![Applied Soft Computing 11 (2011) 2845–2853
Contents lists available at ScienceDirect
Applied Soft Computing
journal homepage: www.elsevier.com/locate/asoc
Economic environmental dispatch using multi-objective differential evolution
M. Basu∗
Department of Power Engineering, Jadavpur University, Block LB, Sector-3, Salt Lake City, Kolkata, West Bengal 700098, India
a r t i c l e i n f o
Article history:
Received 12 September 2006
Received in revised form 18 August 2010
Accepted 28 November 2010
Available online 4 December 2010
Keywords:
Economic environmental dispatch
Multi-objective optimization
Multi-objective differential evolution
a b s t r a c t
Economic environmental dispatch (EED) is an important optimization task in fossil fuel fired power
plant operation for allocating generation among the committed units such that fuel cost and emission
level are optimized simultaneously while satisfying all operational constraints. It is a highly constrained
multiobjective optimization problem involving conflicting objectives with both equality and inequality
constraints. In this paper, multi-objective differential evolution has been proposed to solve EED problem.
Numerical results of three test systems demonstrate the capabilities of the proposed approach. Results
obtained from the proposed approach have been compared to those obtained from pareto differential
evolution, nondominated sorting genetic algorithm-II and strength pareto evolutionary algorithm 2.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
The generation of electricity from fossil fuel releases sulfur
oxides (SOx), nitrogen oxides (NOx), and carbon dioxide (CO2) into
atmosphere. Atmospheric pollution affects not only humans but
also other life-forms such as animals, birds, fish and plants. It also
causes damage to vegetation, acid rain, reducing visibility as well as
causing global warming. The increased concern over environmen-
tal protection and the passage of the clean air act amendments of
1990 have forced the power utilities to reduce their emissions [1].
So today’s concern is to produce electricity not only at the cheapest
possible price, but also at minimum level of pollution.
Several strategies have been proposed to reduce the atmo-
spheric pollution [2]. These include installation of post combustion
cleaning equipment, switching to low emission fuels, replacement
of the aged fuel burners with cleaner ones, and dispatching with
emission considerations. The first three options require installa-
tion of new equipment and/or modification of the existing ones
that involve considerable capital outlay and hence they can be
considered as long-term options. So, latter option is preferred.
The two objectives i.e. cost and emission are conflicting in nature
and they both have to be considered simultaneously to find overall
optimal dispatch. Economic environmental dispatch (EED) serves
to schedule the committed generator outputs with the predicted
load demand so as to optimize both cost and emission simultane-
ously while fulfilling the operating constraints. It is a multiobjective
optimization problem with conflicting objectives because emission
minimization is conflicting with minimum cost of generation.
∗ Tel.: +91 23355813.
E-mail address: mousumibasu@yahoo.com
Different techniques have been reported in the literature per-
taining to EED problem. Nanda et al. treated EED as a multiple,
conflicting objective problem and solved using goal-programming
techniques [3]. In Ref. [4] a linear programming based optimization
procedure has been presented in which the objectives are consid-
ered one at a time. However, many mathematical assumptions have
to be given to simplify the problem. In past decades, the EED prob-
lem was converted to a single objective problem by linear combina-
tion of different objectives as a weighted sum [5,6]. The important
aspect of this weighted sum method is that a set of non-inferior
solutions can be obtained by varying the weights. Unfortunately,
this method cannot be used in problems having a non-convex
pareto-optimal front. To avoid this difficulty, the -constraint
method has been presented in Refs. [7,8]. This method optimizes
the most preferred objective and considers the other objectives
as constraints bounded by some allowable levels. The most obvi-
ous weaknesses of this approach are that it is time-consuming and
tends to find weakly non-dominated solutions. A fuzzy multiobjec-
tive optimization technique for the EED problem has been proposed
in [9]. However, the solutions are sub-optimal and the algorithm
does not provide systematic framework for directing the search
toward pareto-optimal front. A fuzzy satisfaction-maximizing deci-
sion approach was successfully applied to solve the biobjective EED
problem [10]. However, extension of the approach to include more
objectives is a very involved question. An evolutionary algorithm
based approach for evaluating the economic impacts of environ-
mental dispatching and fuel switching has been presented in [11]. A
multiobjective stochastic search technique for the EED problem has
been presented in [12]. However, this technique is computationally
involved and time consuming.
