A New Approach to Economic Dispatching Using

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1/13

This article was downloaded by: [Tripura University]On: 22 February 2013, At: 02:06Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

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A New Approach to EconomicDispatching Using Load Bus

Elimination TechniquesJawad Talaqa

University of Bahrain Department of Electrical

Engineering P.O. Box 32038, Bahrain

Version of record first published: 30 Nov 2010.

To cite this article: Jawad Talaq (2000): A New Approach to Economic Dispatching

Using Load Bus Elimination Techniques, Electric Machines & Power Systems, 28:8,

723-734

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2/13

Electric Machines and Power Systems, 28:723734, 2000

Copyright cs 2000 Taylor & Francis

0731-356X/ 00 $12.00 + .00

A New A pproach to Economic Dispatching UsingLoad Bus Elimination Techniques

JAWAD TALAQ

University of BahrainDepartment of Electrical EngineeringP.O. Box 32038, Bahrain

This paper presents a new technique to solving the economic dispatch problembased on a reduced order model of the original system. Loads are rst mod-

eled by their appropriate voltage-dependent models as load admittances. Loadadmittances are then added to the bus admittance matrix and their respectivebuses are eliminated. The obtained model is a reduced model of the originalsystem. The admittance matrix is of the same order as the number of voltage-controlled buses in the system. The variables of the reduced model are thevoltage-controlled buses angles and active power generations. Newton Raph-son method is used to calculate the angles and active generations of the reduced

model while minimizing the operational cost. Load bus voltages and angles ofthe original system are then calculated by a direct method and load admittancesare modied. The process is repeated until convergence is achieved. The simu-lation is carried out on IEEE 118 bus test system. A comparison between thenew approach and the penalty factors method has been made. It is shown thatoperational cost is improved and solution time is signicantly reduced whencompared to the penalty factors method of economic dispatch.

1 Introduction

Economic dispatch is used in real time to allocate the total generation producedamong all units in the power system in such a way as to minimize the operationalcost. The equal incremental cost criterion for economic dispatch was rst imple-mented by utilities due to its simplicity when transmission losses are neglected.With the developments of power systems, transmission losses could not be ignoredand had to be included in the power balance equation. Transmission losses couldhave been modeled as a constant percentage of total system demand and the equalincremental cost criterion still could be implemented; however, transmission lossesdepend on the allocation of generation among units, and the treatment of losses in

the power balance equation needed to be improved. This led researchers to nd suit-able transmission loss models, and therefore the B-coecients method had emerged.From then, researchers concentrated in nding methods that can be implementedin real time and still meet the required accuracy in modeling transmission lossesand penalty factors of units. References [15] review recent advances in classic eco-nomic dispatch. The optimal power ow is an exact formulation of the networkactive and reactive power mismatches at all buses, and hence led researchers to

Manuscript received in nal form October 7, 1999.Address correspondence to Jawad Talaq.

723

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724 Talaq

further investigate the possibility of real time implementation. Reference [6] con-tains a survey of the optimal power ow literature. Now, optimal power ow is usedin power system planning, and researchers are going on for further improvementthat hopefully it can replace the economic dispatch in real time implementation.

Until then, however, utilities still will rely on the classical economic dispatch for itssimplicity, and further developments that can improve both operational cost andsolution time should be encouraged.

2 T he Equal Incremental Cost C riterion

Traditionally, active power generation is distributed among all generators in such away as to minimize the total operational cost of all generators in the power systemwhile meeting system demand and transmission losses.

The Lagrangian function formulated in this case is:

L =X

i

Ci +

PD + PL o s s

Xi

Pi

!, (1)

where

Ci = a0i + a1iPi + a2iP2

i+ a3iP

3i (2)

is a function of third order representing operational cost of generator i, Pi is theactive generation of unit i, PL o s s is the transmission losses, PD is the total systemdemand, and is the Lagrangian multiplier associated with the power balanceequation.

If transmission losses are ignored, then the familiar equal incremental cost cri-terion for minimum operational cost is obtained:

@Ci@Pi

=@Cj@Pj

= . (3)

3 T he Penalty Factors Method

If transmission losses are included in equation (1), then the following criterion isobtained:

@Ci@Pi

P Fi =@Cj@Pj

P Fj = , (4)

where

P Fi = 1=(1 @PL o s s=@Pi ) (5)

is the penalty factor of unit i.Transmission losses may be related to active generation through the familiar

B-coecients as follows:

PL o s s = PT BP + B0 P + B 00, (6)

where P is the active power generation vector, B is a square matrix, B0 is a rowvector, and B 00 is a constant.

