hybrid algorithm based on ep and lp for security constrained economic dispatch problem

Electric Power Systems Research 76 (2005) 77–85

Hybrid algorithm based on EP and LP for securityconstrained economic dispatch problem

P. Somasundaram∗, R. Lakshmiramanan, K. KuppusamyDepartment of Electrical and Electronics Engineering, Anna University, Chennai 600 025, India

Received 16 June 2004; received in revised form 6 February 2005; accepted 19 April 2005Available online 5 July 2005

Abstract

This paper presents a new hybrid method for solving security constrained economic dispatch problem. The proposed method is developedin such a way that a simple evolutionary programming is applied as a base level search, which can give a good direction to the optimal globalregion and a local search linear programming is used as a fine tuning to determine the optimal solution. The hybrid approach outperforms theevolutionary programming based method and reduces the total computation time. A 10-bus and 66-bus Indian utility systems are considered toillustrate the effectiveness of the proposed method. The investigations reveal that the proposed method is suitable for practical utility systems.©

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2005 Elsevier B.V. All rights reserved.

eywords: Evolutionary programming; Genetic algorithm; Linear programming; Hybrid method; Security constrained economic dispatch

. Introduction

The security constrained economic dispatch (SCED)efers to the case when the system is dispatched such thatt is in the normal secure state relative to a pre-specified con-ingency list. In other words, for any contingency belongingo this list, the redistribution of power flows in the network,ollowing a contingency, will result in a normal system state.

In literature different optimization algorithms have beenroposed to solve the SCED problem. Among them, the most

mportant are non-linear programming[1,2], explicit Newtonpproach[3], successive linear programming (SLP)[4], non-

inear Dantzig–Wolfe decomposition[5], predictor–correctornterior point algorithm[6] and homogeneous linear pro-ramming algorithm[7]. However, the increase of on-line

unctions in modern energy control centres, together withhe growth in the size of power system network, demand foraster and more reliable numerical techniques to solve theptimization problems.

Evolutionary programming (EP) based algorithms areeing increasingly applied for solving power system opti-

mization problems in recent years[8–15]. In reference[9–15] the economic dispatch (ED) problem with vaous constraints have been solved effectively by usingAlthough EP seems to be a good method to solve optimtion problems, it takes large computation time in ordeobtain the solutions.

The rate of solution convergence for ED problem is vfast at the beginning (0–15 iterations) with EP based arithm. Thereafter, it is very slow up to the end of convergeThis results in a very large computation time especially wthe operating and security constraints are included in thproblem formulation. It has also been observed that thecase and contingency case constraint violations are corrsignificantly during first few iterations in EP. In contrast,deterministic SLP method initially suffers from oscillatioand also the model becomes inaccurate when wider or lvariations are allowed in the control variables[4,16]. Butlarger variations in the control variables are required torect the severe constraint violations especially when thecase and contingency case constraints are includedproblem formulation. SLP is a deterministic method andmodel is accurate when the variations in the control varia

∗ Corresponding author.E-mail address: [email protected] (P. Somasundaram).

are small and is very effective in correcting the moderateconstraint violations. The above fact suggests that a hybrid

378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
oi:10.1016/j.epsr.2005.04.005

78 P. Somasundaram et al. / Electric Power Systems Research 76 (2005) 77–85

method with EP algorithm for initial solution and SLP methodfor getting the final solution would be an effective and fastermethod.

In this paper, the application of EP combined with LP forsolving SCED is proposed. Basically, the hybrid method hastwo phases. The first phase employs EP to obtain a near globalsolution, while the second phase employs LP to determinethe final optimal solution. The effectiveness of the proposedmethod is tested on 10-bus system and 66-bus Indian util-ity system. The results obtained with the proposed hybridmethod are compared with those obtained from genetic algo-rithm (GA) and EP.