Over the past few years, several researches have been
made on the development of multiobjective evolutionary search
1568-4946/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.asoc.2010.11.014](https://image.slidesharecdn.com/basu2011-220705064227-5a18a9ba/85/basu2011-pdf-1-320.jpg)
![2846 M. Basu / Applied Soft Computing 11 (2011) 2845–2853
strategies. Strength Pareto evolutionary algorithm (SPEA) [13],
nondominating sorting genetic algorithm II (NSGA II) [14], multiob-
jective evolutionary algorithm (MOEA) [15], multiobjective particle
swarm optimization [16,17], fuzzy clustering-based particle swarm
optimization (FCPSO) [18], etc., constitute the pioneering multiob-
jective approaches that have been applied to solve the EED problem.
Differential evolution (DE) [21,22], a relatively new member
in the family of evolutionary algorithms, was first proposed over
1994–1996 by Storn and Price at Berkeley as a novel approach to
numerical optimization. It is a population-based method and gen-
erally considered as a parallel stochastic direct search optimizer
which is very simple yet powerful. DE has been applied successfully
to various fields of power system optimization [23–26].
This paper proposes multiobjective differential evolution
(MODE) for economic environmental dispatch (EED) problem. This
problem is formulated as a nonlinear constrained multiobjective
optimization problem. In order to show the effectiveness of the pro-
posed approach, three test systems are used in this paper. Results
obtained from the proposed approach have been compared with
those obtained from pareto differential evolution (PDE), nondom-
inated sorting genetic algorithm-II (NSGA-II) and strength pareto
evolutionary algorithm 2 (SPEA 2).
2. Notation
Pi power output of ith unit
Pmin
i
, Pmax
i
lower and upper generation limits for ith unit
PD load demand
PL transmission line losses
Bij transmission loss coefficient
ai, bi, ci, di, ei cost coefficients of ith unit
˛i, ˇi, i, i, ıi emission coefficients of ith unit
N number of generating units
3. Problem formulation
The present formulation treats economic environmental
dispatch (EED) problem as a multi-objective mathematical pro-
gramming problem which attempts to optimize both cost and
emission simultaneously, while satisfying both equality and
inequality constraints. The following objectives and constraints are
taken into account in the formulation of EED problem.
3.1. Objectives
(i) Cost
The fuel cost function of each fossil fuel fired generator, con-
sidering the valve-point effect [27], is expressed as the sum of a
quadratic and a sinusoidal function. The total fuel cost in terms
of real power output can be expressed as
F =
N
i=1
ai + biPi + ciP2
i
+
di sin{ei(Pmin
i
− Pi)}
(1)
(ii) Emission
The atmospheric pollutants such as sulfur oxides (SOx), nitro-
gen oxides (NOx) and carbon dioxide (CO2) caused by fossil-fuel
fired generator can be modeled separately. However, for com-
parison purposes, the total emission of these pollutants which
is the sum of a quadratic and an exponential function [28] can
be expressed as
E =
N
i=1
˛i + ˇiPi + iP2
i
+ i exp(ıiPi)
(2)
3.2. Constraints
(i) Real power balance constraint
The total real power generation must balance the predicted
power demand plus the real power losses in the transmission
lines.
N
i=1
Pi − PD − PL = 0 (3)
where PL is calculated by using B coefficients which can be
expressed in the quadratic form as follows:
PL =
N
i=1
N
j=1
PiBijPj (4)
(ii) Real power operating limits
Pmin
i
≤ Pi ≤ Pmax
i
, i ∈ N (5)
4. Determination of generation level of slack generator
N committed generators deliver their power output subject to
the power balance constraint (3) and the respective capacity con-
straints (5). Assuming the power loading of first (N − 1) generators
are known, the power level of the Nth generator (i.e. the slack gen-
erator) is given by
PN = PD + PL −
N−1
i=1
Pi (6)
The transmission loss PL is a function of all generator outputs
including the slack generator and it is given by
PL =
N−1
i=1
N−1
j=1
PiBijPj + 2PN
N−1
i=1
BNiPi
+ BNNP2
N
+
N−1
i=1
B0iPi + B0NPN + B00 (7)
Expanding and rearranging, Eq. (6) becomes
BNNP2
N +
2
N−1
i=1
BNiPi + B0N − 1
PN
+
⎛
⎝PD +
N−1
i=1
N−1
j=1
PiBijPj +
N−1
i=1
B0iPi −
N−1
i=1
Pi + B00
= 0 (8)
The loading of the slack generator (i.e. Nth) can then be found by
solving Eq. (8) using standard algebraic method.