The B-coecients are functions of the operating point of the power system but

may be assumed constant for a certain range of operating points. Several methods




4/13

A New Approach to Economic Dispatching 725

have been proposed to nd the B-coecients by several authors. One of these isthe one described in [3]:

PL o s s = Re aV(I* ZI), (7)

where Z is the bus impedance matrix and I is the bus current injection vector.Simplifying equation (7) yields

PL o s s =NX

i= 1

NXj= 1

[D ij (PiPj + Q i Qj ) + Cij (Q iPj PiQj )], (8)

where Pi and Q i are active and reactive power injections at bus i, and N is thetotal number of buses in the system:

D ij = R ij cos(ij )=ViVj , (9)

Cij = R ij sin(ij )=Vi Vj . (10)

R ij in equations (9) and (10) are elements of the real part of the impedance matrix.Furthermore, if we assume that bus angle dierences are small, i.e., sinij ~= 0,

then the following simplied equation is obtained [3]:

@PL o s s=@Pi = 2N

Xj = 1

D ijPi . (11)

4 T he Proposed Technique

Load modeling has been applied to the power ow solutions [78]. Bus eliminationtechnique [8] has been applied to the power ow solution without considering theeconomic dispatch problem. The aim here is to apply load modeling and load buselimination techniques to the economic dispatch problem and treat transmission

losses as an exact model. Consider both active and reactive power of loads asexponentially voltage dependent models having the following forms:

PL = P0Va , (12)

QL = Q0Vb, (13)

S*

L= PL + j QL , (14)

where

P0 is the active power specied at nominal voltage.Q0 is the reactive power specied at nominal voltage.

a is the voltage dependent exponential constant of active power model.

b is the voltage dependent exponential constant of reactive power model.

Also we have

S*

L= V

*IL = YL V

2. (15)

This yields

YL = 1V2 [P

0Va j Q 0Vb]. (16)




5/13

726 Talaq

Equation (16) represents the load admittance model. The exponential constants ofthe load model usually are determined from eld tests. The algorithm is capable ofhandling dierent values of exponential constants for the dierent buses; however,three special cases may be obtained from equation (16):

1. Constant admittance load model (a = b = 2.0):

YL = (P0 j Q0 )

2. Constant complex power model (a = b = 0.0):

YL =1

V2(P0 j Q 0)

3. Constant current model (a = b = 1.0):

YL =1

V(P0 j Q0 )

The admittance matrix is modied by adding the load admittances to thediagonal elements that belong to load buses. The load buses are then eliminated toobtain a reduced model as follows:

" ILIG#

=" Y

L0Y

LGYG L YG# " V

LVG#

, (17)"0

IG

#=

"YL YLG

YG L YG

# "VL

VG

#, (18)

where

YL = YL0 + Yd (19)

and Yd is a diagonal matrix containing the load admittance models described by

equation (16).Solving equation (18) yields

VL = Y1

L YLGVG (20)

and

IG = YRVG , (21)

where

YR = YG YG LY1

L YLG (22)

is an admittance matrix of the same order as the number of voltage-controlled buses(NG).

The complex power injections of the reduced model are

S*

G i= V

*

G iIG i = V*

G i

N G

Xj= 1VG j Yij , (23)




6/13


where Yij are the elements of the reduced bus admittance matrix YR and N G isthe number of the voltage-controlled buses in the system.

This yields the active power mismatches of the voltage-controlled buses

DPi = ViN GXj= 1

Vj[G ij cos(ij ) + B ij sin(ij )] Pi . (24)

The augmented Lagrangian function is formulated as follows:

L =N GXi= 1

Ci +N GXi= 1

pi DPi +X 1

2si (Pi Pi lim )

2, (25)

where Ci is the operational cost of unit i as dened in equation (2) and pi is aLagrangian multiplier associated with the active power mismatch of unit i. The lastterm of equation (25) is the sum of quadratic penalty functions imposed on activegeneration of units that exceed their maximum or minimum power limits and si isthe penalty factor corresponding to each penalty function.

The penalty factors, si , for the violating units are increased with iterationsuntil variables are within acceptable tolerances. If a violating unit goes back toits operational range, its si value is set to zero. The reactive power mismatchesneed not be included in the Lagrangian function if they do not violate their limits;

however, reactive generations of units are tested for violations, and if one violatesa limit, then the reactive generation is set to the limit and the voltage is released.Usually, in power ow solutions, PV buses are converted to PQ buses if they violatetheir reactive generation limits. This technique is not implemented here becauseactive generation of units that violate their reactive generation should still remaindispatchable. Instead, the reactive power mismatch of the violated unit is introducedto the Lagrangian function together with its Lagrangian multiplier. This adds two

variables to the Lagrangian function: the voltage and the Lagrangian multiplierassociated with the reactive power mismatch.