2. Problem formulation

The SCED problem can be stated as an NLP problem asfollows:

minFT =∑

j ∈ αGC

(ajP2Gj + bjPGj + cj)

+ asP2Gs + bsPGs + cs (1)

Subject to:the control constraints

P

t

F

w sea fa eterv terss

−t

P

t

F

a

−w ,rc

eb Eqs.(

3. Methodology

EP is a near global stochastic optimization method start-ing from multiple points, which places emphasis on thebehavioural linkage between parents and their offsprings,rather than seeking to emulate specific genetic operatorsas observed in nature to find a solution. Initially, the rateof convergence in EP is very fast and subsequently theconvergence is very slow. On the other hand SLP suffersfrom the oscillatory convergence problem[4,16]. The LPwill be effective only when the magnitudes of constraintviolations are less and corrections in the control variablesare small. In order to overcome these difficulties, a two-phase hybrid method has been proposed in this paper. Inthe first phase, EP is applied for a specified number ofiterations to obtain a near global solution. In the secondphase, solution obtained from EP is taken and the LP prob-lem is formed and is solved to obtain the final optimalsolution.

3.1. Evolutionary programming

3.1.1. Initialization of parent populationThe individuals in a parent are the real power outputs of the

committed generating units excluding the slack bus unit. Thei s:

P

aG mv samew

rentI busg ctionf

f

w intv

P

φ

Gj,min ≤ PGj ≤ PGj,max; j ∈ αGC (2)

he base-case power flow equations

(X, U, C) = 0 (3)

here the state vectorX comprises of the bus voltage phangles and magnitudes. The control vectorU comprises oll the controllable real power generations. The paramectorC includes all the uncontrollable system parameuch as line parameters, loads, etc.

The base-case line flow operating constraints,

φk,max ≤ φk ≤ φk,max; k ∈ αNLL (4)

he slack bus constraint,

Gs,min ≤ PGs ≤ PGs,max (5)

he contingency case power flow equations,

R(XR, U, CR) = 0; R ∈ αNO (6)

nd the contingency case line flow security constraints

φk,max ≤ φRk ≤ φk,max; R ∈ αNO (7)

here the triplet (X, U, C) and (XR, U, CR) characterizeespectively, the given base-case state and theRth post-ontingency state.

The ED problem comprising Eqs.(1)–(5) constitutes thase-case ED problem while ED problem comprising1)–(7) constitutes the SCED problem.

nitial parent population is generated randomly as followConsider the pith parent,Ipi = [Ppi

G1, PpiG2, . . . , P

piGj, . . . ,

piGGC] of the population of sizeNp. The components ofIpi

re generated asPpiGj ∼ UR(PGj, min, PGj, max), j = 1, 2, . . .,

C where UR (PGj, min, PGj, max) denotes a uniform randoariable. The remaining parents are generated in theay.Load flow is run with the unit generations of each pa

pi, pi = 1, 2. . .,Np and the system transmission loss, slackeneration and line flows are evaluated. The fitness fun

or each parent of the population is computed as:

pi = FTpi + k1Plim,piGs + k2

NLL∑k=1

φlim,pik

+ k3

NO∑R=1

NLL∑k=1

φlim,R,pik ; pi = 1, 2, . . . , Np (8)

herek1, k2 and k3 are penalty factors for the constraiolations,FTpi is the total fuel cost for pith parent,

lim,piGs =

{PGs,min − P

piGs, if P

piGs < PGs,min

PpiGs − PGs,max, if P

piGs > PGs,max

(9)

lim,pik =

{∣∣∣φpik

∣∣∣ − φk,max, if∣∣∣φpi

k

∣∣∣ > φk,max

0 otherwise(10)

P. Somasundaram et al. / Electric Power Systems Research 76 (2005) 77–85 79

and

φlim R,pik =

{∣∣∣φR,pik

∣∣∣ − φk,max, if∣∣∣φR,pi

k

∣∣∣ > φk,max

0 otherwise(11)

The values of penalty factorsk1, k2 andk3 are chosen suchthat if there is any constraint violation the fitness functionvalue corresponding to that parent will be ineffective.

