1. Introduction

Power networks are very complex systems formed by

generation units, transmission and distribution networks.

The main goal of all power utilities is to supply highly

reliable power to the customer at lowest cost. At the same

time, the boundaries and restrictions of the generating units

should be also taken into account. This is known as

“Economic Dispatch(ED)” problem [1]. Economic

dispatch is useful to determine the best combination

between the interconnected power plants, and the system

load (demand) to minimize the fuel prices satisfying

equality and inequality restrictions. Several techniques

such as gradient search, lambda iteration, base point

method, dynamic programming etc. are available for

solving problems related to economic dispatch. The last

one is widely used but there are problems regarding

dimensionality. There exists significant non-linearity and

lack of smoothness, due to multiple fuels and ramp rates,

in the input-output characteristics of practical power plants.

It is problematic to solve through both ordinary and

classical ways. So, the wide variety of heuristic methods

such as Bacterial Foraging Algorithm (BFA), Genetic

Algorithm (GA) and Particle Swarm Optimization (PSO)

are general methods to solve these problems.

In past few years, hybrid techniques are observed to be

more proficient to solve non-convex problems. They

Optimization of Non-Convex Economic Dispatch Problem Using

Hybrid Approach Based on Bacterial Foraging and Genetic Algorithm

ABDUL SHAKOOR

Department of Electrical Engineering, University of Engineering and Technology, Taxila, PAKISTAN.

Abstract: Fulfillment of consumer demand is a foremost challenge for all electrical power utilities. An electrical

power system consists of several generating units and each of these units owns a distinct operating set of

operating parameters. A fundamental challenges lies in a fact that there may not be a correlation between

operating costs of these machines and their generated output. Major reason for lack of the correlation includes

the ramp-rate limits, transmission losses and manipulation due to valve-points in the generation units cost

function, and thus, it becomes a non-linear optimization problem. In view of this fact, it creates a rigorous need

to devise a robust solution to cater for such a non-linear optimization scenario. In this paper, bacterial foraging

and genetic algorithms based hybrid technique is used to effectively tackle the economic dispatch problem. The

presented technique incorporates two modifications in the original bacterial foraging algorithm including

differential evolution inspired bacterial movement and genetic algorithm based bacterial reproduction. Where

inspired bacterial movement involves modification in the directional movement for each bacterium in such a

manner that every bacterium tries to improve its direction and position based on differential evolution. The

proposed technique is applied to non-convex dynamic economic dispatch (DED) problem in order to obtain

optimal solution within feasible functional limits while satisfying the load demand at the same time. The

obtained results demonstrate that the proposed hybrid approach outperform the other techniques in term of

optimal solution and significant reduction of computational time in the given scenarios.

Keywords: Economic Dispatch (ED), Dynamic Economic Dispatch (DED), Bacterial Foraging Algorithm

(BFA), Genetic Algorithm (GA), Differential Evolution (DE).

Received: July 7, 2022. Revised: February 28, 2023. Accepted: March 14, 2023. Published: March 24, 2023.

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diminish the search space to locate optimal solution in a

satisfactory computational time. Furthermore, they can

facilitate number of constraints to solve both small and

large scale problems with better quality of solution.

2. Literature Review

The bacterial foraging optimization algorithm was first

proposed by K.M Passino to solve un-constrained

optimization problems [2]. Now a days, the Swarm

optimization techniques are drawn the attention of

researchers considering the fact that they've information

sharing and conveying mechanisms to remedy real world

optimization problems .Amongst swarming based

techniques, bacterial foraging is very promising with set of

advantages related to regional minima, randomness,

direction of movement, attraction/repelling, swarming and

so on. The stand-alone bacterial foraging (BF) experiences

poor convergence attributes for high dimensional issues.

To handle the non-linear and multi-dimensional ED issue,

this disadvantage ought to be treated with the reconciliation

of other EA's. If Economic dispatch is taken in to account,

there are few research papers published on BFA to solve

the ED problem since 2008. Ahmed .Y. Saber, et al. [3], in

order to solve economic dispatch problem, an adaptive

methodology is introduced for the sake of improvement of

searching skill ability of BFA, PSO has been offered.