Taking the rst and second derivatives of the Lagrangian function with respectto the variables and applying Newton Raphson method yields the following matrixequations:

f = W D x, (26)

D x = [D, D p, D P]T , (27)

f = "@L

@

,@L

@p,@L

@P#

T

, (28)

W =

24H JT 0J 0 1

0 1 IIC

35 , (29)

where f is a vector representing the rst derivatives of the Lagrangian function, D xis the incremental variables vector, and W is the Jacobian matrix. Submatrix Hcontains the second derivatives of the Lagrangian function with respect to angles,and submatrix J contains the rst derivatives of the active power mismatches with




7/13

728 Talaq

Figure 1. Flow chart of the proposed algorithm.

respect to angles. IIC is a diagonal submatrix containing the second derivatives ofthe operational cost functions. T denotes the transpose:

H =@2(DP)@2

, (30)

J =@(D P)@

, (31)

IIC =@2C

@P2. (32)

Equation (26) is solved for D x using Newtons method. Once voltage-controlledbus angles are computed, equation (20) is solved for load bus voltages and angles.Load admittances are recalculated using equation (16), and YL of equation (22)

is modied using equation (19). Equation (26) is solved again, and the process isrepeated until convergence is achieved. A ow chart describing the implementationof the proposed algorithm is shown in Figure 1.

5 Results of Simulation

The proposed algorithm has been tested on IEEE 118 bus test system using UNIX-based Mips Millennium Computer System. A comparison between the equal incre-mental cost criterion, the penalty factors method, and the proposed method has

been made for three dierent exponential constants of load modeling. IEEE 118




8/13


bus test system consists of 54 voltage-controlled buses, of which 18 are dispatch-able generators.

The proposed technique consists of major and minor iterations. A major iter-ation refers to the loop in which loads are modeled, eliminated, and the system is

reduced. A minor iteration refers to each Newtons iteration of the reduced model.In the proposed technique, operational cost is minimized while power mismatchesare satised. This means that transmission losses are exactly modeled compared tothe approximate models used for the penalty method. This fact assures least oper-ational cost solutions for the proposed technique compared to the penalty method,even with more updates of the B-coecients. Following are the four categories ofsimulation that have been compared with the proposed technique for each loadmodeling case:

1. The equal incremental cost criterion without losses: Transmission losses areneglected in the power balance equation. System demand assumed constantand units are dispatched followed by a normal power ow. The process con-sists of only one major iteration.

2. The equal incremental cost criterion with losses: First, transmission lossesare neglected and units are dispatched, assuming constant system demand,followed by a normal power ow. This is the rst major iteration. Systemdemand and transmission losses obtained from the rst major iteration areused in the power balance equation, and units are redispatched followed by

another power ow. This is the second major iteration.3. The penalty factors method (two iterations): First, units are dispatched

according to the equal incremental cost criterion and, assuming constantsystem demand, followed by a normal power ow. This is the rst majoriteration. B-coecients are calculated and units are redispatched throughthe penalty factors method followed by another power ow. This is the sec-ond major iteration. The process is repeated for another major iteration byupdating the B-coecients, system demand, and transmission losses.

4. The penalty factors method (three iterations): Same as in 3 above withanother extra update of the B-coecients, system demand, and transmissionlosses.

Operating cost, solution time, and iteration counts are shown in Table 1. Outputgeneration for the three cases, the constant complex power demand, the constantcurrent demand, and the constant load admittance demand are shown in Tables 2,3, and 4.

The operational cost for the constant complex power case of the equal incre-mental cost criterion is taken as basis for the comparison as shown in Table 1. Using

the proposed approach, operational cost has been reduced to 0.978824 pu (2.12%)instead of 0.980297 pu (1.97%) when the penalty factors method is used with twoiterations. The solution time has been considerably reduced to 9.70 seconds insteadof 31.80 seconds, which is just comparable to the ordinary equal incremental costcriterion (8.40 seconds). Operational cost may be further reduced using the penaltyfactors method with more iterations to values comparable with that obtained bythe proposed method, but this will be on the expense of solution time. This is ob-

vious from Table 1, as operational cost has been reduced to 0.978881 pu (2.11%),when the penalty factors method is used with three iterations but solution time in-

creased to 47.20 seconds. T he iteration counts shown in Table 1 refer to the required




9/13

730 Talaq

Table

1

IEEE118bustest

system

operatingcost,s

olutiontime

iterationco

unts

Constantcom

plex

Constantcurrent

Constanta

dmittance

power

loadcase

loa

dcase

loa

dcase

Operating

So

l.

