nsde ieee energy jrnal 40unit

8/13/2019 NSDE IEEE Energy Jrnal 40Unit

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NSDE With Adaptive Crowding Distance For

Combined Non-convex Economic and Emission

Load Dispatch

Nidul Sinha1, SeniorMember IEEE,Biswajit Purkayastha2 and Bipul Syam Purkayastha3

1Nidul Sinha is with the Department of Electrical Engineering, (email:[email protected])2Biswajt Purkayastha is with the Department of Computer Science and Engg., NIT, Silchar, Assam, India-

788010 (e-mail:[email protected]).3Bipul Syam Purkayastha is with the Department of Computer Science, Assam University, Silchar,

Assam, India (e-mail:[email protected]).

AbstractThis paper presents an algorithm based on nondominated sorting differential evolution

algorithm (NSDE) for solving optimal economic emission dispatch problems in electric power

system. Economic emission load dispatch problems are multiobjective optimization problems and is

the right field for testing and application of NSDE algorithm. NSDE is reported to have performed

excellently on many bench mark test cases of multiobjective optimization problems. The

performance of the algorithm is tested on a test case of economic emission load dispatch. The results

are impressive and encouraging.

Index TermsNon-dominated sorting, Differential Evolution, Multiobjective optimization,

Economic Load Dispatch, Emission dispatch.

I. INTRODUCTION

The basic objective of economic load (ED) dispatch in electric power system is to schedule the

outputs of the committed generating units in such a manner so as to meet the load demand at minimum

operating cost while satisfying all unit and system equality and inequality constraints. This makes the ED

problem a large-scale highly non-linear constrained optimization problem.


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The trend of considering more and more realistic conditions and features is on the rise especially after

deregulation. Utilities are giving more stress on any efforts for improving the efficient management of

dispatch resulting into more savings. The realistic models of cost curves are highly nonlinear and they are

approximated as quadratic ones to make the problems solvable by classical methods. However, this

approximation results into huge loss of revenue over the time. On the other hand, consideration of highly

nonlinear characteristics of the units demands for solution techniques having no restrictions on the shape

of the fuel cost curves. The classical gradient based techniques fail in solving these types of problems.

Though dynamic programming (DP) [1] is capable of solving ED problems with inherently nonlinear and

discontinuous cost curves but suffers from intensive mathematical computations. In addition, the

increasing public awareness of the environmental pollution and the passage of the Clean Air Act

amendments of 1990 have forced the utilities to modify their design or operational strategies to

reduce pollution and atmospheric emissions of the thermal power plants [2].

Several strategies to reduce the atmospheric emissions has been proposed and discussed [2 - 4].

These include installation of pollutant cleaning equipment such as gas scrubbers and electrostatic

precipitators, switching to low emission fuels, replacement of the aged fuel burners and generator units

with cleaner and more efficient ones, and emission dispatching. The first three options require installation

of new equipments and/or modification of the existing ones that involve considerable capital investment

and, hence, can be considered as long term options.

The emission dispatching option is an attractive short term alternative in which the emission in

addition to the fuel cost objective are to be minimized. Thus, an economic emission dispatch problem is a

multiobjective optimization problem, which is to determine the allocation of power demand among the

committed generating units, to minimize a number of objectives viz. operating cost, minimal impacts on

environment, etc. subject to physical and technological constraints. In single objective optimization, one

tries to solve for the best solution (called the optimal solution) for one objective only, which is either

maximum or minimum depending upon the objective of the problem. But in multiobjective optimization,


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the objectives are more than one and are of conflicting nature and there may not exist one solution, which

is the best with respect to all objectives. Classical methods scalarize the vector of objectives into a single

objective by summing up the objectives scaled by a weight vector and hence a single solution is available

at every simulation run. In a typical multiobjective problem, a set of optimal solutions is possible. These

solutions are termed as Pareto-optimal solutions or nondominated solutions. These solutions are classified

as superior to the rest of solutions in the search space when all objectives are considered but are inferior to

other solutions in one or more objectives. The rest of the solutions are called dominated solutions. Since

none of the solutions in the non-dominated set is absolutely better than any other, any of them is an

acceptable solution. Thus, the designer in a multiobjective environment should have knowledge of other