Ultimately, a standard test system from IEEE is used to

demonstrate the ability of the proposed strategy and the

effects are contrasted with different methods from recent

literature. K. Vaisakh, et al. [4] presented a hybrid

approach consisting on differential evolution, particle

swarm optimization (DE-PSO) and BFA to treat DED

problem of multiple generating units involving valve-point

effects. The proposed method has been contrasted with

others and seemed expert in two test cases comprising of

five and ten units test models. I.A Farhat et al. [5]

presented an improved bacterial foraging algorithm (IBFA)

to remedy the ED problem on the grounds of the valve-

point effects and transmission losses. To beat the poor

convergence and dimensionality dilemma of BFA, the

basic chemo-tactic step is tuned to have a dynamic

behavior for enhancement of exploration and exploitation

capabilities. Based upon the solution development, BFA

can be more reliant and adaptive. The proposed algorithm

is verified utilizing various test techniques. P.K Hota, et al.

[6] proposed a modified bacterial foraging optimization

algorithm (MBFOA) involving fuzzy logic methodology to

obtain the best promising solution for economic and

emission dispatch and validated on Taiwan power system

of forty generating units. B.K. Panigrahi et al. [7] presented

a bacterial foraging meta-heuristic algorithm for multi-

purpose optimization. In this approach, the most recent

bacterial locations are received by means of chemotaxis.

Furthermore, Pareto optimal front (POF) is chosen through

fuzzy logic sense based sorting. In order to verify the

proficiency of proposed algorithm IEEE 30-bus 6-

generator standard test system is considered and the

outcome are contrasted with the other reported outcome.

Rahmat-Allah Hooshmand, et al. [8] proposed a hybrid

strategy based on Bacterial Foraging Algorithm and

Nelder-Mead technique (BF-NM). Usefulness of the

proposed technique is presented in comparison with

several EA techniques. Total cost obtained as a result of the

proposed technique proved the benefit of the method.

Nicole Pandit, et al. [9] introduced a improved bacterial

foraging algorithm (IBFA) where crossover operation and

parameter automation system is used to improve

computational efficiency. The performance of IBFA is

compared with recently released methods and seems to be

better. Ahmed Yousuf Saber, et al. [10] presented a

modified particle swarm optimization (MPSO) involving

advantages of bacterial foraging (BF) and PSO. The

modified PSO has better exploration and exploitation

capabilities to restrict regional minima. Finally, the results

of present approached from literature are used to exhibit

the effectiveness of the proposed technique. K. Vaisakh, et

al. [11] offered BPSO-DE by the integration of BF with

PSO and DE. The result gives the best foraging

methodology search based on bacteria, that is then updated

at every step of PSO. The solution produced as a result of

BF and PSO is then regulated by the DE operator. Rasoul

Azizipanah-Abarghooee, et al. [12] introduced a novel

bacterial foraging (BF) approach, which involved

initialization using opposition-based and a novel mutation

operator. This operator is utilized in classical bacterial

foraging (BF) to control the pre-mature convergence. In

addition, long step size or short step size may be used for

timely readiness of the bacteria involved in the chemo

tactic step. Zhi Lu et al. [13] proposed a modified Bacterial

foraging technique .In this procedure, a Lamarckian

constraint coping with strategy established procedure is

upgraded for upgrading of bacterial colony. Finally, IEEE

30 Bus system was incorporated for the testing of the

proposed system. Results suggested that this proposed

technique came up with the advantage of catering for the

multi-purpose and non-convex features related to thermal

generators taking both ED and EED issues. G. Wu et al.

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[14] presented a bacterial foraging optimization algorithm

(BFOA).

It performed a study of ED related to a hydropower system.

Based upon the results that showed increased efficiency, it

was concluded that the proposed system was better. Ehab

E. Elattar et al [15]. Offered hybrid bacterial foraging and

genetic algorithm. Proposed method is proven on 5, 10, 30

generation models for non-convex ED. The outcome are

compared with the outcome acquired by means of different

approaches.