Operating

So

l.

Operating

Sol.

cost

time

Iter.

cost

time

Iter.

cost

time

Iter.

(pu

)

(sec

)

counts

(pu

)

(sec

)

counts

(pu

)

(sec)

counts

Equa

lincrementalcost

1.0

00000

8.4

6

0.9

937

50

9.8

6

0.9

87985

11

.4

6

(no

losses)

Equa

lincrementalcost

1.0

00012

12

.7

6,4

0.9

938

72

15

.7

6,4

0.9

88191

18

.9

6,4

(withlosses

)

Pena

lty

factorsmetho

d

0.9

80297

31

.8

6,4,4

0.9

753

25

35

.0

6,4,5

0.9

70686

37

.8

6,4,5

(two

iterations

)

Pena

lty

factorsmetho

d

0.9

78881

47

.2

6,4,4,3

0.9

741

09

51

.4

6,4,5,3

0.9

69634

54

.7

6,4,5,4

(three

iterations

)

Propose

dapproac

h

0.9

78824

9.7

6,3,3,2,2,2,2

0.9

740

61

8.4

6,3,3,2,

2

0.9

69594

4.0

6




10/13


Table

2

IEEE

118bustestsystem

econom

icdispatchresu

ltsfo

rtheconstantcomp

lexpower

loa

dcase

Equa

lincrementalc

ost

Pena

lty

factorsmethod

Propose

d

(no

losses)

(withlosses)

(two

iterations)

(three

iterations)

app

roac

h

Gen

Output

Out

put

Output

Pena

lty

Output

Pena

lty

Ou

tput

no

(pu

)

(pu

)

(pu

)

factor

(pu

)

factor

(pu

)

1

2.6

48308

2.78

9004

2.4

91929

0.9

79561

2.4

25627

0.9

80862

2.5

15780

2

3.0

00000

3.00

0000

3.0

00000

0.9

93083

3.0

00000

0.9

92414

3.0

00000

3

3.0

00000

3.00

0000

2.8

98995

1.0

08289

2.8

59776

1.0

08374

2.9

19261

4

4.7

24726

3.01

8913

1.9

05124

1.0

00000

2.8

07982

1.0

00000

2.8

87783

5

1.8

18647

1.93

1203

1.7

03315

0.9

78853

1.6

85921

0.9

77559

1.6

65509

6

2.2

37294

2.46

2407

2.9

64136

0.9

45336

2.9

27559

0.9

44105

2.9

07532

7

1.1

82206

1.27

6003

1.8

69217

0.9

15163

1.8

63007

0.9

13247

1.8

61620

8

3.0

00000

3.00

0000

3.0

00000

0.9

49162

3.0

00000

0.9

47706

3.0

00000

9

0.0

00000

0.00

0000

0.5

66501

0.9

29570

0.5

30774

0.9

28873

0.5

01174

10

2.0

60386

2.23

6255

2.8

45772

0.9

36016

2.7

73667

0.9

36635

2.7

31307

11

0.1

98770

0.36

1094

0.5

94663

0.9

40489

0.5

64251

0.9

40274

0.5

48163

12

1.8

64411

2.05

2006

2.6

25525

0.9

39073

2.5

37073

0.9

40169

2.4

97523

13

1.3

00752

1.43

1632

2.1

37909

0.9

21830

2.0

20119

0.9

25968

1.9

76541

14

2.6

37294

2.86

2407

0.7

53693

1.0

42669

0.7

37510

1.0

40614

0.7

46670

15

3.0

00000

3.00

0000

3.0

00000

1.0

00179

3.0

00000

1.0

07192

3.0

00000

16

1.5

16013

1.72

4451

1.7

75624

0.9

60674

1.5

56930

0.9

66464

1.5

75372

17

3.0

00000

3.00

0000

2.6

46669

0.9

98249

2.5

36044

1.0

00830

2.5

13478

18

1.2

15920

1.33

8265

0.6

86765

1.0

06544

0.6

42112

1.0

07069

0.6

30012

Totaloutput(pu

)