3.1.2. MutationAn offspring populationIoi is generated from each parent

Ipi as

Ioi = [PoiG1, P

oiG2, . . . , P

oiGj, . . . , P

oiGGC];

oi = pi + Np; pi = 1, 2, . . . , Np (12)

wherePoiGj are generated as:

PoiGj = P

piGj + N(0, σ2

j ); j = 1, 2, . . . , GC (13)

subject to

PoiGj =

{PGj,min if Poi

Gj < PGj,min,

PGj,max if PoiGj > PGj,max,

(14)

N(0,σ2j ) represents a normal random variable with mean zero

and standard deviationσj. The standard deviationσj decidest g toj

σ

w nc nv

off allig

ringi ionso

3-

is l.T est

W

W

where the competitorfr is selected at random from amongthe 2Np trial solutions,ui is uniform random number rangingover [0, 1] andq is the arbitrarily chosen competition numberwhich is normally taken asNp.

The 2Np trial solutions are ranked in descending orderof their weights and the firstNp trial solutions are taken asthe next parent population. The mutation, competition andselection are repeated until the specified number of genera-tion/iteration is reached.

3.2. Linear programming

The formulation of LP problem is described as follows:Linearizing Eqs.(1)–(7) around the base-case operating

state (X, U, C), the incremental cost function is obtained as

�FT =∑

j ∈ αGC

(2ajPoGj + bj)�PGj+(2asP

oGs + bs)�PGs

(17)

The slack bus real power variation�PGs is a dependentvariable and can be expressed in terms of�PGj’s. Assumingthat the change in transmission loss due to correction�PG’sis negligible,�PGs is expressed as

�PGs = −(�PG1 + �PG2 + �PG3 + . . . + �PGGC) (18)

So

�

w

γ

T

[

w lu-a werfl sa t andt s,t

[

w s‘ tantm -c

[

fort as

[

he width of the normal distribution curve correspondinth generating unit.σj is computed as:

j = β × fpi

fmax(PGj,max − PGj,min) (15)

hereβ is a scaling factor,fpi the value of the fitness functioorresponding toIpi andfmax is the maximum fitness functioalue among the parent population.

The value ofσj is chosen according to the relative valuepi so that the width of the normal distribution curve is smf fpi is low and vice-versa. Hence, iffpi is low the offspringenerated is nearer to the parent and vice-versa.

The fitness function value corresponding to each offsps computed by running a load flow with the unit generatf each offspring and using Eq.(8).

.1.3. Competition and selectionThe 2Np trial solutions, namelyNp solutions correspond

ng to parent populationIpi, pi = 1, 2, . . ., Np and their off-pringIoi, oi = Np + 1, Np + 2, . . ., 2Np compete for survivahe weight (Wi) for each individual in the population decid

he survival. The weightWi is computed as

i =q∑

t=1

wt ; i = 1, 2, . . . , 2Np (16)

t =1 if ui >

fi

fr + fi

0 otherwise

ubstituting for�PGs from Eq.(18)in Eq.(17)the linearizedbjective function becomes

FT =∑

j ∈ αGC

γj �PGj (19)

here

j = (2ajPoj + bj) − (2asP

oGs + bs) (20)

he linearized power flow equation is given by

FX]�X = −[FU ]�U (21)

here [FX] and [FU] are matrices of partial derivatives evated at the starting state. For simplifying the linearized poow model, the well-knownP–Q decoupling principle ipplied. Since the voltage profile is assumed constan

he control vectorU comprises of only real power injectionhe linearized power flow model(21) reduces to

H ]�δ = �P (22)

here [H] is submatrix of the Jacobian [FX]. This is termed aincremental state-control relation’. Substituting the cons

atrix [B′] [2], for the Jacobian submatrix [H], the stateontrol relation for the base-case state becomes

B′]�δ = �P (23)

In a similar way, the incremental state-control relationheRth postulated contingency state could be obtained

[B′] + [�B′R]]�δR = �P (24)


where [�B′R] is the change in the base-case constant matrix[B′] created by theRth line outage. The incremental Eqs.(23)and(24) will be used whenever the incremental state vector�δ is to be expressed in terms of�P in the development ofthe LP model.

