Chapter-1

Economic Load Dispatch

1.1 Introduction

Electrical power systems are designed and operated to meet the continuous variation of power demand. In

power system, minimization of the operation cost is very important. Economic Load Dispatch (ELD) is a

method to schedule the power generator outputs with respect to the load demands, and to operate the

power system most economically, or in other words, we can say that main objective of economic load

dispatch is to allocate the optimal power generation from different units at the lowest cost possible while

meeting all system constraints.

There are many conventional methods that are used to solve economic load dispatch problem such as

Lagrange multiplier method, Lambda iteration method and Newton- Raphson method. In the conventional

methods, it is difficult to solve the optimal economic problem if the load is changed. It needs to compute

the economic load dispatch each time which uses a long time in each of computation loops. Therefore this

problem requires computational process where the total required generation is distributed among the

generation units in operation, by minimizing the selected cost criterion, subject it to load and operational

constraints as well.

1.2 Load Dispatching

The operation of a modern power system has become very complex. It is necessary to maintain frequency

and voltage within limits in addition to ensuring reliability of power supply and for maintaining the

frequency and voltage within limits it is essential to match the generation of active and reactive power

with the load demand. For ensuring reliability of power system it is necessary to put additional generation

capacity into the system in the event of outage of generating equipment at some station. Over and above it

is also necessary to ensure the cost of electric supply to the minimum. The total interconnected network is

controlled by the load dispatch centre. The load dispatch centre allocates the MW generation to each grid

depending upon the prevailing MW demand in that area. Each load dispatch centre controls load and

frequency of its own by matching generation in various generating stations with total required MW

1

demand plus MW losses. Therefore, the task of load control centre is to keep the exchange of power

between various zones and system frequency at desired values.

1.3 Generator Operating Cost

The total cost of operation includes the fuel cost, cost of labour, supplies and maintenance. Generally,

costs of labour, supplies and maintenance are fixed percentages of incoming fuel costs. The power output

of fossil plants is increased sequentially by opening a set of valves to its steam turbine at the inlet. The

throttling losses are large when a valve is just opened and small when it is fully opened.

Fig 1 Simple model of a fossil plant

Figure 1 shows the simple model of a fossil plant dispatching purposes. The cost is usually approximated

by one or more quadratic segments. The operating cost of the plant has the form shown in Figure 2. So, the

fuel cost curve in the active power generation, takes up a quadratic form, given as:

(1)

Figure.2 Operating costs of a fossil fired generator

2

1.4 Economic Load Dispatch Problems

The ELD may be formulated as a nonlinear constrained optimization problem. Three different types of

ELD problems have been formulated and solved by DE/BBO approach.

1.4.1 ELDQCTL

ELD with quadratic cost function and transmission loss:-38 Generator system

The objective function Ft of ELD problem may be written as

(2)

Where is cost function of the ith generator, and is usually expressed as a quadratic polynomial; ,

and are the cost coefficients of the ith generator; m is the number of committed generators; Pi is the

power output of the ith generator. The ELD problem consists in minimizing subject to following

constraints.

Real Power Balance Constraint:

(3)

The transmission loss may be experssed using B-coefficients as

(4)

Generator Capacity Constraints: The power generated by each generator shall be within their lower limit

and upper limit . So that

(5)

3

1.4.2 ELDPOZR

ELD with quadratic cost function prohibited operating zones and Ramp rate limits:-3 Generator System

The objective function of this type of ELD problem is same as mentioned in (2). Here the objective

function is to be minimized subject to the constraints of (3), (5), and ramp-rate limits as mentioned below.

Ramp Rate Limit Constraints: The power generated, Pi, by the ith generator in certain interval may not

exceed that of previous interval by more than a certain amount , the up-ramp limit and neither may

it be less than that of the previous interval by more than some amount , the down-ramp limit of the

generator. These give rise to the following constraints.

As generation increases

(6)

As generation decreases

(7)

and

(8)

Prohibited Operating Zone: The prohibited operating zones are the range of output power of a generator

where the operation causes undue vibration of the turbine shaft. Normally operation is avoided in such

regions. Hence mathematically the feasible operating zones of unit can be described as follows:

(9)

4

where j represents the number of prohibited operating zones of unit i. is the upper limit and is

the lower limit of the jth prohibited operating zone of the ith unit. Total number of prohibited operating

zone of the ith unit is .