5. Principle of multiobjective optimization
Most of the real-world problems involve simultaneous opti-
mization of several objective functions. These functions are non-
commensurable and often competing and conflicting objectives.
Multi-objective optimization having such conflicting objective
functions gives rise to a set of optimal solutions, instead of one
optimal solution because no solution can be considered to be bet-
ter than any other with respect to all objective functions. These
optimal solutions are known as pareto-optimal solutions.
Generally, multi-objective optimization problem consisting of
a number of objectives and several equality and inequality con-](https://image.slidesharecdn.com/basu2011-220705064227-5a18a9ba/85/basu2011-pdf-2-320.jpg)
![M. Basu / Applied Soft Computing 11 (2011) 2845–2853 2847
straints can be formulated as follows:
Minimize fi(x), i = 1, . . . , Nobj (9)
Subject to
gk(x) = 0 k = 1, . . . , K
hl(x) ≤ 0 l = 1, . . . , L
(10)
where fi is the ith objective function, x is a decision vector.
6. Multi-objective differential evolution
Differential evolution (DE) is a type of evolutionary algorithm
[21,22] for optimization problems over a continuous domain. DE is
exceptionally simple, significantly faster and robust. The basic idea
of DE is to adapt the search during the evolutionary process. At
the start of the evolution, the perturbations are large since parent
populations are far away from each other. As the evolutionary pro-
cess matures, the population converges to a small region and the
perturbations adaptively become small. As a result, the evolution-
ary algorithm performs a global exploratory search during the early
stages of the evolutionary process and local exploitation during the
mature stage of the search. In DE the fittest of an offspring competes
one-to-one with that of corresponding parent which is different
from other evolutionary algorithms. This one-to-one competition
gives rise to faster convergence rate. In multi-objective differential
evolution (MODE) [19,20], a pareto-based approach is introduced
to implement the selection of the best individuals. Firstly, a pop-
ulation of size, NP, is generated randomly and objective functions
are evaluated. At a given generation of the evolutionary search, the
population is sorted into several ranks based on non-domination.
Secondly, DE operations are carried out over the individuals of the
population. Trial vectors of size NP are generated and objective
functions are evaluated. Both the parent vectors and trial vectors
are combined to form a population of size 2NP. Then, the ranking
of the combined population is carried out followed by the crowd-
ing distance calculation. The best NP individuals are selected based
on its ranking and crowding distance. These individuals act as the
parent vectors for the next generation. The algorithm of MODE can
be described in the following steps:
Step 1. Generate box, R, of size NP. Parent vectors of size NP is ran-
domly generated and kept in R.
Step 2. Classify these vectors into fronts based on nondomination
[20] as follows:
(a) Create new empty box R/ of size NP.
(b) Compare each vector with all other vectors in R.
(c) Start with i = 1.
(d) If ith vector is not dominated by any other vector in R,
keep ith vector in R/ and go to (f).
(e) If ith vector is dominated by any other vector in R, go to
(f).
(f) Increment i by one. If i ≤ NP, go to (d) otherwise go to
(g).
(g) R/ now contains a sub-box (of size ≤NP) of nondomi-
nated vectors, referred to as the first front or sub-box.
Assign it a rank number equal to one (Irank = 1).
(h) Create subsequent fronts or sub-boxes of R/ with the
vectors remaining in R and assign these Irank = 2, 3, . . ..
Finally, all NP vectors are in R/ into one or more fronts.
