0 0

0

I I O 0 0 0 0 0

0

0 I O

0 0 0 0 0

Jh

0 0 0 0 0 i

0

JhT H ,

O O O I O O O O ~ T

0 0 0 0 0 0 0 0 - J g

0

0

0

0

0

0

0

0

Jg'

0

(23a)

191



/= I /= I

Hki ), H,(xV, Hh(xk)

= Hessian matrices of the objective

function,'equality and inequality constraints, respectively. D,

Si 'Zi ,

0 2 = S i i Zi

+

Z j , 0 3

=

S, Z3 and 0 = S i i

(Z, + Z J .

Zi , Z2,

Z3, Z , are diagonal matrices defined by components z i ,

z2,

z3, z4 .

ndg

is the number o f equality constraints variables.

a)

Predictor step

The affine-scaling direction dwaJ is first computed by

[l

11

J ( W k

- 1

- 2 2 - z ,

-

3

-z3 - z 4

- I x - s , + x u

- s l - s 2 - h 1 + h u

- Vf (x )+ J g ( x ) T y - J h ( x ) T z 2 i T z 4

-s3 -s4

- X I + x u

-

~ ( x ) - s ,

h"

g(x)

J F ( W k

dwaJ s used to obtain an estimate for the independent vector

of the corrector step and for the barrier parameter pa[ T o

estimate the value of pd, the primal and dual step sizes along

the affine-scaling direction

(a, ';a / / )

are computed by

(1 6),

like the maximum primal and dual step sizes, respectively. An

estimate for the complementarity gap is given by

Paf

=

( z :

+

a$AzZpf j T (s: + a ; AsZpf)

+

( z f

+ a f A+ +

z i

+ a f h ~ ; fT

(s,

+ a5As;f)+

(z,"

+ a z f A z f f ) T ( s : AS$)+

(z,"

+

a:f

kif

+

z t

+

a$

An estimate for pay is obtained by

(25)

) T ( s t

+

a;fAstf)

where ASi < AS, J and ASJJ are diagonal matrices

defined by com ponents

As//,

s*af,

A.93aJ

and As4@,espectively

Since both the predictor and corrector steps use the s a m e

matrix (JF), the add itional comp utational effort for the

predictor-corrector method consists in the solution of a linear

system for computation of Awaf. The technique proposed by

Mehrotra is well-accepted and used for the computation

of

better search directions in the IPM. However, better yet results

can be obtained by adding another corrector step to the

process. The resulting method is called multiple predictor-

corrector (MPC). The use of MPC method may improve the

convergence performance, resulting in a smaller number ad

iterations [12]. The m-th corrector step consists of solving

b)

Corrector step

The direction dw is computed by

S ; ' ( p f e - A q f & Z f f ) - Z 1

Yil(p fe -

As;f (AzZff +

A @ ) )

-

2 -

1

s;'(paf

e -

hS, f;f)

-

z3

y;'(paf

e - AS;~

A Z ; ~

A$ ))

-

3- 4

- s 3 - s 4 - x 1 i x ' J

- 1 x - s , + x u

-sI

-

2

-

h'

+

h"

-

V f ( ) g ( x ) T y

-

Jh(x)T

z2

-

Tz4

-

h(x)

- + h"

when tn = 0,

Awo

= Awaf. The number of corrections tn for

iteration k varies, if tn =

1

the MF'C method corresp onds to the

conventional predictor-corrector method. In this paper, the

corrector (m

+

1) is computed only if it is possible to obtain a

decrease in the complementarity gap at iteration m (p ' <p

m - i )

and m is less than a maximum iteration count M (M is

typically equal to 5). I f p

2

p m- at iteration m, th e process is

interrupted and the search direction

Aw

= Aw m-i is used. Test

results indicate that larger gains are obtained in further

iterations, as oppos ed to the beg inning o f the iterative process.

IV. COMPUTER

ESTRESULTS

The proposed methodology has been implemented using the

MATLAB version

6.1.

All simulations were conducted on a

PC Pentium

IV.

The IEEE 30,

57,

118,

and 300-bus test

systems have been used in the simulations.

In

Table I it is

shown relevant information for all systems used in the tests.

Also, the initial active power loss (Ploss) s shown.