1.4.3 ELDVPL

ELD with valve-point loading effects and without transmission loss:-40Generator system

Real input-output characteristics of a generator display higher-order nonlinearities and discontinuities due

to valve-point loading in fossil fuel burning plant. The valve-point loading effect has been modelled in as a

recurring rectified sinusoidal function, such as the one show in figure 3

Fig. 3 Operating cost characteristics with valve point loading

The generating units with multi-valve steam turbines exhibit a greater variation in the fuel cost functions.

The valve-point effects introduce ripples in the heat-rate curves. In ELD with “Valve point loadings”, the

objective function Ft is represented by a more complex formula, given as Ft is given by

(10)

5

The objective of ELDVPL is to minimize Ft of (10) subject to the constraints given in

(3) and (5) as in ELDQCTL. Transmission loss is not considered. Here is zero.

6

Chapter-2

Biogeography Based Optimization

2.1 Introduction

BBO is based on the science of biogeography. Biogeography describes how species migrate from one

island to another, how new species arise, and how species become extinct. Its aim is to elucidate the reason

of the changing distribution of all species in different environments over time .The environment of BBO

corresponds to an archipelago, where every possible solution to the optimization problem is an Island or

Habitat. Each solution feature is called a suitability index variable (SIV). The goodness of each solution is

called its habitat suitability index (HSI), where a high HSI of an island means good performance on the

optimization problem, and a low HSI means bad performance on the optimization problem. Improving the

population is the way to solve problems in the heuristic algorithms. The method to generate the next

generation in BBO is by immigrating solution features to other islands, and receiving solution features by

emigration from other islands. Then mutation is performed for the whole population in a manner similar to

mutation in GAs.

2.2 Basic Procedure

The basic procedure of BBO is as follows:

1) Define the island modification probability, mutation probability, and elitism parameter. Island

modification probability is similar to crossover probability in GAs. Mutation probability and

elitism parameter are the same as in GAs.

2) Initialize the population ( n islands)

3) Calculate the immigration rate and emigration rate for each island. Good solutions have high

emigration rates and low immigration rates. Bad solutions have low emigration rates and high

immigration rates.

4) Probabilistically choose the immigration islands based on the immigration rates. Use roulette wheel

selection based on the emigration rates to select the emigrating islands.

5) Migrate randomly selected SIVs based on the selected islands in the previous step.

6) Probabilistically perform mutation based on the mutation probability for each island.

7) Calculate the fitness of each individual island.

8) If the termination criterion is not met, go to step 3; otherwise, terminate.

7

Mathematically the concept of emigration and immigration can be represented by a probabilistic model.

Let us consider the probability Ps that the habitat contains exactly S species at t. Ps changes from time t to

time t +∆t as follows:

(11)

Fig 4. Species model of a single habitat.

where λs and μs are the immigration and emigration rates when there are S species in the habitat as given in

Fig. 4. This equation holds because in order to have S species at time (t +∆t ), one of the following

conditions must hold:

1) there were S species at time t , and no immigration or emigration occurred between t and t +∆t;

2) there were S-1 species at time t, and one species immigrated;

3) there were S+1 species at time t, and one species emigrated.

(12)

8

If time ∆t is small enough so that the probability of more than one immigration or emigration can be

ignored then taking the limit of (11) as ∆t→ 0 gives (12). From the straight-line graph of Fig. 4, the

equation for emigration rate μk and immigration rate λk for k number of species can be written as per the

following way:

(13)

(14)

When value of E=I, then combining (13) and (14)

(15)

In BBO, as discussed there are two main operators, the migration and the mutation. With the migration

operator, BBO can share the information among solutions. Especially, poor solutions tend to accept more

useful information from good solutions. This makes BBO be good at exploiting the information of the

current population. Details about the two operators are given below.

2.3 MIGRATION

In BBO algorithm a population of candidate solution can be represented as vectors of

real numbers. Each real number in the array is considered as one (SIV). Using this

SIV, the fitness of each set of candidate solution, i.e., HSI value can be evaluated. In

an optimization problem high HSI solutions represent better quality solution, and low

HSI solutions represent an inferior solution. The emigration and immigration rates of

each solution are used to probabilistically share information between habitats. With

probability, known as habitat modification probability, each solution can be modified

based on other solutions. According to BBO if a given solution is selected for

modification, then its immigration rate is used to probabilistically decide whether or

9

not to modify each suitability index variable (SIV) in that solution. After selecting the

SIV for modification, emigration rates of other solutions are used to select which

solutions among the habitat set will migrate randomly chosen SIVs to the selected

solution. In order to prevent the best solutions from being corrupted by immigration

process, some kind of elitism is kept in BBO algorithm. Here, best habitat sets, i.e.,

those habitats whose HSI are best, are kept as it is without migration operation after

each iteration. This operation is known as elitism operation.