The linearized model developed in the control space isgiven by:

determine: the control vector�PGto minimize:

�FT =∑

j ∈ αGC

γj �PGj (25)

Subject to:the control constraints

PGj,min − PoiGj ≤ �PGj ≤ PGj,max − Poi

Gj;

j = 1, 2, . . . , GC (26)

the base-case line flow operating constraints

−φk,max − φok ≤ �φk ≤ φk,max − φo

k ; k = 1, 2, . . . , NLL

(27)

the slack bus constraint

PGs,min − PoiGs ≤ −(�PG1+�PG2+�PG3 + . . . + �PGGC)

a

−w tings ion.T so sea

[

w ci-d stantm

[

I cys

[

� -l hodt asi or-r too

3.3. Computational procedure

The overall procedure of the proposed hybrid method isdescribed as follows:

Step 1: Read the system data.Step 2: Solve the SCED problem through EP. EP will termi-

nate when the suboptimal solution is obtained after afew specified number of iterations, which is normallychosen in the range 10–15.

Step 3: Use the solution from step 2 to formulate the LPmodel and solve the LP problem to obtain the finalsolution. The above steps are clearly illustrated inFig. 1.

3.4. Solution approach used

First, the ED problem with base case constraints is solvedusing the proposed hybrid method. Then taking this solutionas initial condition, the ED problem with security constraints

Fig. 1. Flow chart for the proposed method.

≤ PGs,max − PoGs (28)

nd the contingency case line flow security constraints

φk,max − φok ≤ �φR

k ≤ φk,max − φok ; R ∈ αNO (29)

hereφ◦ is the line phase angle under the given operatate and�φ is the change permitted during the correcthe incremental line phase angle�φ is expressed in termf incremental control vector�P�φ is related to bus phangle as:

�φ] = [M][�δ] (30)

here [M] is the bus incidence matrix which gives the inence of lines in the network to the buses. Using the conatrix [B′], �φ is expressed as

�φ] = [M][B′]−1[�P ] (31)

n a similar way, [�φR] for the Rth postulated contingentate can be obtained as

�φR] = [M][[ B′] + [�B′R]]−1

[�P ] (32)

Replacing the incremental line phase angle�φ in (27)andφR in (29) using Eqs.(31) and(32), we get the LP prob

em. The above LP problem is solved using Big-M meto obtain�PG’s. The solution obtained from EP is takennitial solution for the LP problem. The control vector is cected asP∗

G = PoG + �PG and a power flow is conducted

btain the optimal state.


Table 1Penalty factors

Penalty factors

k1 k2 k3

10-bus system 25 50 10066-bus system 50 250 500

is solved using the proposed hybrid method. However, theproblem has also been solved directly by taking the base caseand security constraints using the proposed hybrid method. Ithas been found that the former method is better than the laterin terms of the computational effort. Hence, it is proposed toimplement the former approach.

4. Simulation results and discussions

The proposed algorithm has been tested on 10-bus[17]and 66-bus Indian utility[18] systems to assess the perfor-mance of the algorithm. The simulations were carried out on aPentium III, 850 MHz processor. The 10-bus system consistsof 5 generators, 13 lines and a total demand of 3.4 pu. Theload data, line data and generation cost data for the system aretaken from Ref.[17]. The 66-bus Indian utility system con-sists of 4 generators, 93 lines and a total demand of 1250 MW[18]. For both systems the objective function is total fuelcost and the fuel cost curve of the units are represented byquadratic cost functions. The line flow limits are taken to bethe same in the base case and postulated contingency cases.