The focus of this paper is to implement bacterial foraging

and genetic algorithms based hybrid approach for non-

convex economic dispatch problem considering the ramp

rate limits and valve point loading effects. For the sake of

validation of the research work, the results of the research

are compared with many other techniques.

The paper organization as follows: Brief introduction and

literature review in section-1. Mathematical model for non-

convex economic dispatch problem is formulated in

Section-2. Proposed hybrid algorithm is presented in

section-3. Results and comparison with different

techniques are in section-4. Section-5 concludes the whole

work.

3. Non-convex Economic

Dispatch Problem Formulation

In order to determine the optimal load of all the linked

generating units, the economic dispatch problem is

designed. The goal is to limit the cost function subject to

the system constraint. In order to formulate the ED

problem, NG is defined as the number of committed units,

PD as the total load demand, PGi as active power generation

for unit i, Fi(PGi) and  as Operational and total cost for

unit i over the dispatch period.

 

1

:

G

N

r i Gi

i

Minimize F F P





(1)

Subject to:

Load balance equation

10

G

N

Gi D

iPP





(2)

Generating unit capacity limits

min max , 1 , 2, ,

Gi Gi Gi G

P P P i N   

(3)









 are upper and lower operational limits for

generator i.

Transmission Losses For energy systems: Mostly electrical

energy is transmitted over long transmission lines, the

values of the network losses must be monitored, as these

affect the output of the generator. It is estimated that for

practical systems the losses can be 5% to 10% of the total

energy generation [16]. So the function is expressed in

equation (1) must be minimized while satisfying the

equilibrium equation (4) of the energy.

10

G

N

Gi D L

iP P P

  



(4)

PL is the actual power loss of the system which is calculated

by equation (5), known as George's formula [17].

11

GG

NN

L Gi ij Gj

iJ

P P B P





(5)

In order to obtain the most accurate losses, a constant and

a linear and must be added to the equation (5) known as

Kron loss formula [17] in equation (6).

0 00

1 1 1

G G G

N N N

L Gi ij Gj i Gi

i J i

P P B P B P B

  

  

  

(6)

The B-coefficients vary with the condition of the system

operation. Although it is considered that they are constant

parameters.

Valve Point Loading Effects: The actual input-output

characteristics are highly non-convex due to opening and

closing of fuel valve. In order to represent the valve points

are included in the fuel cost function as follows:

 

2

min

**

sin

i Gi i i Gi i Gi

i i Gi Gi

F P a b P c P

e f P P

   



(7)

Where  are fuel cost coefficient of generator

i.  are valve point loading effects,



 



 are upper

and lower limits for generator i.

Ramp Rate Limits (RRL):

 

1

it i

it

P P UR





(8)

 

1it i

it

P P DR



(9)

Where and  are the ramp-up and ramp-down rate

for the  generator. So, the limits of the capacity of the

unit are modified as:

 

min

1

max ,

Gi i Gi

it

P P DR P



(10)

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 

max

1

min ,

Gi Gi i

it

P P P UR





(11)

4. Essential Background of

Hybrid Techniques

4.1 Bacterial Foraging

Bacterial foraging algorithm (BFA) which is inspired by

the foraging behavior of Escherichia coli (E. coli) was first

proposed by K.M. Passino.

In the nature, the living organisms try to maximize ingested

energy (E) due the measure of time (T) they spend seeking

wild food assets. The foraging species also perform an

optimization task to maximize the function E/T. This is

important for the survival of the species. Foraging task

involves the search of a food source, make the decision to

enter and find wild food and determinate which is the best

moment for seeking a better and a new food source. These

tasks vary between one specie to another and affected by

different internal aspects of the organisms such as food

type, metabolic qualities. They are affected by external

aspects like the weather and geography. Such tasks are also

carried out at the same time with other tasks like seeking

of safe shelter or territory. As indicated by ideal foraging

theory, foraging can be planned as an optimization

problem.