2.4 MUTATION

It is well known that due to some natural calamities or other events HSI of natural habitat might get

changed suddenly. In BBO such an event is represented by mutation of SIV and species count probabilities

are used to determine mutation rates. The probabilities of each species count can be calculated using the

differential equation of (12). Each habitat member has an associated probability, which indicates the

likelihood that it exists as a solution for a given problem. If the probability of a given solution is very low,

then that solution is likely to mutate to some other solution. Similarly if the probability of some other

solution is high, then that solution has very little chance to mutate. So it can be said that very high HSI

solution and very low HSI solutions have less chance to create more improved SIV in the later stage. But

medium HSI solutions have better chance to create much better solutions after mutation operation.

Mutation rate of each set of solution can be calculated in terms of species count probability using the

following equation:

where is a user-defined parameter. This mutation scheme tends to increase diversity among the habitats.

Without this modification, the highly probable solutions will tend to be more dominant in the total habitat.

This mutation approach makes both low and high HSI solutions likely to mutate, which gives a chance of

improving both types of solutions in comparison to their earlier value. Few kind of elitism is kept in

mutation process to save the features of a solution, so if a solution becomes inferior after mutation process,

then previous solution (solution of that set before mutation) can be reverted back to that place again if

needed. In ELD problem, if a solution is selected for mutation, then it is replaced by a randomly generated

10

new solution set. Other than this, any other mutation scheme that has been implemented for GAs can be

implemented for BBO.

11

Chapter-3

Differential Evolution (DE)

3.1 Introduction

Differential evolution (DE) is technically population based Evolutionary Algorithm, capable of handling

non-differentiable, nonlinear and multi-modal objective functions. DE generates new offspring by forming

a trial vector of each parent individual of the population. The population is improved iteratively, by three

basic operators: mutation, crossover, and selection. A brief description of different steps of DE algorithm

is given below.

Fig.5 General Evolutionary Algorithm Procedure

3.2 DE Procedure

3.2.1. Initialization:

The population is initialized by randomly generating individuals within the boundary constraints

; (16)

where “rand” function generates random values uniformly in the interval [0, 1]; is the size of the

population; D is the number of decision variables. and are the lower and upper bound of the

jth decision variable, respectively.

12

3.2.2. Mutation

As a step of generating offspring, the operations of “Mutation” are applied. “Mutation” occupies quite an

important role in the reproduction cycle. The mutation operation creates mutant vectors by

perturbing a randomly selected vector with the difference of two other randomly selected vectors

and at the kth iteration as per the following equation:

; (17)

, and are randomly chosen vectors at the kth iteration and .

, and are selected anew for each parent vector. is known as “Scaling factor” used to

control the amount of perturbation in the mutation process and improve convergence.

3.2.3. Crossover/Recombination

Crossover represents a typical case of a “genes” exchange. The trial one inherits genes with some

probability. The parent vector is mixed with the mutated vector to create a trial vector, according to the

following equation:

(18)

where . are the jth individual of ith target vector, mutant vector,

and trial vector at iteration, respectively. is a randomly chosen index that

guarantees that the trial vector gets at least one parameter from the mutant vector even if .

. is the “Crossover constant” that controls the diversity of the population and aids the algorithm

to escape from local optima.

13

3.2.4. Selection

Selection procedure is used among the set of trial vector and the updated target vector to choose the best.

Selection is realized by comparing the cost function values of target vector and trial vector. Selection

operation is performed as per the following equation:

; (19)

3.3 DE Algorithm

The pseudo-code of the DE algorithm is shown below:

DE Algorithm with Strategy 1

1) Generate the initial population P

2) Evaluate the fitness for each individual in P

3) while the termination criterion is not satisfied

4) for i=1 to NP

5) Select uniform randomly

6) jrand = randint(1,D)

7) for j=1 to D

8) if

9)

10) else

11)

12) end

13) end

14) end

15) for i=1 to NP

14

16) Evaluate the offspring

17) if is better than

18) =

19) end

20) end

21) end

Where D is the number of decision variables. NP is the size of the parent population P. is the jth

variable of the solution . is the offspring. randint(1,D) is a uniformly distributed random integer

number between 1 and D . Many schemes of creation of a candidate are possible. Here Strategy 1 has been

mentioned in the algorithm.