4.1. ED with base case constraints: 10-bus and 66-busIndian utility systems

4.1.1. EP based algorithmboth

t naltyfT ialp -b 3t linefl itsl for

TL sec

P n

1

Table 3Lines having limit violation in the initial population with base caseconstraints—66-bus system

Parent Lines having limit violation

1 20, 21 22, 23, 24, 252 323 324 20, 21 22, 23, 24, 25, 335 32, 33, 346 33, 567 328 209 –

10 32

66-bus system. The fitness function convergence characteris-tics of 10-bus and 66-bus systems for four trial runs (i.e. theprogram is executed four times) are given inFigs. 2 and 3,respectively. It is seen fromFigs. 2 and 3that the fitnessfunction converges smoothly to the optimum value withoutany abrupt oscillations. The convergence is reached after 50iterations for 10-bus system and 200 iterations for 66-bussystem. Even though limit violations are corrected the num-ber of iterations are prohibitively large. It can also be notedthat the convergence after 10th iteration is very slow. Theoptimal generation schedule and fuel cost obtained from EPfor 10-bus and 66-bus systems are given in third column ofTables 4 and 5, respectively.

F casec

Fig. 3. Convergence characteristics of fitness function for ED with base caseconstraints using EP algorithm—66-bus system.

The base-case ED problem is solved using EP forest systems. A population size of 10 is chosen. The peactors chosen for both the systems are given inTable 1.he line flow limit violations for each parent of the initopulation are given inTables 2 and 3for 10-bus and 66us system, respectively. It can be seen fromTables 2 andhat there are several lines having limit violations. Theow limit violations are found in the range of 0–227% ofine limits for 10-bus system and 0–98% of its line limits

able 2ines having limit violation in the initial population with base caonstraints—10-bus system

arent Lines having limit violatio

1 3, 5, 82 3, 6, 8, 10, 11, 12, 133 3, 5, 8, 124 3, 4, 5, 8, 95 5, 86 3, 5, 87 3, 6, 8, 9, 10, 11, 12, 138 3, 6, 10, 11, 12, 139 3, 5, 80 2, 4, 5, 8, 9

ig. 2. Convergence characteristics of fitness function for ED with baseonstraints using EP algorithm—10-bus system.


Table 4Optimum generation schedule for ED with base case constraints—10-bus system

Output (pu) GA EP Proposed hybrid method

EP—after 10th iteration LP solution

PG1 1.068 1.106 0.964 1.0594PG2 0.576 0.587 0.566 0.580PG3 0.516 0.518 0.512 0.516PG4 0.417 0.372 0.612 0.407PG5 0.852 0.844 0.774 0.866Loss (pu) 0.028 0.028 0.028 0.028Fuel cost (pu) 140.54 140.54 140.65 140.54Computation time (s) 0.300 0.185 0.049 0.008

Total 0.0571

Table 5Optimum generation schedule for ED with base case constraints—66-bus system

Output (MW) GA EP Proposed hybrid method


PG1 443.25 443.26 433.26 442.83PG2 443.08 443.07 445.45 443.33PG3 281.41 281.43 303.61 281.94PG4 120.00 120.00 105.61 119.66Loss (MW) 37.74 37.74 37.94 37.75Fuel cost (Rs/h) 716547.80 716547.62 721744.19 716547.81Computation time (s) 5.50 3.62 0.27 0.06

Total 0.33

4.1.2. Hybrid methodThe suboptimal solution obtained from EP at the end of

10th iteration for the 10-bus and 66-bus systems are given infourth column ofTables 4 and 5, respectively. The suboptimalsolution of 10-bus system has one line with limit violationand 66-bus system has two lines with limit violation. The lineflow limit violations are found to be 11% of its line limit for10-bus system and 0–3% of its line limit for 66-bus system.The solution obtained from EP after 10th iteration is takenand the LP problem is formed as explained in Section3.2and the solution is obtained using Big-M method. There isno line limit violation in the optimal schedule obtained fromthe LP solution. The optimal generation schedule and fuel

cost obtained from the proposed hybrid method for 10-busand 66-bus systems are given in fifth column ofTables 4 and 5,respectively. The computation time taken for the 10-bus and66-bus systems by the proposed method are only 30 and 9%of the time taken by EP method and only 19 and 6% of thetime taken by GA.