Commonly the animals that live in groups perform

cooperative tasks. In this activity, the individuals share

information with other group members. The information

sharing can be possible through sound signals, chemical

signals or body language. Through social foraging an

individual may obtain higher rates of energy gain. Other

advantages of grouping individuals are the ease of driving

and the facility of protecting each other from predators.

Some examples of social foraging are: Wolves, fishes, and

ants. The food search strategy is the biggest part of the

foraging and many foragers follow the same strategy used

by predators. A bacterial foraging operates through

locomotion, chemotexis and evolution process.

Locomotion is the nonstop rotation alternates between two

modes: swimming and tumbling. During swimming, the

counterclockwise rotation of the flagellum pushes the body

forward. When it propels clockwise, the bacteria tumbles

and moves in random direction. The displacement is small

during tumbling.

Chemotexis is the movement of a bacterium in a direction

corresponding to a gradient concentration of a particular

substance. E. coli swim to head to nourished places. They

have been observed to be attracted to serine or aspartate.

On the other hand, they tumble to avoid unpleasant areas

usually containing metal ions Nickel, Cobalt, amino acids

and organic acids.The foraging behavior of E.coli is

generally observed in chemotaxis (swimming or tumbling)

in relation to the chemicals in the medium.In general, the

concentration of desirable chemicals is directly

proportional to the rate of swimming and indirectly

proportional to tumbling [2].In an impartial domain where

neither attractive nor dangerous substances exist, the

bacteria movement alternates between swimming and

tumbling. In homogenous environments, where both

desirable and toxic substances exist, the bacteria will swim

more and will tumble less. Note that the availability of food

source will not inhibit the organisms to seek for food hence

they will continue to look for nourishment. They will swim

as long as the gradient concentration is favorable. If they

come in contact with adverse substances, they will tumble

but will still swim to climb back to the positive

concentration gradient.

Evolution process in E. coli is occurred at a mutation rate

of approximately 10-7 per gene per generation. The genes

change through the process of conjugation where DNA

attributed to fitness and fertility are passed on to the next

generation. In other words, characteristics favorable to its

survival are inherited by succeeding generations. When the

environment is adverse or had sudden or slow changes,

elimination, dispersion or both occurs. The population can

be eliminated partially or totally. Dispersal drives the

population to another part of the environment which can

either be beneficial or disadvantageous. Either event has a

two-sided impact on the chemotactic process.

4.2 Genetic Algorithm (GA)

Genetic algorithms are search strategies that utilize

processes found in regular natural development. At every

generation, another population is made by selecting an

individual as per their physical fitness in the problem

domain. Selection, crossover and mutation are the three

fundamental operations employed in genetic algorithms.

The selected solutions are modified through these

operations and the most appropriate issue is selected to be

passed on to succeeding generations. Genetic algorithms

simultaneously consider multiple points on the search

distance. They have been found to provide a rapid

convergence to a near optimum solution in many cases of

problems.

Differential development (DE) is also belong to the class

of genetic algorithms (GAs) which utilize bio-inspired

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operations of selection, crossover, and mutation on a

population to limit an objective function through the span

of progressive generations. An initial mutant parameter

vector is made through the selection of three random

individuals from the population.DE utilizes real values

rather than bit-string encoding, and arithmetic operations

rather than logical operations in mutation compare to

exemplary GAs. Let NP denote the number of individuals

in the population. In order to generate the initial

population, NP guess for the optimal values of the

parameter vector by either choosing values between upper

and lower limits or user defined. Every generation includes

making of another population from the present population

individuals {x_i | I = 1, . . . ,NP}, where i is population

index.

5. Hybrid Bacterial Foraging,

Genetic Algorithm, and Differential

Evaluation (HBFA-DE-GA)

The Inspired movement for bacterium is accomplished

using differential mutation as follow; an initial mutant

parameter vector  is made by selecting the three

individuals from the population, current member ( ), and

two random members and  from population. Then 

is generated as.