FLOWCHART OF DE ALGORITHM:

15

3.4 DE/BBO APPROACH

DE has been found to yield better and faster solution, satisfying all the constraints, both for uni-modal and

multi-modal system, using its different crossover strategies. But when system complexity and size

increases, DE method is unable to map its entire unknown variables together in a better way. Due to

presence of crossover operation in Evolutionary based algorithms, many solutions whose fitness are

initially good, sometimes lose their quality in later stage of the process. In BBO there is no crossover-like

operation; solutions get fine tuned gradually as the process goes on through migration operation.

This gives an edge to BBO over techniques mentioned above. In a nut shell, DE has good exploration

ability in finding the region of global minimum. Similarly, BBO has good exploitation ability in global

optimization problem. In order to utilize both the properties of DE and BBO for solution of complex

optimization problems, a hybrid technique called DE/BBO has been developed. Proposed DE/BBO

approach is described below:

3.4.1. Hybrid Migration Operator

Hybrid migration operator is most important step in DE/BBO algorithm. In this algorithm child population

takes new features from different sides. These are mutation operation of DE, migration operation of

BBO and corresponding parents of offspring. The core idea of the proposed hybrid migration operator

is based on two considerations. Here, due to this hybridization good solutions would be less destroyed,

while poor solutions can accept a lot of new features from good solutions. In this sense, the current

population can be exploited sufficiently.

Algorithm for the Hybrid Migration operator of DE/BBO

1) for i=1 to NP

2) Select uniform randomly

3)

16

4) for j=1 to D

5) if rand(0,1)<

6) if randj(0,1)>CR or

7)

8) else

9) Select with probability

10)

11) end

12) else

13)

14) end

15) end

16) end

3.4.2. Main Procedure of DE/BBO

Introducing previously mentioned hybrid migration operator of BBO into DE has developed one new

algorithm known as DE/BBO. The structure of proposed DE/BBO is also very simple. Detail hybrid

method is given in below:

The Main DE/BBO algorithm

1) Generate the initial population P

2) Evaluate the fitness for each individual in P

3) While the termination criterion is not satisfied

4) For each individual calculate species count probability

5) For each individual calculate Immigration rate and emigration rate

6) Modify the updated population with the Hybrid Migration operation of Algorithm 3

7) for i=1 to NP

17

8) Evaluate the offspring

9) if is better than

10)

11) end

12) end

13) end

3.4.3. DE/BBO Algorithm for ELD Problem

In this section, a new approach, DE/BBO algorithm is described for solving the ELD problems.

Step 1) For initialization, choose number of Generator units m, population size N. Specify maximum and

minimum capacity of each generator, power demand, B-coefficients matrix for calculation of transmission

loss. Initialize DE parameters like Crossover Probability CR, Scaling Factor F. Also initialize the BBO

parameters like max immigration rate , max emigration rate , lower bound for immigration probability per

gene, upper bound for immigration probability per gene, etc. Set maximum number of Iteration.

Step 2) Initialize the Population P. Since the decision variables for the ELD problems are real power

generations, they are used to represent each element of a given population set. Each element of the

Population matrix is initialized randomly within the effective real power operating limits. Each population

set of the population matrix should satisfy equality constraint (3) using the concept of slack generator as

mentioned before. Each individual population set of the population matrix represents a potential solution

to the given problem.

Step 3) Calculate the fitness value for each population set of the total population P for given emigration

rate , immigration rate . Fitness value represents the fuel cost of the generators in the power system for

a particular power demand.

Step 4) Probabilistically perform hybrid migration operation on those elements of population matrix P,

which are selected for migration. Perform hybrid migration operation on the Population set.

18

Step 5) After migration operation, new offspring population set (P’) is generated. In ELD problems these

represent new modified generation values of generators (PG.) Equality constraint (3) is satisfied using

concept of slack generator.

Step 6) Fitness value of each newly generated population set (offspring matrix P’) is recomputed, i.e., fuel

cost of each power generation set.