Step 3. To calculate the crowding distance, Ii,dist, for the ith vector
in any front, F, of R/, sort all the vectors in front, F, according
to each objective function value in ascending order of mag-
nitude. The crowding distance of the ith vector in its front
F is the average side-length of the cuboid formed by using
the nearest neighbors as the vertices. Assign large values
of crowding distance Idist to the boundary vectors (vectors
with smallest and largest function values).
The following procedure is adopted to identify the bet-
ter of the two vectors. Vector i is better than vector j (i) if
Ii,rank Ij,rank or (ii) if Ii,rank = Ij,rank and Ii,dist Ij,dist.
Step 4. Take a new empty box R// of size NP. Perform DE operations
over NP vectors in R/ to generate NP trial vectors and store
these vectors in R//.
(a) Select a target vector, i in R/.
(b) Start with i = 1.
(c) Choose two vectors, r1 and r2 at random from the NP
vectors in R/. Find the vector difference between these
two vectors and multiply this difference with the scaling
factor Fs to get the weighted difference.
(d) Choose a third random vector r3 from the NP vectors in R/
and add this vector to the weighted difference to obtain
the noisy random vector.
(e) Perform crossover between the target vector and noisy
random vector to find the trial vector. This is carried out
by generating a random number and if random number
CR (crossover factor), copy the target vector into the
trial vector else copy the noisy random vector into the
trial vector and put it in box R//.
(f) Increment i by one. If i ≤ NP, go to (c) otherwise go to
Step 5.
Step 5. Copy all NP parent vectors from R/ and all NP trial vectors
from R// into box R///. Box R/// has 2NP vectors.
(a) Classify these 2NP vectors into fronts based on non-
domination and calculate the crowding distance of each
vector. Take the best NP vectors from box R/// and put
into box R////.
This completes one generation. Stop if generation number is
equal to maximum number of generations. Else copy NP vectors
from box R//// to the starting box R and go to Step 2.
7. Parameter selection
In differential evolution, scaling factor (SF), crossover constant
(CR), and population size (NP) are the three control parameters.
Proper selection of control parameters is important for the perfor-
mance of the algorithm. The scaling factor controls the amount of
perturbation in the mutation process. Its value lies in the range of [0,
1.2]. Lower value of SF results premature convergence while higher
value of SF tends to slow down convergence speed. The crossover
constant whose value is in the range of [0, 1], controls the diversity
of the population. The diversity for searching the solution space
depends on the population size. However, large population slows
down convergence speed.
The optimal control parameters are problem dependent. Gener-
ally, parameter tuning is used to select control parameters. Through
testing, parameter tuning adjusts the control parameters until the
best settings are determined.
8. Implementation of MODE for EED problem
In this section, an algorithm based on MODE for solving EED
problem is described below.
8.1. Initialization
Let pk = [P1, P2, . . ., Pi, . . ., PN]/ be the kth parent population to be
evolved and k = 1, 2, . . ., NP. The elements of pk are real power out-
puts of the committed N generating units. The real power output
of the ith unit is determined by setting Pi∼U(Pmin
i
, Pmax
i
), where
i = 1, 2, . . ., N. U(Pmin
i
, Pmax
i
) denotes a uniform random variable
ranging over [Pmin
i
, Pmax
i
]. Each population should satisfy the con-](https://image.slidesharecdn.com/basu2011-220705064227-5a18a9ba/85/basu2011-pdf-3-320.jpg)
![2848 M. Basu / Applied Soft Computing 11 (2011) 2845–2853
Table 1
Generation, cost, emission and CPU time of six-unit system for PD = 1200 MW.
DE MODE PDE NSGA-II SPEA 2
Economic dispatch Emission dispatch EED EED EED EED
P1 (MW) 84.4354 125.0000 108.6284 107.3965 113.1259 104.1573
P2 (MW) 93.3638 150.0000 115.9456 122.1418 116.4488 122.9807
P3 (MW) 225.0000 201.1816 206.7969 206.7536 217.4191 214.9553
P4 (MW) 209.9995 199.5454 210.0000 203.7047 207.9492 203.1387
P5 (MW) 325.0000 287.6191 301.8884 308.1045 304.6641 316.0302
P6 (MW) 314.9998 286.8137 308.4127 303.3797 291.5969 289.9396
Cost ($) 64,083 65,991 64,843 64,920 64,962 64,884
Emission (lb) 1345.6 1240.7 1286.0 1281.0 1281.0 1285
CPU time (s) 8.32 8.56 3.09 3.52 5.42 7.05
straints given by (3), and (5). Evaluate the value of F and E of each
population.