 

12

.

i i i i

v x F x x  

(12)

0 1Where F

Genetically inspired reproduction instead of simple

bacterial reproduction is introduced as follows; bacteria are

kept sorted according to their fitness and split into fittest

50% and worst 50% then fittest and worst are recombined

using heuristic cross over operator [18] as:

 

1 1 51 *Offspring P P P



  

(13)

Where β is random value (0-1) .P1 is fittest parent and P51

is worst parent.P2 and P52 is next pair for recombination

vice versa. Offspring’s than mutated using dynamic

mutation operator [19] as.

 

, , 0

, , 1

U

j j j

jL

j j j

x k x x

x

x k x x



  



  





(14)

Where k is generation number. L and U are lower and

upper limits for variable, is random number (0, 1), G is

the highest number of generations, b is degree of

dependency on iteration number.

*( , ) (1 / )* b

jj

k x x k G



  

(15)

5.1 Economic Dispatch using Proposed

BFA-DE-GA Algorithm

1. Initialization of parameters:

 Number of bacteria (Nb)

 Number of chemotexic steps(Nch)

 Number of elimination dispersal steps(Ned)

 Number of reproduction steps(Nre)

 Probability of mutation(Pm)

 Probability of crossover (Pc)

 Scaling factor (SF)

 Genetic algorithm iterations(GA iter)

 Differential evolution iterations(DE iter)

2. Initialization of system parameters:

 Population matrix (X)

 Machines data matrix (H)

 Load demand matrix(Ld)

Loop (iter: 1→T)

3. Elimination/ dispersal

Loop (l: 1→L)

4. Reproduction

Loop (k: 1→K)

5. Chemotaxis

Loop (j: 1→J)

6. Bacterium population

Loop (i: 1→I)

 Calculate the initial fitness of the  bacterium

using fitness function using Eq. (16).

1

FITB FITB Penalty Function



(16)

 Save as LAST_FITB

4. Differential Evolution inspired movement.

Loop (DE iter: 1→Maximum DE iter)

 Randomly select two bacteria from population

 Apply differential evolution (DE) mutation using

Eq. (12).

 Calculate the fitness of the resultant bacterium

using Eq. (16).

 If resultant bacterium is better than the 

bacterium then swim in the same direction.

 Calculate the fitness of bacterium at new position

using Eq. (16).

 Save as FITB.

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 If FITB <LAST_FITB, move bacterium into same

direction.

End of Loop (DE iter)

End of Loop (j)

5. Genetic Reproduction:

Loop (GA iter: 1→Maximum GA iter)

 Sort bacteria in ascending order according to their

fitness.

 Split bacteria in fittest 50% and worst 50%.

 Select one bacterium from fittest population list

and one from worst population list.

 Apply Genetic cross-over and mutation between

them using Eq. (13, 14).

 Calculate the fitness of the resultant bacterium

using Eq. (16).

 If resultant bacterium is better than fittest

member, replace worst with resultant.

 Else if, resultant bacterium is not better than

fittest member, replace worst with fittest.

End of Loop (GA iter)

End of Loop (k)

6. Elimination-Dispersal:

 Eliminate bacteria according to Ped.

 Randomly disperse the bacteria in optimization

domain to keep the bacteria size constant.

End of Loop (l)

End of Loop (iter)

Table 1: Parameters for hybrid BFA-DE-GA

The graphical illustration of Hybrid BFA-DE-GA is given

in Figure 1.

Figure 1: Hybrid BFA-DE-GA flow chart

6. Experimental Setup and Case Studies

In this research work hybrid BFA-DE-GA algorithm is

implemented using Visual studio C++ and executed on an

Intel ® Core ™ i5 CPU 2.50 GHz, 4GB RAM,PC. In order

to check the consistency of algorithm 50 independent runs

are conducted with random initial solutions for each run.

Results are contrasted with different strategies reported in

literature. The parameters used for various test cases have

been shown in Table 1.