Step 7) Perform selection operation between parent population (P) and newly generated offspring (P’)

based on their fitness values as per (19).

Step 8) Go to step 3 for the next iteration. This loop can be terminated after a predefined number of

iterations.

19

20

Chapter-4

Numerical Example and Simulation Results

4.1 Test Case 1

A 3 generators system with ramp rate limit and prohibited operating zone is considered here. The load

demand is 300 MW. Results obtained from proposed DE/BBO, BBO, and other methods have been

presented in Table III. The convergence characteristic is shown in Fig.6.

The input data is given below:

Table I. Generating units capacity and coefficients

Unit

1 50 250 0.00525 8.663 328.13

2 5 150 0.00609 10.04 136.91

3 15 100 0.00592 9.76 59.16

Table II. Generating units ramp rate limits and prohibited zones

UnitProhibited Zones

1 215.0 55.0 95.0 [105,117][165,177]

2 72.0 55.0 78.0 [50,60][92,102]

3 98.0 45.0 64.0 [25,32][60,67]

Table III. Comparison Among Different Methods After 50 Trials

21

(Three-Generator System, PD=300 MW )

Output(MW) DE/BBO BBO APSO GA 2PHASE N.N

207.637 207.9926 200.528 194.26 165

87.2833 86.0125 78.2776 50 113.4

15.0000 16.0723 33.9918 79.62 34

Total Power 309.920 310.0774 312.797 323.89 312.45

Ploss9.9204 10.0774 12.8364 24.011 12.45

Total Cost($/hr) 3619.7565 3260.1748 3634.3127 3737.20 3652.6000

Av. Cost($/hr) 3619.7568 3620.1799 3634.3127 - -

Time(sec.) 0.015 .017 - - -

Fig.6 Convergence characteristic of three-generator system, PD=300 MW

4.2 Test Case 2

22

A system with 40 generators with valve point loading is used here. The load demand is 10 500 MW. Transmission loss has not been considered here. The result obtained from proposed DE/BBO method has been compared with BBO, ICA_PSO, SOH_PSO, and other methods. Their best solutions are shown in Table IV. Its convergence characteristic is shown in Fig 7.

Table IV. Best Power Output For 40-Generator System (PD=10500 MW)

Output

(MW)

DE/

BBO

BBO ICA_

PSO

Output

(MW)

DE/

BBO

BBO ICA_

PSO

P1110.7998 111.0465 110.80 P21

523.2794 523.417 523.28

P2110.7998 111.5915 110.80 P22

523.2794 523.2795 523.28

P397.3999 97.60077 97.41 P23

523.2794 523.3793 523.28

P4179.7331 179.7095 179.74 P24

523.2794 523.3225 523.28

P587.9576 88.30605 88.52 P25

523.2794 523.3661 523.28

P6140.00 139.9992 140.0 P26

523.2794 523.4362 523.28

P7259.5997 259.6313 259.60 P27

10.00 10.05316 10.00

P8284.5997 284.7366 284.60 P28

10.00 10.01135 10.00

P9284.5977 284.7801 284.60 P29

10.00 10.00302 10.00

P10130.00 130.2484 130.00 P30

97.00 88.47754 96.39

P11168.7998 168.8461 168.80 P31

190.00 189.9983 190.00

P1294.00 168.8239 94.00 P32

190.00 189.9881 190.00

P13214.7598 214.7038 214.76 P33

190.00 189.9663 190.00

P14394.2794 304.5894 394.28 P34

164.7998 164.8054 164.82

P15394.2794 394.2761 394.28 P35

200.00 165.1267 200.00

P16304.5196 394.2409 304.52 P36

200.00 165.7695 200.00

P17489.2794 489.2919 489.28 P37

110.00 109.9059 110.00

P18489.2794 489.2919 489.28 P38

110.00 109.9971 110.00

P19511.2794 511.2997 511.28 P39

110.00 109.9695 110.00

P20511.2794 511.3073 511.28 P40

511.2794 511.2794 511.28

121420.89 121426.95 121413.2

23

Fig 7 Convergence characteristic of 40-generator system (with valve-point loading, Pd=10500MW)

4.3 Test Case 3

A system with 38 generators is taken here. Fuel cost characteristics are quadratic. The load demand is 6000 MW. The result obtained using proposed DE/BBO algorithm has been compared with BBO, PSO_TVAC, and other methods and is shown in Table V. Its convergence characteristic is shown in Fig. .