Pareto-optimal solutions. The DE algorithm is a population based algorithm like GAs using the similar

operators; crossover, mutation and selection. The main difference between GA and DE is that GAs rely

mostly on crossover while DE relies mostly on mutation operation. The algorithm uses mutation operation

as a search mechanism and selection operation to direct the search toward the prospective regions in the

search space. Mutation in DE uses differences of randomly sampled pairs of solutions in the population

and greediness may be embedded in it in a controlled way. The DE algorithm also uses a non-uniform

crossover that can take child vector parameters from one parent more often than it does from others. By

using the components of the existing population members to construct trial vectors, the recombination

(crossover) operator efficiently shuffles information about successful combinations, enabling the search

for a better solution space. The highlights of DE are:

A population of solution vectors is successively updated by addition, subtraction, and componentswapping, until the population converges, hopefully to the optimum.

No derivatives are used. Very few parameters to set. A simple and apparently very reliable method.


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DE is reported [22 - 26] to be the only algorithm, which consistently found the optimal solution, and often

with fewer function evaluations than the other direct search methods on benchmark nonlinear functions.

The attributes for better performance of DE are:

Simple vector subtraction to generate random direction.

More variation in population (because solution has not converged yet) leads to more varied searchover solution space.

Size and direction Annealing versus self-annealing.Very few works are reported on the performance of DE algorithm on highly nonlinear power system

problems. The number of reported impressive performances of DE on benchmark mathematical functions

have encouraged us in applying this method on highly nonlinear economic emission load dispatch

problems. The key factor behind DE is a new scheme for generating trial parameter vectors. DE generates

new parameter vectors by adding the weighted difference vector between two population members to a

third member. The values of the weights play vital rule in the success of the algorithm. The weights are

dependent on the nature and size of the objective functions. There is no explicit rule or guideline for

proper choice of their values. Usually they are determined through experimentations for a particular

problem.

Different techniques have been reported in the literature for solving environmental/economic dispatch

(EED) problems. Granelli et al [5] have reduced the EED problem into a single objective one by treating

the emission as a constraint with a permissible limit. This formulation, however, has the severe difficulty

in getting the trade-off relations between cost and emission. Farag et al [6] have proposed a linear

programming based optimization method in which the objectives are considered one at a time.

Unfortunately, the EED problem is highly nonlinear and a multimodal optimization problem. In other

research direction, the multiobjective EED problem was converted into a single objective problem by


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linear combination of different objectives as a weighted sum [7 - 9]. The important aspect of this method

is that a set of non-inferior (or Pareto-optimal) solutions can be obtained by varying the weights.

However, this requires multiple runs as many times as the number of desired Pareto optimal solutions.

Further, this method cannot be used to find Pareto optimal solutions in problems having non-convex

Pareto optimal front. In addition, there is no rational basis for determining adequate weights and the

objective function so formed may lose significance due to combining non-commensurable objectives. To

avoid this difficulty, the -constraint method for multiobjective optimization was presented in Refs. [10 -

11]. This method is based on optimization of the most preferred objective and considering the other

objectives as constraints bounded by some allowable levels. These levels are then altered to generate the

entire Pareto-optimal set. The most obvious weaknesses of this approach are that it is time-consuming and

tends to find weakly nondominated solutions. Over the past decade, a number of multiobjective

evolutionary algorithms (MOEAs) have been suggested [13 - 17]. The potential of evolutionary

algorithms (EAs) for solving multiobjective optimization problems was hinted as early as the late 1960s

by Rosenberg. But the implementation of multiobjective evolutionary algorithm (MOEA) was delayed

until mid-1980s. Since, EAs work with a population of solutions, hence it is possible for EA to provide a

diverse set of solutions for a multi-criterion environment. Moreover, EAs are less susceptible to the shape

or continuity of the Pareto front (i.e. they can easily deal with discontinuous and concave Pareto fronts),

on the other hand, mathematical programming techniques have these as known problems.

Deb et al have proposed nondominated sorting GA (NSGA) [16] for solving multiobjective problems.

NSGA is a popular non-domination based genetic algorithm for multi-objective optimization. It is a very

effective algorithm but has been generally criticized for its computational complexity, lack of elitism and

for choosing the optimal parameter value for sharing parameter share. A modified version of it, NSGA- II,

was developed by Deb et al [17], which has a better sorting algorithm, incorporates elitism and no sharing

parameter needs to be chosen a priori. Very recent addition to evolutionary algorithms is the Differential


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Evolution which is reported to have performed impressively on highly non linear optimization problems.