4.2. ED with security constraints: 10-bus and 66-busIndian utility systems

4.2.1. EP based algorithmFour single line outages of lines numbered 4, 11, 12 and

13 are considered for 10-bus system. Five single line outages

Table 6Lines having limit violation in the initial population with security constraints—10-bus system

Parent Lines having limit violation

Base case Contingency case

Outage of line 4 Outage of line 11 Outage of line 12 Outage of line 13

1 – 8, 12 12 3, 5, 7, 10, 13 8, 10, 122 12 12 5, 12 3, 5, 7, 13 123 5, 8, 12 8, 12 2, 5, 8, 12 3, 4, 5, 7, 13 5, 8, 124 6, 7 7, 10, 13 3, 6, 7, 10 7, 10, 13 6, 105 8, 12, 13 8, 12, 13 3, 8, 10, 12, 13 3, 5, 7, 10, 13 8, 10, 126 5, 12 8, 9, 12 2, 5, 9, 12 2, 3, 4, 5, 6, 7 3, 5, 12

–5, 123, 10

1 5, 8,

7 – 78 5, 12 8, 129 10, 13 3, 7, 10, 130 5, 8, 12 8, 12

– 6, 103, 4, 5, 7, 13 5, 8, 123, 7, 10, 13 6, 10

12 3, 4, 5, 7, 10, 13 5, 8, 12


Tabl

e7

Line

sha

ving

limit

viol

atio

nin

the

initi

alpo

pula

tion

with

secu

rity

cons

trai

nts—

66-b

ussy

stem

Par

ent

Line

sha

ving

limit

viol

atio

n

Bas

eca

seC

ontin

genc

yca

se

Out

age

oflin

e21

Out

age

oflin

e22

Out

age

oflin

e23

Out

age

oflin

e25

Out

age

oflin

e56

1–

22,3

420

,21,

3424

,25,

3423

20,3

3,34

220

,21,

22,2

3,24

,25,

33,3

4,56

22,2

3,24

,25,

33,3

4,56

20,2

1,23

,24,

25,3

3,34

,56

20,2

1,22

,24,

25,3

3,34

,56

20,2

1,22

,23,

33,3

4,56

20,2

1,22

,23,

24,2

5,33

,34

3–

2220

,21

24,2

523

33,3

44

20,2

2,24

,25

2220

,21

24,2

523

20,2

2,24

,25,

33,3

45

20,2

1,22

,23,

24,2

5,34

22,2

3,24

,25,

3420

,23,

24,2

5,34

24,2

5,34

20,2

3,34

20,2

1,22

,23,

24,2

5,33

,34

632

,33,

34,5

622

,32,

33,3

4,56

20,2

1,32

,33,

34,5

624

,25,

32,3

3,34

,56

23,3

2,33

,34,

5633

,34

7–

2220

,21

24,2

523

33,5

48

22,2

3,24

,25,

3422

,23,

24,2

5,34

20,2

1,23

,24,

25,3

420

,24,

25,3

320

,21,

22,2

3,34

20,2

1,22

,23,

24,2

5,34

920

,21,

22,2

3,24

,25,

33,3

4,56

22,2

4,25

,33,

34,5

620

,21,

24,2

5,33

,34,

5624

,25,

33,3

4,56

20,2

3,33

,34,

5620

,21,

22,2

3,24

,25,

33,3

410

20,2

1,22

,23,

24,2

522

,23,

24,2

520

,21,

23,2

4,25

20,2

2,24

,25

20,2

1,22

,23

20,2

1,22

,23,

24,2

5,33

,34

Fig. 4. Convergence characteristics of fitness function for ED with securityconstraints using EP algorithm—10-bus system.

of lines numbered 21, 22, 23, 25 and 56 are considered for 66-bus system. The line flow limit violations in the base case andcontingency cases with each parent of the initial populationfor 10-bus and 66-bus systems are given inTables 6 and 7,respectively. It can be observed fromTables 6 and 7that thereare several lines having limit violation. The line flow limitviolations are found in the range of 0–70% of its line limitsfor 10-bus system and 0–163.58% of its line limits for 66-bus system. The fitness function convergence characteristicsof 10-bus and 66-bus systems for four trial runs are given inFigs. 4 and 5, respectively. It is seen fromFigs. 4 and 5thatthe fitness function converges smoothly to the optimum valuewithout any abrupt oscillations. The convergence is reachedafter 50 iterations for 10-bus system and 200 iterations for66-bus system, which is prohibitively large. It can also benoted that the convergence after 10th iteration is very slow.The optimal generation schedule and fuel cost obtained fromEP for 10-bus and 66-bus systems are given in third columnof Tables 8 and 9, respectively.