38

.404727

38

.42

3640

37

.465

838

37

.468353

37.4

77726

Losses

(pu

)

1.7

24727

1.74

3640

0.7

85

838

0

.788353

0.7977263

Totalloa

d(pu

)

36

.68




11/13

732 Talaq

Table

3

IE

EE118bustestsystem

econom

icdispatchresultsfortheconstantcurrentloa

dcase

Equa

lincrementalcost

Pena

lty

factorsmetho

d

Propose

d

(no

losses)

(with

losses)

(two

iter

ations)

(three

iterations)

app

roac

h

Gen

Output

Output

Output

Pena

lty

Output

Pena

lty

Ou

tput

no

(pu

)

(p

u)

(pu

)

factor

(pu

)

factor

(pu

)

1

2.6

48308

2.771172

2.4

80031

0.9

79685

2.4

13588

0.9

80996

2.5

02934

2

3.0

00000

3.000000

3.0

00000

0.9

93097

3.0

00000

0.9

92396

3.0

00000

3

3.0

00000

3.000000

2.8

91147

1.0

08247

2.8

52101

1.0

08321

2.9

10766

4

4.5

06134

3.014110

1.9

38106

1.0

00000

2.7

99179

1.0

00000

2.8

74410

5

1.8

18647

1.916938

1.6

93776

0.9

78977

1.6

76291

0.9

77689

1.6

55924

6

2.2

37294

2.433876

2.9

45675

0.9

45412

2.9

10642

0.9

44129

2.8

90394

7

1.1

82206

1.264115

1.8

60466

0.9

15298

1.8

54993

0.9

13326

1.8

53963

8

3.0

00000

3.000000

3.0

00000

0.9

49074

3.0

00000

0.9

47559

3.0

00000

9

0.0

00000

0.000000

0.5

53624

0.9

29654

0.5

19818

0.9

28864

0.4

90652

10

2.0

60386

2.213966

2.8

28636

0.9

36200

2.7

59700

0.9

36686

2.7

17809

11

0.1

98770

0.288125

0.5

85584

0.9

40612

0.5

56641

0.9

40290

0.5

40652

12

1.8

64411

2.028230

2.6

06553

0.9

39289

2.5

21934

0.9

40232

2.4

83118

13

1.3

00752

1.415044

2.1

16378

0.9

22487

2.0

04543

0.9

26302

1.9

62918

14

2.6

37294

2.833876

0.7

36971

1.0

42752

0.7

28309

1.0

40387

0.7

37329

15

3.0

00000

3.000000

3.0

00000

1.0

00916

3.0

00000

1.0

07248

3.0

00000

16

1.5

16013

1.698033

1.7

40770

0.9

61432

1.5

34594

0.9

66757

1.5

54360

17

3.0

00000

3.000000

2.6

29004

0.9

98497

2.5

24167

1.0

00808

2.5

02359

18

1.2

15920

1.322759

0.6

81477

1.0

06333

0.6

38166

1.0

06763

0.6

26485

Totaloutput(pu

)

38

.186134

38.200244

37

.288197

37

.294667

37.3

04076

Losses

(pu

)

1.7

05317

1.726982

0.75

5894

0.7

80144

0.7

89341

Totalloa

d(pu

)

36

.480817

36.473262

36

.512304

36

.514522

36.5

14735




12/13


Table

4

IEEE118bustestsystemec

onom

icdispatchresu

lts

fortheconstanta

dm

ittance

loa

dcase

Equa

lincrementalcost

Pena

lty

factorsmetho

d

Propose

d

(no

losses)

(with

losses

)

(two

iter

ations)

(three

iterations)

app

roac

h

Gen

Output

Output

Output

Pena

lty

Output

Pena

lty

Ou

tput

no

(pu

)

(p

u)

(pu

)

factor

(pu

)

factor

(pu

)