Very few works (in fact none) are reported on the performance of NSDE on highly nonlinear

multiobjective power system problems. The number of reported impressive performances of DE and

nondominated sorting based GA has induced us in developing a method fusing DE and nondominated

sorting, naming it NSDE for solving the multiobjective optimization problems of power systems.

In view of the above, the main objective of the present work is to develop a program based on NSDE

and study its performance in solving the optimal operation of combined economic and emission dispatch.

The rest of the paper is organized as follows: In Section II the mathematical problem formulation is

briefly reviewed. Multiobjective is described in Section III and section IV describes the implementation

of NSDE. Section V presents the experimental results and discussions. Conclusions are drawn in Section

VI.

II. PROBLEM FORMULATION

In this section the optimization problem is formulated as a minimization of sum of fuel costs of all

generators including minimization of SO2 and NOx emission subject to constraints as described.

Problem objectives

A. Minimization of fuel cost

1) Fuel Cost:The generator cost curves are represented by quadratic functions. F(Pi),the total fuel cost

which is a function of power output in Rs./hr, can be expressed as:

|))(sin(|)( min2

1 iiiiiiiii

n

ii PPfePcPbaPF +++= = (1)

where n is the number of generators, ai, bi, ci are the fuel cost coefficients, ei and fi are the fuel cost

coefficients with valve point effects and Piis the power output of the ith unit.


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2) Emissions: The total tonn/hr emissionE(Pi)of atmospheric pollutants such as sulphur dioxide, SO2

and nitrogen oxides, NOxcaused by burning of fuel in thermal units can be expressed as:

=

++=n

i

iiiiii PPPE1

3)()( (2)

where i,i, iare the coefficients of the characteristics of emission by the ith

generating unit.

B. Subject to

1) Power balance constraint: The total power generated by the units must be equal to the sum of total

load demand and total real power loss in the transmission lines. Hence the constraint is:

01

==

lossD

n

i

i PPP (3)

where PDis the total load demand and Plossis the total transmission loss. In this work the power loss is not

considered. However, it may be calculated by an iterative algorithm or by using directly B-loss matrix of

the power system.

2) Generation capacity constraints: The real power output of generating units must be restricted within

their respective lower and upper bounds as follows:

maxmin jjj

PPP for j=1,2,. n (4)

where Pjmin and Pjmaxare the minimum and maximum power outputs of the jth

unit.

III MULTIOBJECTIVE OPTIMIZATION

Given a single objective function f: = S n, for x , the value,f*=f(x*), is called a

global minimum (in case of single objective) iff

x : f(x*) f(x) (5)

where x* is a global minimal solution(s), f is the objective function and is the set of feasible solutions

which is again a subset of solution space, S. The problem of determining the global minimum solution is

called the global optimizationproblem.


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However, many real-world problems have more than one objective functions, sometimes conflicting,

to be optimized simultaneously. They do not have one particular solution but multiple numbers of them

and a particular solution cannot claim to be better than the other. A decision maker has to choose one of

the solutions depending on the system requirements. These solutions are called Pareto-Optimal solutions.

A general multi-objective optimization problem (MOOP) consists of a number of objectives along with

a number of equality and inequality constraints. A MOOP can be formulated as follows [20]:

Minimize F(x) =fm(x) m = 1,2,,M (6)

wherefmis the m-thobjective function and M is the number of objectives.

Subject to following equality and inequality constraints:

hk(x) = 0 k=1,2,,K

gj(x) 0 j =1,2,,J

where KandJare the number of equality and inequality constraints respectively. A solution x is a vector

of n (dimensional) decision variables (optimization parameters) : x = (x1,x2, ., xn), each component

is chosen from some universe of feasible solution and the lower and upper bound of each xi is defined

as

xi(L)

xi xi(U)

i=1,2,..., n

A solution in objective space u is said to dominate v, (represented by u p v), if u is better or equal

tov in all attributes (objectives) and strictly better in at least one attribute, ie

(: ifi u () if v ) and (: jfj u () jf< v ) (7)

where i, j{1,2, , M}.

A solution u is said to bePareto-optimalwith respect to , iff there is nov for which v p

u.