6.1 Non-convex Systems

Proposed algorithm is tested on standard IEEE test system

compromise of 5, 10 and 30 generation units,the ramp rate

Start

Initialize variables and

generate initial

population

If

FITB>LAST_FITB

l=l+1

K=k+1

j=j+1

i=i+1

D

C

B

A

B

C

D

Elimination dispersal

loop counter

l=1

Reproduction loop

counter

k=1

Chemotaxic loop counter

J=1

For each bacterium

i=1

Compute fitness value of

current bacterium

Save as LAST_FITB

Move the bacterium in

the direction whose bias

is determined by DE

Calculate fitness of

bacterium

Save as FITB

Save the bacterium in the

list of healthy bacterium

If

i=Nb

If

J<Nc

Reproduction biased on

Genetic cross-over and

mutation

If

K<Nre

Elimination and dispersal

If

L<Ned

End

E

No

Yes

Parameter

Value

Number of bacteria

120

Number of

reproduction steps

4

Elimination-dispersal

steps

2

Generation of GA

100

Generation of DE

300

Scaling factor for DE

0.8

Crossover probability

of DE

0.8

Elimination/dispersal

probability

0.25

Adaptive parameters

5-Units

10-Units

30-Units

Number of

chemotexic steps

50

100

200

Crossover probability

0.8

0.9

Mutation probability

0.1

0.15

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limits and valve point loading effects are considered for all

test cases.

6.1.1 Case-1: 5-Generation Units Test System

This case provides the solution for a 5 generation units test

system by taking into account the transmission losses as

well. Unit data and load demand pattern for case-1 is

adapted from [18] and transmission loss coefficients from

[19].

Table 2: Cost & computational time comparison for case-

1.

Table 2 shows the cost & computational time for 5-

generation units test system. The proposed technique has

achieved reduction in cost of $3243.65/day as compared

with CGNM, $820.47/day as compared with GA,

$343.48/day as compared to AIS, and $211.29/day as

compared to PSO. However if compared to ABC

$3.88/day. Proposed technique

Figure 2: Cost Comparison for 5-units Test Case.

Figure 2 gives a graphical comparison of the optimal costs,

the cost of proposed hybrid algorithm are far less than the

existing AI techniques. The best scheduling for 5-units test

case is given in Table 3. Figure-3 provides a distribution of

the cost function for 50 independent runs for 5-unit test

system.

Table 3: The best generation schedule of 5-unit system

using Hybrid BF-DE-GA approach.

Figure-3 Distribution of the cost function for 50

independent runs for 5-unit test system.

Hr.

P1

P2

P3

P4

P5

PL

1

10.02

20.04

30.02

124.50

229.41

3.99

2

34.97

20.02

30.00

124.91

229.51

4.41

3

64.96

30.76

30.00

124.94

229.52

5.19

4

75.00

26.75

30.22

174.94

229.55

6.45

5

75.00

20.51

30.33

209.83

229.55

7.21

6

75.00

31.41

70.32

209.83

229.57

8.13

7

64.75

20.01

110.30

209.81

229.51

8.38

8

74.99

36.00

112.75

209.82

229.53

9.10

9

75.00

66.00

119.68

209.84

229.55

10.07

10

66.61

95.98

112.66

209.78

229.52

10.55

11

75.00

103.97

112.74

209.82

229.52

11.04

12

74.97

124.68

112.68

209.82

229.58

11.72

13

64.03

98.54

112.66

209.82

229.52

10.56

14

49.62

98.55

112.66

209.83

229.52

10.17

15

19.62

92.26

112.63

209.21

229.50

9.21

16

10.01

75.79

112.67

159.21

229.52

7.20

17

10.02

87.77

112.52

124.86

229.53

6.68

18

40.00

108.67

112.68

125.01

229.52

7.89

19

70.00

125.00

113.25

125.32

229.55

9.13

20

74.99

122.08

112.66

175.31

229.50

10.55

21

45.00

92.94

112.61

209.81

229.52

9.88

22

15.00

96.01

112.54

159.81

229.48

7.84

23

10.00

96.34

72.56

124.77

229.49

6.16

24

10.05

70.86

32.66

124.90

229.51

4.98

Total Cost = 44041.952539($)

Method

Cost($/24h)

Time(min)

CGNM [18]

47285.6

NA

GA [20]

44862.42

3.32

AIS [21]

44385.43

4.00

PSO [20]

44253.24

3.55

ABC [20]

44045.83

3.29

HBFA-DE-GA

44041.95

0.40

42000

43000

44000

45000

46000

47000

48000

Total Cost($/24h)

Method

Cost Comparison for Case-1

CGNM

GA

AIS

PSO

ABC

HBFA-DE-GA

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6.1.2 Case Study 2: 10-Generation Units Test System

without Network Losses

The ten-unit test system with non-smooth fuel cost function

is used for DED problem. Unit data is taken from [18] and

load demand pattern for case-2 is given in Table 3.