1

2.6

48308

2.754576

2.4

68611

0.9

79806

2.4

62075

0.9

81126

2.4

90715

2

3.0

00000

3.000000

3.0

00000

0.9

93111

3.0

00000

0.9

92387

3.0

00000

3

3.0

00000

3.000000

2.8

83650

1.0

08208

2.8

44777

1.0

08272

2.9

02705

4

4.3

02682

3.010645

1.9

66801

1.0

00000

2.7

90022

1.0

00000

2.8

61717

5

1.8

18647

1.903661

1.6

84980

0.9

79073

1.6

67481

0.9

77788

1.6

47095

6

2.2

37294

2.407321

2.9

28447

0.9

45471

2.8

94889

0.9

44140

2.8

74375

7

1.1

82206

1.253050

1.8

52280

0.9

15414

1.8

47502

0.9

13389

1.8

46782

8

3.0

00000

3.000000

3.0

00000

0.9

48981

3.0

00000

0.9

47413

3.0

00000

9

0.0

00000

0.000000

0.5

41589

0.9

29722

0.5

09556

0.9

28848

0.4

80749

10

2.0

60386

2.193220

2.8

12663

0.9

36360

2.7

46629

0.9

36725

2.7

05114

11

0.1

98770

0.276055

0.5

77113

0.9

40715

0.5

49524

0.9

40295

0.5

33597

12

1.8

64411

2.006101

2.5

88889

0.9

39476

2.5

07786

0.9

40282

2.4

69567

13

1.3

00752

1.399605

2.0

96512

0.9

23077

1.9

90097

0.9

26598

1.9

50152

14

2.6

37294

2.807321

0.7

22288

1.0

42779

0.7

19733

1.0

40164

0.7

28589

15

3.0

00000

3.000000

3.0

00000

1.0

01548

3.0

00000

1.0

07282

3.0

00000

16

1.5

16013

1.673446

1.7

08939

0.9

62103

1.5

13945

0.9

67014

1.5

34761

17

3.0

00000

3.000000

2.6

12813

0.9

98701

2.5

13128

1.0

00775

2.4

91959

18

1.2

15920

1.308327

0.6

76597

1.0

06120

0.6

34488

1.0

06467

0.6

23178

Totaloutput(pu

)

37

.982682

37.993328

37

.122172

37

.131633

37.1

41055

Losses

(pu

)

1.6

88615

1.711393

0.76

6963

0.7

72540

0.7

81589

Totalloa

d(pu

)

36

.294067

36.281930

36

.355209

36

.359093

36.3

59466




13/13

734 Talaq

Newtons iterations for each major iteration. For the constant complex power case,it took seven major iterations for the proposed approach solution. The requiredNewtons iterations for this case are 6, 3, 3, 2, 2, 2, and 2.

For the constant current load model, the reduction in operational cost is

0.974061 (2.59%) compared to 0.975325 (2.46%) of the penalty factors method.The solution time has also been considerably reduced. For the constant admittanceload model case, the solution time, as expected, is lowest (4.0 seconds) becausethere is no need for updating load admittances as they are constants.

6 Conclusions

A new technique to solving the economic dispatch problem has been proposed.The technique uses a reduced order model of the original system. The reducedorder model is obtained by eliminating all load buses from the system after beingmodeled appropriately. The variables of the model are the voltage-controlled busesangles and active generation. The eect of dierent load modeling on generatoractive generation and operational cost has been studied. A comparison between theproposed technique and the classic economic dispatch has been made. Results ofthe simulation on IEEE 118 bus test system shows that improvement in operationalcost and solution time is achieved by using the proposed technique.

References

[1] Happ, H. H., 1977, Optimal Power Dispatch: A Comprehensive Survey, IEEE Trans.on Power App. and Syst., Vol. PAS-96, No. 3, pp. 841854.

[2] Aoki, K., and Satoh, T ., 1984, New Algorithms for Classic Economic Dispatch, IEEETrans. on Power Appt. and Syst., Vol. PAS-103, No. 6, pp. 14231431.

[3] Wenyuan, L., 1985, An On-Line Economic Power Dispatch Method With Security,Electric Power Systems Research, Vol. 9, pp. 173181.

[4] Lin, C. E., Hong, Y. Y., and Chuko, C. C., 1987, Real-Time Economic Dispatch,IEEE Trans. on Power Syst., pp. 968972.

[5] Chowdhury, B. H., and Rahman, S., 1990, A Review of Recent Advances in EconomicDispatch, IEEE Trans. on Power Syst., Vol. 5, No. 4, pp. 12481259.

[6] Huneault, M., and Galiana, F. D., 1991, A Survey of the Optimal Power Flow Liter-ature, IEEE Trans. on Power Syst., Vol. 6, No. 2, pp. 762770.

[7] El-Hawary, M. E., and Dias, L. G., 1987, Incorporation of Load Models in Load FlowStudies: Form of Model Eects, IEE Proceedings, Vol. 134, Part C, No. 1, pp. 2730.

[8] Talaq, J. H., 1995, Modeling and Elimination of Load Buses in Power Flow Solutions,IEEE Trans. on Power Syst., Vol. 10, No. 3, pp. 11541158.



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A New Approach to Economic Dispatching Using