Fig.8 Convergence characteristic of 38-generator system (Pd=6000MW)

Table V. Best Power Output for 38-generator system (Pd=6000MW)Output(MW) DE/BBO BBO PSO_TVAC New_PSO

24

P1426.606060 422.230586 443.659 550.000

P2426.606054 422.117933 342.956 512.263

P3429.66314 435.779411 433.117 485.733

P4429.663181 445.481950 500.00 391.083

P5429.663193 428.475752 410.539 443.846

P6429.663164 428.649254 492.864 358.398

P7429.663185 428.119288 409.483 415.729

P8429.663168 429.900663 446.079 320.816

P9114.000000 115.904947 119.566 115.347

P10114.000000 114.115368 137.274 204.422

P11119.768032 115.418622 138.933 114.000

P12127.072817 127.511404 155.401 249.197

P13110.000000 110.000948 121.719 118.886

P1490.000000 90.0217671 90.924 102.802

P1582.000000 82.000000 97.941 89.039

P16120.000000 120.038496 128.106 120.000

P17159.598036 160.303835 189.108 156.562

P1865.000000 65.0001141 65.00 84.265

P1965.000000 65.0001370 65.00 65.041

P20272.000000 272.999591 267.422 151.104

P21272.000000 271.872680 221.383 226.344

P22260.000000 259.732054 130.804 209.298

P23130.648618 125.993076 124.269 85.719

P2410.000000 10.4134771 11.535 10.000

P25113.305034 109.417723 77.103 60.000

P2688.0669159 89.3772664 55.018 90.489

P2737.5051018 36.4110655 75.000 39.670

P2820.000000 20.0098880 21.682 20.000

P2920.000000 20.0089554 29.829 20.995

P3020.000000 20.000000 20.326 22.810

P3120.000000 20.000000 20.000 20.000

P3220.000000 20.033959 21.840 20.416

P3325.000000 25.0066586 25.620 25.000

P3418.000000 18.0222107 24.261 21.319

P358.00000000 8.00004260 9.667 9.122

P3625.000000 25.0060660 25.000 25.184

P3721.7820891 22.0005641 31.642 20.000

P3821.0621792 20.6076309 29.935 25.104

Cost

($/h)

9417235.786391673 9417633.6376443729 9500448.307 9516448.312

CONCLUSION

25

The DE/BBO method has been successfully implemented to solve different convex

and non-convex ELD problems. It has been observed that the DE/BBO has the ability

to converge to a better quality solution and possesses better convergence

characteristics and robustness than ordinary BBO. It is also clear from the results

obtained by different trials that the proposed DE/BBO method can avoid the

shortcoming of premature convergence exhibited by other optimization techniques.

Due to these properties, the DE/BBO method in future can be tried for solution of

complex power system optimization problems.

26

References

1. Aniruddha Bhattacharya and Pranab Kumar Chattopadhyay, “Hybrid Differential Evolution With Biogeography-Based Optimization for Solution of Economic Load Dispatch”, IEEE Trans. Power Syst., vol. 25, no. 4, Nov. 2010

2. P. H. Chen and H. C. Chang, “Large-scale economic dispatch by genetic algorithm,” IEEE Trans. Power Syst., vol. 10, no. 4, pp. 1919–1926, Nov. 1995.

3. N. Sinha, R. Chakrabarti, and P. K. Chattopadhyay, “Evolutionary programming techniques for economic load dispatch,” IEEE Trans. Evol. Comput., vol. 7, no. 1, pp. 83–94, Feb. 2003.

4. M. Sydulu, “A very fast and effective non-iterative “Lamda Logic Based” algorithm for economic dispatch of thermal units,” in Proc. IEEE Region 10 Conf. TENCON, Sep. 15–17, 1999, vol. 2, pp. 1434–1437.

5. D. Simon, “Biogeography-based optimization,” IEEE Trans. Evol. Comput., vol. 12, no. 6, pp. 702–713, Dec. 2008

6. Aniruddha Bhattacharya and Pranab Kumar Chattopadhyay, “Biogeography–Based optimization for Different Economic Load Dispatch Problems,” IEEE Trans. On Power Systems, Vol. 25, No.2, pp 1064-1077, May 2010

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