8.2. Classification of parent population
Classify these NP populations into fronts based on nondomi-
nation and calculate the crowding distance of each population as
described in Section 6. Sort these NP populations according to front
level and crowding distance.
8.3. Mutation
For each target population, a noisy population is generated by
adding the weighted difference vector between two parent popu-
lation members to a third member. For each target population p
g
k
at gth generation the noisy population p
/g
k
is obtained by
p
/g
k
= p
g
r1
+ SF (p
g
r2
− p
g
r3
), k ∈ NP (11)
where p
g
r1
, p
g
r2
and p
g
r3
are selected randomly from NP populations at
gth generation and r1 /
= r2 /
= r3 /
= k. The noisy population should
satisfy the constraints given by (3), and (5). Calculate the value of F
and E of each noisy population.
8.4. Crossover
Perform crossover for each target population p
g
k
with its noisy
population p
/g
k
and create a trial population p
//g
k
such that
p
//g
k
=
p
/g
k
, if ≤ CR
p
g
i
, otherwise
, i ∈ NP (12)
where is an uniformly distributed random number within [0, 1].
8.5. Classification of combined populations
Now, there are NP parent population and NP trial population.
Total populations are 2NP. Classify these 2NP populations into fronts
based on nondomination and calculate the crowding distance of
each population as described in Section 5. Select the best NP popu-
lation.
This completes one generation. Stop if generation number is
equal to maximum number of generations. Else copy best NP pop-
ulation and go to 8.3.
9. Simulation results
The proposed method has been applied to three test systems. In
order to show the effectiveness of the proposed MODE approach,
PDE, NSGA-II and SPEA 2 have been applied to solve the prob-
lem. The proposed MODE, DE, PDE, NSGA-II and SPEA 2 algorithms
have been implemented in MATLAB 7 on a PC (Pentium-IV, 80 GB,
3.0 GHz).
Fuel cost and emission objectives are minimized individually
by using differential evolution (DE) in order to explore the extreme
points of the trade-off surface. In DE, population size, maximum
number of generation, scaling factor and crossover factor have been
selected as 200, 100, 0.75 and 1.0 respectively for all three test
systems under consideration.
MODE has been applied to optimize both cost and emission
objectives simultaneously. For comparison, PDE, NSGA-II and SPEA
2 have been applied to solve EED problem. The population size,
maximum number of generation, scaling factor and crossover fac-
tor have been selected as 20, 30, 0.75 and 1.0 in the proposed MODE
and PDE for all three test systems.
In case of NSGA-II and SPEA 2, the population size, maximum
number of generations, crossover and mutation probabilities have
been selected as 20, 30, 0.9 and 0.2 for all the three test systems.
9.1. Test System 1
This test system consists of six generating units with quadratic
cost and emission level functions. Unit data and loss coefficients
have been found in Appendix A.
It is seen that under the cost minimization criterion, fuel cost
is 64,083 $ and emission is 1345.6 lb. But cost increases to 65,991 $
and emission decreases to 1240.7 lb for the case of emission mini-
mization. MODE has been applied to solve EED problem. In case of
MODE, fuel cost is 64,843 $ which is more than 64,083 $ and less
than 65,991 $ and emission is 1286.0 lb which is less than 1345.6 lb
and more than 1240.7 lb. Same problem has been solved by using
PDE, NSGA-II and SPEA 2. Results obtained from DE, proposed
MODE and best compromise solution of last generation obtained
from PDE, NSGA-II and SPEA 2 are shown in Table 1.
Fig. 1 depicts cost and emission convergence for six-unit system.