For a given MOOP, F(u), the Pareto Optimal Set P* is defined as:

P* = {u | v and F(v) p F(u) }


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For a given MOOP, F(u) and the Pareto Optimal Set P*, the Pareto front PF* is defined as:

PF* = {F(u) = (f1(u),f2(u), ,fk(u)) | uP*}

IV. IMPLEMENTATION OF NSDE

A. Initial Population

An initial population, Pof sizeNn, where N is the number of individuals (chromosomes) and n is

the number of power generators (units), is created. Initially a gene of each individual Pi,jis determined by

setting its value randomly such that its value lies between the upper and lower limits ofjth generator, i.e.

Pi,j~ U(Pjmin, Pjmax), j= 1, 2, . . . , n.

B. Non-dominated Sort

After the initial population P is generated, a non-dominated sorting of the population is done into

different fronts. For each solutionp, two attributes are found:

1) Domination count np, the number of solutions, which dominate this solution , and2) Sp, a set of solutions, which are dominated by solutionp.

All solutions in the first non-dominated front will have their domination count npas zero. Now, for

each solution pwith np=0, each member (q Sp) is found and its domination count is reduced by one. In

doing so, if for any member the domination count becomes zero, it (q) is included in a separate set Q. The

members of this set Qbelong to the second non-dominated front. This process continues until all fronts

are identified. The algorithm of non-dominated sort is given below. Fig. 1 depicts the solutions classified

into fronts after the method is applied.

Algorithm

i initialize Fi =

For eachp P

Sp=

np= 0


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for each q P

if (pp q) then

Sp= Sp {q}

else if (qp

p) then

np= np + 1

if (np= 0) then

prank= 1

F1= F1 {p}

i= 1

while Fi 0

Q =

for each pFi

for each qSp

nq= nq - 1

if nq= 0 then

qrank= i+ 1

Q = Q {q}

i = i + 1

Fi= Q

C. Density Estimation

To get an estimate of the density of solutions surrounding a particular solution in the population, the

average distance of two points on either side of the point under consideration along each of the objectives

is calculated. A cuboid is formed by taking the nearest solutions on either side. The quantity idistanceserves


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as an estimate of the perimeter of the cuboid and is called the crowding distance. Fig. 2 depicts the

crowding distance.

The computation of crowding-distance requires sorting the population according to each

objective function

value in ascending order of magnitude. Then, for each objective function, the solutions with smallest

and largest

Fig. 1. Solutions are sorted on non domination and domination basis

Fig:2. Figure showing distance of neighbours of point i considering both the objectives.


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function values are assigned an infinite distance value for the objective under consideration. All other

solutions having function value intermediate between minimum and maximum values mentioned above

are assigned a distance value equal to the absolute normalized difference in the function values of two

adjacent solutions. This calculation is continued with other objective functions. The overall crowding-

distance value for each solution is calculated as the sum of individual distance values corresponding to

each objective. Each objective function is normalized before calculating the crowding distance. The

algorithm cited below shows the for a particular solution i. crowding-distance computation of all solutions

within a non-dominated set I.

Algorithm for calculation of crowding-distance:

l=| I |

for each i, set I [i]distance= 0

for each objective m

I = sort ( I , m )

I [1]distance= I [l]distance=

For i = 2to ( l 1 )

I[i]distance =I[i]distance + (I[i+1].m - I[i-1].m)/(fmmax

- fmmin

)

In the above algorithm, I [i].mrefers to the mth objective function value of the ith individual in the set I

and the parametersfmmax

and fmmin

are the maximum and the minimum value of the mth objective.

After all population members in the set I are assigned a distance metric, the extent of proximity of two

solutions with other solutions can be compared.

D. Selection Algorithm

Non-dominated sorting based selection is used for selecting the population for the next generation.

Initially, a combined populationRt= Pt Qt is formed, where Pt is the parent population and Qt is

the new population obtained after applying DE operators. The populationRt is of size 2N. The population


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Rt is sorted according to non-domination. Then crowding distance is calculated for each individual. Since

Nchromosomes are selected for next generation (i.e. Pt+1) from 2N chromosomes of combined population

Rt, elitism is ensured. Now, solutions belonging to the non-dominated set F1 are the best solutions in the

combined population and must be emphasized more than any other solution during selection. While

selecting N solutions from fronts starting with F1 the following cases are considered within a front .