Table 4: Cost & computational time comparison for case-

2.

Table 4 shows that cost and computational time

comparison for case-2.The cost $13303/day, $10392/day,

$8891/day, $3537/day, and $30.38/day less than EP-

SQP,DGPSO,PSO-SQP,AIS, and HIGA respectively.

Computational time for 10-units test case is smallest as

compared to all other Techniques.

Figure 4: Cost Comparison for 10-units Test Case 2.

The graphical representation for optimal cost is shown in

figure 4, evident that costs are reduced by significant

amount. Distribution of cost function for 50 independent

runs for case-2 is given in figure 5, which demonstrate that

$ 1018443/day is minimum cost and $ 1020915.20/day is

maximum cost. Best generation scheduling for case-2 at

optimal cost of $ 1018443/day is given in Table 7.detail of

power generation by each generator toward economical

operation is also given in table 7.

Table 5: Hourly load demand for test cases.

Figure 5 Distribution of the cost function for 50

independent runs for 10-unit test system.

6.1.3 Case Study 3: 30-Generation Units Test System

The data of the thirty-unit test system are obtained by

tripling the ten-unit system of Case-2, and Non convexity

of the test system is enhanced by varying the system

Method

Cost ($/24h)

Time(min)

EP-SQP [22]

1031746.00

20.51

DGPSO [23]

1028835.00

15.39

PSO-SQP [24]

1027334.00

16.37

AIS [25]

1021980.00

19.01

HIGA [26]

1018473.38

3.53

HBFA-DE-GA

1018443.00

0.79

Hours

5-Units

10-Units

30-Units

1

410

1036

3108

2

435

1110

3330

3

475

1258

3774

4

530

1406

4218

5

558

1480

4440

6

608

1628

4884

7

626

1702

5106

8

654

1776

5328

9

690

1924

5772

10

704

2072

6216

11

720

2146

6438

12

740

2220

6660

13

704

2072

6216

14

690

1924

5772

15

654

1776

5328

16

580

1554

4662

17

558

1480

4440

18

608

1628

4884

19

654

1776

5328

20

704

2072

6216

21

680

1924

5772

22

605

1628

4884

23

527

1332

3996

24

463

1184

3552

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parameters. Load demand pattern for case 3 can be found

in Table 5.

Table 6: Cost and Computational Time Comparison for

30-units Test System.

Table 6 shows that when proposed technique applied

to 30-generators test system gives very favorable results as

compared to other AI techniques. A reduction of

$43307/day is visible when proposed algorithms is

compared to IPSO, $38846.59/day as compared to CE,

$17234/day reduction when compared with ICPSO,

$8172.068/day as compared to HIGA, $7698/day and

$2972/day cost reduction as compared with EAPSO and

HGABFA respectively.

Figure 6: Cost compression for 30-units test case without

losses.

The graphical representation of 30-generation test system

is shown in figure-6, which evident that results obtained

using proposed methods are much better than other

reported methods. Distribution of cost function for 50

independent runs for case-3 is given in figure 7.

Figure 7: Distribution of the cost function for 50

independent runs for 30-unit test system.

7. Conclusion

A novel hybrid methodology is proposed in this paper,

which is based upon HBF, GA and DE .for the solution of

a non-convex DED problem is solved considering the valve

point loading effects and the ramp rate limits. The proposed

technique is tested on IEEE standard test systems in order

to verify the proficiency. Finally proposed method is

compared with the other evolutionary computational

techniques for 5, 10 and 30 units.