The following values were set: po=

0.01,

y = 0.99995, =

0.2, nd M = 5. A su mmary o f the simulations is presented in

Table 11, where

x

is the number of primal v ariables,

xlu

is the

number o f primal variables with lower and upper bound, nec

and

nic

are the number of equality and inequality constraints,

respectively,

PIoss

represents the minim um active power loss.

In Table

111

are presented the details of the convergerice

evolution

of

some values during for the IEE E 300-bus system.

In Table 111may be noted that the primal and dual functions

errors decrease continuously during the i terat ions.

An important feature of IPM is that the number of i terat ions

192



does not depend on the size or the number of variables of the

optimization problem (see Table 11), being a very robust tool

to be used for several power system applications. The results

obtained for the active power

loss

minimization problem are

compatible with other results reported in the literature.

TABLE

CHARACTERISTICSF TEST SYSTEMS

Dados IEEE30 IEEE 57 IEEEI 18 IEEE300

nb 30 57

118

300

CHARACTERISTICSF TEST SYSTEMS

Dados IEEE30 IEEE 57 IEEEI 18 IEEE300

nb 30 57

118

300

ng 6 7 54 69

nsh

4 5 12 12

nl 37 65 177 36 1

nt 4 15 9 50

Ploss(Mw) 17.90 28.46 133.36 413.11

TABLE

1

RESULTS OBTAINED FOR TEST SYSTEMS

Dados IEE E30 IEEE57

lEEEl18

IEEE300

X 63 128 244 649

xlu

34 72 127 350

nec

49 101 169 518

nic

10 12 66

81

iterations 8 7 9 12

PIOSS(Wj

16.53 24.56 122.84 379.43

TABLE

ll

SOLUTION OF IEEE 3oo-BUS TEST SYSTEM

iter a

n a k d

pinfk dinfk

P k P k

0 0.0000

0.0000

2.574eMl

1.657etOO 4.352e-01 2.OOOe-03

1 0.3449 0.2643

I

,598eM I 1.180e+00 3.849.~-01 1.778e-03

2 0.4922

0.0671 7.828eM0

I.O02e+OO

3.646e-01 1.600e-03

3 0.8145

0.2393 1.195e+00

6.832e-01 2.870e-01 1.354e-03

4 0.8576

0.6896 1.745e-01

2.043e-01 1.120e-01 5.333e-04

5 0.7505 0.9051 4.344e-02

1.972e-02

1.574e-02 7.520e-05

6 0.4052

0.7360 2.962e-02

5.543e-03 5.262e-03 2.5 18e-05

7 0.7883

0.871 1 7.697e-03

8.743e-04 1.239e-03 5.949e-06

8 0.5768

0.5972 3.725e-03

3.877e-04 5.400e-04 2.596e-06

9 0.4469 0.3205 2.997e-03 1.789e-04 3.547e-04 1.706e-06

10 0.7068

0.7042 1.991e-03

5.225e-05 1.059e-04 5.095e-07

11 0.9164 0.9445 4.032e-04

7.143e-06

6.584e-06 3.168e-08

12 0.9738

0.9965

1.176e-05 1.375e-06

6.794e-08 1.584e-09

V. ONCLUSION

A

model for active power loss minimization has been

presented. This model uses a methodology based on an

optimal power

flow

in which the objective function minimizes

the active power output of the slack generator. The OPF has

been solved with a multiple predictor-corrector interior point

method,

of

the family of higher order interior point methods,

enhanced with an optimal step length computation appro ach.

The optimal computation of the primal and dual step sizes

permits the minimization of the primal and dual objective

function errors, respectively, assuring a continuous decreas e

of

the primal and dual objective function during the iterations of

the interior point method procedure.

The

IEEE 30-bus, IEEE 57-bus, IEEE

11

&bus and IEEE

300-bus systems have been used to test the methodology.

Results of the tests have indicated that the convergence is

facilitated and the number of iterations may be small.

REFERENCES

M.

IliG,

F.D. Galiana and L.H. Fink, Power Sysfems Restructuring,

Kluwer Academic Publishers, 1998.

N. Karm arkar, “A new polynom ial-time algorithm for linear

programming,” Combinaforica , pp. 373-395, 1984.

V.H. Quintana, G.L. Torres and J. Medina-Palomo, “Interior-point

methods and their applications to power systems:

A

classification of

publications and software codes,”

IEEE Trans. Power Systems,

vol. 15,

pp. 170-176, Feb. 2000.