1) There may be more than one chromosome having zero crowding distance and/or

2) Different solutions having a crowding distance less than , a threshold value.}

Case 1 is an indication of duplicate chromosomes and the case 2 where chromosomes are having a

crowding distance less than is an indication of close proximity of solutions which, if accepted, will

result into cluster of solutions which is undesired. Our algorithm selects only one solution in case of

duplicate chromosomes and rejects chromosomes having crowding distance less than .

If the number of solutions selected from front F1 are less than N, the remaining (y) members of the

population Pt+1 are chosen from subsequent nondominated fronts in the order of their ranking. Thus,

solutions from the set F2 are chosen next, followed by solutions from the set F3 and so on till N number

of solutions are selected. While selecting, we are accepting solutions from best to worst front (F 1, F2,

..), but not accepting all solutions of any particular front, there is a chance of not getting all N

chromosomes even from all the fronts (having 2N chromosomes). In such case, we are filling up the

population by duplicating the accepted solutions. The new population Pt+1 of size N is now used for

selection, crossover, and mutation to create a new population Qt+1 of sizeN.

Algorithm for Selection:

Rt= Pt Qt

F = sort on non-dominated method (Rt)

% F = (F1, F2, .. ), Fi is ith front %


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Pt+1 = and i= 1 % iis front index %

while |Pt+1 |


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The algorithms are implemented in Matlab command line for solution of the test case of combined

economic and emission dispatch problem. The programs were run on a 3.2 GHz, Pentium-IV, with 512

MB RAM PC.

H. Main algorithm

Set the lower and upper bounds of variables

generate Nn size initial population

% N = popsize, n = no. of variables depending on bounds%

evaluate chromosomes on objective function (P0 )

% P0 is initial pop of size N(n+M), M=no. of objectives%

apply NDS and CD sort(P0 )

% Now pop size is N(n+M+2), front & CD column added%

t=1 % t=gen number %

do

. apply DE crossover and mutation (Pt )

. combine parent & offspring to form Rt, of size =2N

. apply NDS and CD sort(Rt)

. select best N chromosomes from Rt for next gen

. t=t+1

until ( t max gen )

where NDS=Non-Dominated Sort, CD=Crowding Distance,

DE=Differential Evolution


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V. NUMERICAL TESTS

The performance of the proposed algorithm is experimented on a test case adopted from Ref. [19] with

modifications.