The distribution of 20 nondominated solutions obtained in the last
generation of proposed MODE, PDE, NSGA-II and SPEA2 for six-unit
system is shown in Fig. 2. It is seen from Fig. 2 that no objective (i.e.
cost or emission) can be further improved without degrading the
other.
9.2. Test System 2
This test system consists of ten generating units with nons-
mooth fuel cost and emission level functions. Unit data and loss
coefficients have been given in Appendix A.
During cost minimization, fuel cost is 1.11500 × 105 $ and
emission is 4581.00 lb. But cost increases to 1.16400 × 105 $ and
emission decreases to 3923.40 lb in case of emission minimiza-
tion. In case of EED by using MODE, fuel cost is 1.1348 × 105 $
which is more than 1.11500 × 105 $ and less than 1.16400 × 105 $
and emission is 4124.90 lb which is less than 4581.00 lb and more
than 3923.40 lb. Results obtained from DE, proposed MODE and](https://image.slidesharecdn.com/basu2011-220705064227-5a18a9ba/85/basu2011-pdf-4-320.jpg)
![2852 M. Basu / Applied Soft Computing 11 (2011) 2845–2853
Table A3 (Continued )
Unit Pmax
i
(MW) Pmin
i
(MW) ai ($/h) bi ($/MWh) ci ($/(MW)2
h) di ($/h) ei (rad/MW) ˛i (ton/h) ˇi (ton/MWh) i (ton/(MW)2
h) i (ton/h) ıi (1/MW)
28 10 150 1055.100 3.33 0.52124 120 0.077 360 −1.11 1.8420 0.9936 0.04060
29 10 150 1055.100 3.33 0.52124 120 0.077 360 −1.11 1.8420 0.9936 0.04060
30 47 97 148.890 5.35 0.01140 120 0.077 50 −1.89 0.0850 0.9936 0.04060
31 60 190 222.920 6.43 0.00160 150 0.063 80 −2.08 0.0121 0.9142 0.04540
32 60 190 222.920 6.43 0.00160 150 0.063 80 −2.08 0.0121 0.9142 0.04540
33 60 190 222.920 6.43 0.00160 150 0.063 80 −2.08 0.0121 0.9142 0.04540
34 90 200 107.870 8.95 0.00010 200 0.042 65 −3.48 0.0012 0.6550 0.02846
35 90 200 116.580 8.62 0.00010 200 0.042 70 −3.24 0.0012 0.6550 0.02846
36 90 200 116.580 8.62 0.00010 200 0.042 70 −3.24 0.0012 0.6550 0.02846
37 25 110 307.450 5.88 0.01610 80 0.098 100 −1.98 0.0950 1.4200 0.06770
38 25 110 307.450 5.88 0.01610 80 0.098 100 −1.98 0.0950 1.4200 0.06770
39 25 110 307.450 5.88 0.01610 80 0.098 100 −1.98 0.0950 1.4200 0.06770
40 242 550 647.830 7.97 0.00313 300 0.035 220 −2.68 0.0151 0.5035 0.02075
The transmission loss formula coefficients of six-unit system
are:
B =
⎡
⎢
⎢
⎢
⎢
⎣
0.000140 0.000017 0.000015 0.000019 0.000026 0.000022
0.000017 0.000060 0.000013 0.000016 0.000015 0.000020
0.000015 0.000013 0.000065 0.000017 0.000024 0.000019
0.000019 0.000016 0.000017 0.000071 0.000030 0.000025
0.000026 0.000015 0.000024 0.000030 0.000069 0.000032
0.000022 0.000020 0.000019 0.000025 0.000032 0.000085
⎤
⎥
⎥
⎥
⎥
⎦
The transmission loss formula coefficients of ten-unit system
are:
B =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0.000049 0.000014 0.000015 0.000015 0.000016 0.000017 0.000017 0.000018 0.000019 0.000020
0.000014 0.000045 0.000016 0.000016 0.000017 0.000015 0.000015 0.000016 0.000018 0.000018
0.000015 0.000016 0.000039 0.000010 0.000012 0.000012 0.000014 0.000014 0.000016 0.000016
0.000015 0.000016 0.000010 0.000040 0.000014 0.000010 0.000011 0.000012 0.000014 0.000015
0.000016 0.000017 0.000012 0.000014 0.000035 0.000011 0.000013 0.000013 0.000015 0.000016
0.000017 0.000015 0.000012 0.000010 0.000011 0.000036 0.000012 0.000012 0.000014 0.000015
0.000017 0.000015 0.000014 0.000011 0.000013 0.000012 0.000038 0.000016 0.000016 0.000018
0.000018 0.000016 0.000014 0.000012 0.000013 0.000012 0.000016 0.000040 0.000015 0.000016
0.000019 0.000018 0.000016 0.000014 0.000015 0.000014 0.000016 0.000015 0.000042 0.000019
0.000020 0.000018 0.000016 0.000015 0.000016 0.000015 0.000018 0.000016 0.000019 0.000044
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
References