For 5-units test system, results are compared with SA, GA,

AIS, PSO and ABC.

For 10-units test system results are compared with EP-

SQP, DGPSO, PSO, SQP, AIS and HIGA.

For 30-units test system results are compared with IPSO,

CE, ICPSO, HIGA, EAPSO and HGABF.

The results assured that the proposed hybrid approach

outperform the other techniques in term of cost and

significant reduction of computational time in all the test

cases.

Future work can involve the efforts to solve the economic

dispatch problems with security restraints and the restricted

operation areas. Moreover, RE resources like wind and

solar plants can also be taken into consideration.

Method

Cost($/24h)

Time(min)

IPSO [27]

3090570.00

NA

CE [28]

3086109.59

NA

ICPSO [29]

3064497.00

NA

HIGA [26]

3055435.068

NA

EAPSO [30]

3054961.00

NA

HGABF [15]

3050235.00

9.35

HBFA-DE-GA

3047263.00

4.52

3020000

3030000

3040000

3050000

3060000

3070000

3080000

3090000

3100000

Total Cost($/24h)

Method

Cost Comparison for Case-3

IPSO

CE

ICPSO

HIGA

EAPSO

HGABF

HBFA-DE-GA

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Table 7: Best scheduling of 10-generation units test system

Hour

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

PT

1

150.01

135.03

193.62

60.10

122.89

122.65

129.67

47.03

20.01

55

1036

2

226.63

135.01

191.41

60.02

122.88

122.45

129.60

47.00

20.01

55

1110

3

303.24

214.98

182.88

60.01

122.87

122.43

129.59

47.00

20.01

55

1258

4

379.87

222.26

196.88

60.01

172.73

122.63

129.58

47.02

20.01

55

1406

5

456.53

222.27

194.31

60.04

172.77

122.48

129.59

47.01

20.01

55

1480

6

456.52

222.27

274.30

60.05

222.60

140.64

129.60

47.01

20.02

55

1628

7

456.48

302.16

286.71

60.01

222.58

122.44

129.60

47.02

20.01

55

1702

8

456.53

309.55

303.19

109.99

222.60

122.51

129.59

47.00

20.04

55

1776

9

456.50

389.54

323.31

120.43

222.63

159.98

129.60

47.00

20.01

55

1924

10

456.49

460.00

320.86

170.42

222.64

159.99

129.60

47.00

50.00

55

2072

11

456.95

459.99

339.98

220.41

224.98

159.99

129.63

47.00

52.07

55

2146

12

456.49

459.97

339.99

267.34

222.60

159.99

129.59

76.99

52.06

55

2220

13

456.48

396.83

302.51

241.48

222.60

159.99

129.67

85.35

22.08

55

2072

14

456.48

396.80

294.21

191.49

172.70

122.40

129.60

85.31

20.02

55

1924

15

379.86

396.79

283.37

180.81

122.81

122.45

129.59

85.31

20.00

55

1776

16

303.25

316.80

317.75

130.81

73.01

122.47

129.59

85.31

20.01

55

1554

17

226.62

309.52

288.26

120.36

122.87

122.45

129.60

85.32

20.00

55

1480

18

303.27

309.55

120.41

172.76

122.47

129.64

85.32

20.03

55

1628

19

379.88

389.55

300.91

120.44

172.73

122.58

129.59

85.32

20.02

55

1776

20

456.57

460.00

312.49

170.43

222.60

159.98

129.59

85.32

20.01

55

2072

21

456.51

396.80

315.21

120.43

222.60

122.53

129.60

85.31

20.00

55

1924

22

379.88

316.81

275.72

70.48

172.76

122.44

129.67

85.24

20.02

55

1628

23

303.25

236.85

196.62

60.02

122.87

122.47

129.59

85.32

20.01

55

1332

24

226.70

222.28

189.25

60.11

73.21

122.48

129.62

85.32

20.03

55

1184

Total Cost = 1018142.725815($/24h)

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Policy)

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stages from the formulation of the problem to the

final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflict of interest to declare that is

relevant to the content of this article.

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