iSO2 Emission Co-efficients

C0 C

1 C

2

1 0.026625 0.004764 2.21E-09

2 0.026625 0.004764 2.21E-09

3 0.020387 0.005121 8.37E-09

4 0.022512 0.006251 8.38E-09

5 0.026625 0.004764 2.21E-09

6 0.020387 0.00461 8.37E-09

7 0.142664 0.004658 3.18E-09

8 0.142664 0.004658 3.18E-09

9 0.142664 0.004658 3.18E-09

10 0.142664 0.004658 3.18E-09

11 0.154545 0.003515 3.17E-09

12 0.154545 0.003515 3.17E-0913 0.134964 0.005251 7.89E-10

14 0.134964 0.005251 7.89E-10

15 0.134964 0.005251 7.89E-10

16 0.134964 0.005251 7.89E-10

17 0.134964 0.005251 7.89E-10

18 0.134964 0.005251 7.89E-10

19 0.134964 0.005251 7.89E-10

20 0.134964 0.005251 7.89E-10

21 0.134964 0.005251 7.89E-10

22 0.134964 0.005251 7.89E-10

23 0.134964 0.005251 7.89E-10

24 0.134964 0.005251 7.89E-10

25 0.134964 0.005251 7.89E-10

26 0.134964 0.005251 7.89E-10

27 0.023141 0.003813 2.18E-09

28 0.023141 0.003813 2.18E-09

29 0.023141 0.003813 2.18E-09

30 0.026625 0.004764 2.21E-09

31 0.010387 0.00461 8.37E-09

32 0.010387 0.00461 8.37E-09

33 0.010387 0.00461 8.37E-09

34 0.010727 0.00514 8.72E-09

35 0.010727 0.00514 8.72E-0936 0.010727 0.00514 8.72E-09

37 0.026625 0.004764 2.21E-09

38 0.026625 0.004764 2.21E-09

39 0.026625 0.004764 2.21E-09

40 0.27845 0.005025 7.64E-10

TABLE I

TABLE SHOWING SO2 EMISSION CO-EFFICIENTS OF ith

GENERATOR FOR EMISSIONFUNCTION C0+ C1 Pi+ C2 Pi

3


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iNOx Emission Co-efficients

C0 C1 C2

1 0.10571 1.30E-03 1.49E-09

2 0.10571 1.30E-03 1.49E-09

3 0.020417 4.07E-04 1.45E-08

4 0.021151 4.07E-04 1.46E-08

5 0.10571 1.30E-03 1.49E-09

6 0.020417 4.07E-04 1.45E-08

7 0.022132 4.02E-04 1.48E-08

8 0.022132 4.02E-04 1.48E-08

9 0.022132 4.02E-04 1.48E-08

10 0.022132 4.02E-04 1.48E-08

11 0.019372 4.01E-04 1.40E-08

12 0.019372 4.01E-04 1.40E-08

13 0.32163 3.00E-05 6.15E-09

14 0.32163 3.00E-05 6.15E-09

15 0.32163 3.00E-05 6.15E-09

16 0.32163 3.00E-05 6.15E-0917 0.32163 3.00E-05 6.15E-09

18 0.32163 3.00E-05 6.15E-09

19 0.32163 3.00E-05 6.15E-09

20 0.32163 3.00E-05 6.15E-09

21 0.32163 3.00E-05 6.15E-09

22 0.32163 3.00E-05 6.15E-09

23 0.32163 3.00E-05 6.15E-09

24 0.32163 3.00E-05 6.15E-09

25 0.32163 3.00E-05 6.15E-09

26 0.32163 3.00E-05 6.15E-09

27 0.104511 1.26E-03 1.45E-09

28 0.104511 1.26E-03 1.45E-0929 0.104511 1.26E-03 1.45E-09

30 0.10571 1.30E-03 1.49E-09

31 0.020417 4.07E-04 1.45E-08

32 0.020417 4.07E-04 1.45E-08

33 0.020417 4.07E-04 1.45E-08

34 0.020821 4.09E-04 1.47E-08

35 0.020821 4.09E-04 1.47E-08

36 0.020821 4.09E-04 1.47E-08

37 0.10571 1.30E-03 1.49E-09

38 0.10571 1.30E-03 1.49E-09

39 0.10571 1.30E-03 1.49E-09

40 0.00285

3.183266-

04 1.72E-09

TABLE II

TABLE SHOWING NOx EMISSION CO-EFFICIENTS OF ith GENERATOR FOR EMISSION

FUNCTION C0+ C1 Pi+ C2 Pi3


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iOutput limits Fuel co-efficients