4.2.2. Hybrid methodThe suboptimal solutions for 10-bus and 66-bus systems

obtained from EP at the end of 10th iteration are given infourth column ofTables 8 and 9, respectively. The suboptimalsolutions of 10-bus and 66-bus systems have two lines withl er 0%o ed

F urityc

imit violations. The line flow limit violations are found in thange of 0–14% of its line limits for 10-bus system and 0–1f its line limits for 66-bus system. The solution obtain

ig. 5. Convergence characteristics of fitness function for ED with seconstraints using EP algorithm—66-bus system.


Table 8Optimum generation schedule for ED with security constraints—10-bus system

Output (pu) GA EP Proposed hybrid method


PG1 1.096 1.103 1.080 1.095PG2 0.647 0.654 0.635 0.648PG3 0.602 0.601 0.605 0.602PG4 0.383 0.379 0.348 0.386PG5 0.697 0.687 0.758 0.693Loss (pu) 0.025 0.025 0.026 0.025Fuel cost ($/h) 140.63 140.638 140.60 140.632Computation time (s) 2.80 1.59 0.22 0.05

Total 0.27

Table 9Optimum generation schedule for ED with security constraints—66-bus system

Output (MW) GA EP Proposed hybrid method


PG1 816.801 816.801 793.187 816.721PG2 176.655 176.656 193.698 177.204PG3 179.685 179.684 189.453 180.210PG4 120.000 119.999 115.504 118.988Loss (MW) 43.139 43.142 41.844 43.126Fuel cost (Rs/h) 1225415.85 1225415.87 1175618.75 1225415.75Computation time (s) 340.0 235.9 6.86 0.66

Total 7.52

from EP after 10th iteration is taken and LP problem is formedas explained in Section3.2and the solution is obtained usingBig-M method. There is no line limit violation in the optimalschedule obtained at the end of single LP move. The optimalgeneration schedule and fuel cost obtained from the proposedhybrid method for 10-bus and 66-bus systems are given infifth column ofTables 8 and 9, respectively. The computationtime taken for the 10-bus and 66-bus systems by the proposedmethod are only 17 and 3.2% of the time taken by EP methodand only 10 and 2.2% of the time taken by GA.

5. Conclusions

This paper presents a new hybrid method for solving theSCED problem using EP combined with LP. In the proposedalgorithm probabilistic EP and the deterministic LP tech-niques are combined to solve the SCED problem. The hybridmethod eliminates the prohibitively large computation timeby exploiting the initial fast convergence rate of EP algorithmand accurate LP model resulting from smaller variation ofcontrol variables. The results obtained with the sample sys-tems reveals that the proposed method is fast and reliable andsuitable for practical systems.

A

aa

fpi fitness function of theith parent individualfoi fitness function of theith offspring individualFT total fuel cost�FT incremental change in fuel costGC the total number of buses having controllable real

power generationsNLL the total number of limiting linesNO number of single line outagesNp population sizePGj real power output ofjth generatorPGs slack bus real power generationPGj,max maximum generation limit ofjth generatorPGj,min minimum generation limit ofjth generator�PGj incremental change in generation ofjth unit

Greek lettersαGC set containing the numbers of buses having control-

lable real power generationsαNLL set containing the numbers of limiting linesαNO set containing the numbers of single-line outagesβ scaling factorδ bus phase angle�δ incremental change in bus phase angleΦk line phase angle ofkth lineΦ

�

σ

ppendix A. Nomenclature

j, bj, cj cost coefficients of thejth generators, bs, cs cost coefficients of the slack generator

k,max maximum line phase angle limit ofkth lineΦk incremental change in line phase angle ofkth

linestandard deviation


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hybrid algorithm based on ep and lp for security constrained economic dispatch problem

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