G.L Torres and V .H. Quintana, “An Interior-point methods for non-linear

optimal power flow u sing voltage rectangular coordinates,” IEEE Trans.

Power Sysfems, vol. 13,110. 4, pp, 121 1-1218, Nov. 1998 .

S. Granville, “Optimal reactive dispatch through interior point methods,”

IEEE Trans. Power Sysfems, vol. 9, pp.136-146, Feb. 1996.

H. Wei, H. Sasaki, J. Kubokawa and R. Yohoyama, “An interior point

methods for power systems weighted nonlinear L1 norm static state

estimation,”

IEEE Trans. Power Systems,

vol. 13, no. 2, pp. 617-623,

May 1998.

G.D. Irisarri, X. Wang, J. Ton g and S. Mokhtari, “Maximum loadability

of power systems using interior point method nonlinear optimization,”

IEEE Trans. Power Systems, vol. 1 2, no.

1,

pp. 162 -172, Feb. 1997.

X. Wang, G.C. Ejebe, J. Tong and J.G. Waight, “Preventivekorrective

control for voltage stability using direct interior point method,” IEEE

Trans. Power Systems, vol. 13, no. 3, pp. 878-883, Aug. 1998.

J. Medina, V.H. Quintana, A.J. Conejo and F.P. Thoden, “A com parison

of

interior-point codes for medium-term hydro-thermal coordination,”

IEEE Trans. Power Systems, vol. 13, no. 3, pp. 836-843, Aug. 1998 .

X . Yan and V.H. Quintana, “An efficient predictor - corrector interior

point algorithm for security-constrained economic dispatch,” IEEE

Trans. Power Systems, vol. 12, no. 2, pp. 803-810, May 19 97.

S. Mehrotra, “On the implem entation of a primal-dual interior point

method”, SIAM Joumal on Optimization, vol. 2, pp. 575-601, 1992.

T.J. Carpenter, I.J. Lustig, J.M. Mulvey and D.F. Shanno, “Higher-order

predictor-corrector interior point methods with applications to quadratic

objectives,” SIAM Journal on Optimization, vo1.3, no.4, pp.696-725,

1993.

R. Fletcher and S. Leyffer, “Nonlinear programming without a penalty

function,” Uni. of Dun dee, Dundee, U.K., Numeric. Anal. Rep. N A/I71 ,

Sept. 22, 1997.

A.V. Fiacco and G .P. McCormick, Nonlinear programming: Sequential

unconstrainedminimization echniques, John Wiley & Sons, 1968.

Marcos

J.

Rider was born in Lima, Peru in 1975. He received the B .S. (with

honors) and P.E. degrees from National University of Engineering (UNI),

Lima, Peru, and the M.S. degree from Federal University of MaranhPo

(UFMA), Brazil, all in electrical engineering. Currently he is working towards

a Ph.D. degree in electrical engineering at State University of Campinas

(UNICAM P), Brazil. His research areas are the developme nt of methodologies

of optimization and ap plications of artificial intelligence in power systems.

V. Leonard0 Paucar

(M’90-SM’99) was

born

in Lima, Peru. He received the

B.S. (first-class honors) and P.E. degrees from UNCP o f Peru, the M .S. degree

from Catholic University of Chile and the Ph.D. degree from State University

of Campinas (UNICAMP), Brazil, all in electrical engineering. He is a

Professor at National U niversity of Engineering (UNI), Peru, and at present is

a V isiting Professor at Fe deral University of MaranhPo (UFMA), Brazil. His

research interests include the applications of artificial intelligence techniques

in power system s security and electricity markets.

Ariovaldo V. Garcia

received his B.S. and Ph.D. degrees in electrical

engineering in 1974 and 1981, respectively, from State University of

Campinas (UNICAM P), Brazil, where he is currently an Associate Professor

of Electrical Engineering. He has served as a consultant for a number of

organizations. His general research interests are in the areas of planning and

control of electrical power systems.

Manfred F. Bedrifiana

was bom in Lima, Peru. He received the

B.S.

degree

in electrical engineering (first-class honors) from National University

of

Engineering (UNI), L ima, Peru. Currently he is working towards a M.S degree

in electrical engineering at Federal Unive rsity of MaranhPo (UFMA ), Brazil.

His research areas are electricity markets and security assessment of electrical

energy systems.

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lescope-2003-a higher order interior point method to minimize active power loss

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