Min Max a b c e f

1 36 114 94.7 6.73 0.0069 100 0.084

2 36 114 94.7 6.73 0.0069 100 0.084

3 60 120 309.54 7.07 0.0203 100 0.084

4 80 190 369.03 8.18 0.0094 150 0.063

5 47 97 148.89 5.35 0.0114 120 0.077

6 68 140 222.33 8.05 0.0114 100 0.084

7 110 300 287.71 8.03 0.0036 200 0.042

8 135 300 391.98 6.99 0.0049 200 0.042

9 135 300 455.76 6.6 0.0057 200 0.042

10 130 300 722.82 12.9 0.0061 200 0.042

11 94 375 635.2 12.9 0.0052 200 0.042

12 94 375 654.69 12.8 0.0057 200 0.042

13 125 500 913.4 12.5 0.0042 300 0.035

14 125 500 1760.4 8.84 0.0075 300 0.035

15 125 500 1728.3 9.15 0.0071 300 0.035

16 125 500 1728.3 9.15 0.0071 300 0.035

17 220 500 647.85 7.97 0.0031 300 0.03518 220 500 649.69 7.95 0.0031 300 0.035

19 242 550 647.83 7.97 0.0031 300 0.035

20 242 550 647.81 7.97 0.0031 300 0.035

21 254 550 785.96 6.63 0.003 300 0.035

22 254 550 785.96 6.63 0.003 300 0.035

23 254 550 794.53 6.66 0.0028 300 0.035

24 254 550 794.53 6.66 0.0028 300 0.035

25 254 550 801.32 7.1 0.0028 300 0.035

26 254 550 801.32 7.1 0.0028 300 0.035

27 10 150 1055.1 3.33 0.5212 120 0.077

28 10 150 1055.1 3.33 0.5212 120 0.077

29 10 150 1055.1 3.33 0.5212 120 0.07730 47 97 148.89 5.35 0.0114 120 0.077

31 60 190 222.92 6.43 0.0016 150 0.063

32 60 190 222.92 6.43 0.0016 150 0.063

33 60 190 222.92 6.43 0.0016 150 0.063

34 90 200 107.87 8.95 0.0001 200 0.042

35 90 200 116.58 8.62 0.0001 200 0.042

36 90 200 116.58 8.62 0.0001 200 0.042

37 25 110 307.45 5.88 0.0161 80 0.098

38 25 110 307.45 5.88 0.0161 80 0.098

39 25 110 307.45 5.88 0.0161 80 0.098

40 242 550 647.83 7.97 0.0031 300 0.035

TABLE III

TABLE SHOWING FUEL-COST CO-EFFICIENTS OF ith GENERATOR FOR COST FUNCTION ai+ bi Pi

+ Ci Pi2+ | eisin(fi(Pimin Pi))| WITH VALVE POINT LOADING

The non-dominated solutions corresponding to the Pareto front is obtained. The results of the extreme

cases (minimum cost, minimum SO2 emission, minimum NOx emission) are reported in Table-IV. The


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results demonstrate that the algorithm is well competent to find the non-dominated solutions for the

economic emission dispatch problem even when there are more than two objective functions. The

algorithm is capable of finding non-dominated solutions even for more than three objective functions.

Total load demand = 6050 MW

Population size = 100

Penalty factor, = 1000

Number of units = 40

Number of objectives = 3

i

Units output on various situations

(MW)

Case 1 Case 2 Case 3

1 114.00 36.00 114.00

2 114.00 114.00 53.27

3 112.50 66.67 94.89

4 80.00 81.63 147.37

5 59.46 97.00 49.49

6 105.00 68.00 140.007 189.93 180.57 110.00

8 135.00 135.00 135.00

9 215.95 135.00 135.00

10 130.00 130.00 130.00

11 95.92 309.67 120.70

12 94.00 375.00 94.00

13 125.00 125.00 204.17

14 125.00 125.00 195.31

15 125.00 129.13 265.01

16 125.00 125.00 244.14

17 220.00 220.00 220.0018 220.00 220.00 220.00

19 242.00 242.00 242.00

20 242.00 242.00 242.00

21 254.00 254.00 254.00

22 254.00 254.00 254.00

23 254.00 254.00 254.00

24 254.00 254.00 254.00

25 254.00 254.00 254.00


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26 254.00 254.00 254.00

27 10.00 150.00 10.00

28 10.00 10.00 22.50

29 10.00 117.52 71.01

30 59.59 97.00 72.75

31 190.00 60.00 109.50

32 172.15 60.00 97.7933 187.44 80.43 135.17

34 90.00 90.00 90.00

35 90.00 90.00 90.00

36 167.01 106.56 90.00

37 92.11 110.00 48.42

38 54.91 44.76 25.00

39 104.71 110.00 95.60

40 418.33 243.07 411.92

Cost

(Rs./hr)

76240.64

(Min cost)97508.14 81886.12

SO2(Ton/hr)

34.15 33.29(Min SO2)

34.22

NOX

(Ton/hr)9.58 10.39

9.23

(Min NOX)

TABLE IV

TABLE SHOWING OUTPUT OF GENERATORS DURING MIN COST, MINEMISSION

VI. CONCLUSION

In this paper programs based on NSDE algorithm are developed in Matlab command line and

their performances are tested on a test case of 40 units for optimum combined economic and emission

dispatch problem. The results demonstrate that the algorithm based on NSDE is well competent and

efficient in solving multiobjective optimization problem like combined economic emission dispatch

problem specifically when more than two or three conflicting objective functions are to be optimized.


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Fig:3 3-D graph shown is the result obtained after optimization of 3 variables: Cost, SO2and NOXemission.

ACKNOWLEDGMENT

The authors would like to thank NIT Silchar and MHRD, Govt. of India, for their supports for carrying

out the research works.

VII.REFERENCES

[1] Zi-Xiong Liang and J. D. Glover,A zoom feature for a dynamic programming solution to economic

dispatch including transmission losses,IEEE Trans. on Power Systems, vol. 7, no. 2, pp. 544-549,May.

1992.