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- 1. Applied Soft Computing 11 (2011) 2845–2853 Contents lists available at ScienceDirect Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc Economic environmental dispatch using multi-objective differential evolution M. Basu∗ Department of Power Engineering, Jadavpur University, Block LB, Sector-3, Salt Lake City, Kolkata, West Bengal 700098, India a r t i c l e i n f o Article history: Received 12 September 2006 Received in revised form 18 August 2010 Accepted 28 November 2010 Available online 4 December 2010 Keywords: Economic environmental dispatch Multi-objective optimization Multi-objective differential evolution a b s t r a c t Economic environmental dispatch (EED) is an important optimization task in fossil fuel fired power plant operation for allocating generation among the committed units such that fuel cost and emission level are optimized simultaneously while satisfying all operational constraints. It is a highly constrained multiobjective optimization problem involving conflicting objectives with both equality and inequality constraints. In this paper, multi-objective differential evolution has been proposed to solve EED problem. Numerical results of three test systems demonstrate the capabilities of the proposed approach. Results obtained from the proposed approach have been compared to those obtained from pareto differential evolution, nondominated sorting genetic algorithm-II and strength pareto evolutionary algorithm 2. © 2010 Elsevier B.V. All rights reserved. 1. Introduction The generation of electricity from fossil fuel releases sulfur oxides (SOx), nitrogen oxides (NOx), and carbon dioxide (CO2) into atmosphere. Atmospheric pollution affects not only humans but also other life-forms such as animals, birds, fish and plants. It also causes damage to vegetation, acid rain, reducing visibility as well as causing global warming. The increased concern over environmen- tal protection and the passage of the clean air act amendments of 1990 have forced the power utilities to reduce their emissions [1]. So today’s concern is to produce electricity not only at the cheapest possible price, but also at minimum level of pollution. Several strategies have been proposed to reduce the atmo- spheric pollution [2]. These include installation of post combustion cleaning equipment, switching to low emission fuels, replacement of the aged fuel burners with cleaner ones, and dispatching with emission considerations. The first three options require installa- tion of new equipment and/or modification of the existing ones that involve considerable capital outlay and hence they can be considered as long-term options. So, latter option is preferred. The two objectives i.e. cost and emission are conflicting in nature and they both have to be considered simultaneously to find overall optimal dispatch. Economic environmental dispatch (EED) serves to schedule the committed generator outputs with the predicted load demand so as to optimize both cost and emission simultane- ously while fulfilling the operating constraints. It is a multiobjective optimization problem with conflicting objectives because emission minimization is conflicting with minimum cost of generation. ∗ Tel.: +91 23355813. E-mail address: mousumibasu@yahoo.com Different techniques have been reported in the literature per- taining to EED problem. Nanda et al. treated EED as a multiple, conflicting objective problem and solved using goal-programming techniques [3]. In Ref. [4] a linear programming based optimization procedure has been presented in which the objectives are consid- ered one at a time. However, many mathematical assumptions have to be given to simplify the problem. In past decades, the EED prob- lem was converted to a single objective problem by linear combina- tion of different objectives as a weighted sum [5,6]. The important aspect of this weighted sum method is that a set of non-inferior solutions can be obtained by varying the weights. Unfortunately, this method cannot be used in problems having a non-convex pareto-optimal front. To avoid this difficulty, the -constraint method has been presented in Refs. [7,8]. This method optimizes the most preferred objective and considers the other objectives as constraints bounded by some allowable levels. The most obvi- ous weaknesses of this approach are that it is time-consuming and tends to find weakly non-dominated solutions. A fuzzy multiobjec- tive optimization technique for the EED problem has been proposed in [9]. However, the solutions are sub-optimal and the algorithm does not provide systematic framework for directing