[2] A. A. El-Keib, H. Ma and J. L. Hart,Economic dispatch in view of the Clean Air Act of 1990,IEEE

Trans. on Power Systems, vol. 9, no. 2, pp. 972-978, 1994.

[3] J. H. Talaq, F. El-Hawary, M. El-Hawary ,A summary of environmental / economic dispatch

algorithms,IEEE Trans. on Power Systems, vol. 9, no. 3, pp. 1508-1516,1994.

[4] J. S. Helsin and B. F. Hobbs,A multiobjective production costing model for analyzing emission

dispatching and fuel switching,IEEE Trans. on Power Systems, vol. 4, no. 3, pp. 836-842, 1989.

[5] G. P. Granelli, M. Montagna, G.L. Pasini and P. Marannino, Emission constrained dynamic

dispatch,Electric Power Syst. Res., vol. 24, pp. 56-64, 1992.


23/25

[6] A. Farag, S. Al-Baiyat and T.C. Cheng,Economic load dispatch multiobjectiveoptimization

procedures using linear programming techniques, IEEE Trans. On Power Systems, vol. 10, no. 2, pp.

731-738, 1995.

[7] J. Zahavi and L. Eisenberg,Economic-environmental power dispatch, IEEE Trans. On Syst., Man,

Cybern., SMC-5, no. 5, pp. 485-489, 1985.

[8] J. S. Dhillon, S.C. Parti and D.P. Kothari,Stochastic economic emission load dispatch, Electric

Power Syst. Res., vol. 26, pp. 179-186, 1993.

[9] C.S. Chang, K.P. Wong and B. Fan,Security-constrained multiobjective generation dispatch using

bicriterion global optimization,IEE Proc.- Gener. Transm. Distrib., vol. 142, no. 4, pp. 406-414, 1995.

[10] A. A. Abou El-Ela, M.A. Abido,Optimal operation strategy for reactive power control, Modelling,

Simulation and Control, Part -A, vol. 41, pp. 19-40, AMSE Press, 1992.

[11] T. Hsiao, H.D. Chiang, C.C. Liu and Y.L. Chen,A computer package for optimal multi-objective

VAR planning in large scale power systems, IEEE Trans. On Power Syst., vol. 9, no. 2, pp. 668-676,

1994.

[12] C.A.C. Coello, A.D. Christiansen,MOSES: a multiobjective optimization tool for engineering

design,Engineering Optimization, vol. 31, no. 3, pp. 337-368, 1999.

[13] C.M. Fonseca and P.J. Fleming,An overview of evolutionary algorithms in multiobjective

optimization,IEEE Trans. On Evol. Comput., vol. 3, no. 1, pp. 1-16, 1995.

[14] C.A.C. Coello,A comprehensive survey of evolutionary-based multiobjective optimization

techniques, Knowledge Inf. Syst., vol. 1, no. 3, pp. 269-308, 1999.

[15] M. M. Raghuwanshi and O. G. Kakde,Survey on multiobjective evolutionary and real coded genetic

algorithms,In Proc.of the 8th Asia Pacific Symposium on Intelligent and Evolutionary Systems, pp. 150-

161, 2004.

[16] N. Srinivas and Kalyanmoy Deb,Multiobjective optimization Using


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Nidul Sinha was born in Tripura, India, in 1962. He received the B.E. degree inelectrical engineering from Calcutta University, in 1984 and M.Tech. degree in power

apparatus and systems from Indian Institute of Technology, New Delhi, in 1989. He

received his Ph.D. degree in electrical engineering from Jadavpur University. His

research interests include application of soft computing techniques to operation, controland economics of electrical power systems, deregulation and optimization. He is a

reviewer of IEEE TPWRS, TPWRD, PESL, IEE Part-C, and EPSR.

Bipul Syam Purkayastha received his B.Sc., M.Sc., M. Phil. and Ph.D. degrees from

NEHU, Shillong, India. His research interests include soft computing, bioinformatics,image processing and combinatorial optimization.

Biswajit Purkayastha received his B.Tech. degree in Electronics from JNTU, AP, India

and M. Tech. in Computer Science and Engg. from IIT, Kharagpur, India. He receivedhis Ph.D. degree in 2009 from the Department of Computer Science, Assam University,

Silchar, India. His research interests include evolutionary algorithms, soft computing,

optimization, and computer vision.

nsde ieee energy jrnal 40unit

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