the search toward pareto-optimal front. A fuzzy satisfaction-maximizing deci- sion approach was successfully applied to solve the biobjective EED problem [10]. However, extension of the approach to include more objectives is a very involved question. An evolutionary algorithm based approach for evaluating the economic impacts of environ- mental dispatching and fuel switching has been presented in [11]. A multiobjective stochastic search technique for the EED problem has been presented in [12]. However, this technique is computationally involved and time consuming. Over the past few years, several researches have been made on the development of multiobjective evolutionary search 1568-4946/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2010.11.014
- 2. 2846 M. Basu / Applied Soft Computing 11 (2011) 2845–2853 strategies. Strength Pareto evolutionary algorithm (SPEA) [13], nondominating sorting genetic algorithm II (NSGA II) [14], multiob- jective evolutionary algorithm (MOEA) [15], multiobjective particle swarm optimization [16,17], fuzzy clustering-based particle swarm optimization (FCPSO) [18], etc., constitute the pioneering multiob- jective approaches that have been applied to solve the EED problem. Differential evolution (DE) [21,22], a relatively new member in the family of evolutionary algorithms, was first proposed over 1994–1996 by Storn and Price at Berkeley as a novel approach to numerical optimization. It is a population-based method and gen- erally considered as a parallel stochastic direct search optimizer which is very simple yet powerful. DE has been applied successfully to various fields of power system optimization [23–26]. This paper proposes multiobjective differential evolution (MODE) for economic environmental dispatch (EED) problem. This problem is formulated as a nonlinear constrained multiobjective optimization problem. In order to show the effectiveness of the pro- posed approach, three test systems are used in this paper. Results obtained from the proposed approach have been compared with those obtained from pareto differential evolution (PDE), nondom- inated sorting genetic algorithm-II (NSGA-II) and strength pareto evolutionary algorithm 2 (SPEA 2). 2. Notation Pi power output of ith unit Pmin i , Pmax i lower and upper generation limits for ith unit PD load demand PL transmission line losses Bij transmission loss coefficient ai, bi, ci, di, ei cost coefficients of ith unit ˛i, ˇi, i, i, ıi emission coefficients of ith unit N number of generating units 3. Problem formulation The present formulation treats economic environmental dispatch (EED) problem as a multi-objective mathematical pro- gramming problem which attempts to optimize both cost and emission simultaneously, while satisfying both equality and inequality constraints. The following objectives and constraints are taken into account in the formulation of EED problem. 3.1. Objectives (i) Cost The fuel cost function of each fossil fuel fired generator, con- sidering the valve-point effect [27], is expressed as the sum of a quadratic and a sinusoidal function. The total fuel cost in terms of real power output can be expressed as F = N i=1 ai + biPi + ciP2 i + di sin{ei(Pmin i − Pi)} (1) (ii) Emission The atmospheric pollutants such as sulfur oxides (SOx), nitro- gen oxides (NOx) and carbon dioxide (CO2) caused by fossil-fuel fired generator can be modeled separately. However, for com- parison purposes, the total emission of these pollutants which is the sum of a quadratic and an exponential function [28] can be expressed as E = N i=1 ˛i + ˇiPi + iP2 i + i exp(ıiPi) (2) 3.2. Constraints (i) Real power balance constraint The total real power generation must balance the predicted power demand plus the real power losses in the transmission lines. N i=1 Pi − PD − PL = 0 (3) where PL is calculated by using B coefficients which can be expressed in the quadratic form as follows: PL = N i=1 N j=1 PiBijPj (4) (ii) Real power operating limits Pmin i ≤ Pi ≤ Pmax i , i ∈ N (5) 4. Determination of generation level of slack generator N committed generators deliver their power output subject to the power balance constraint (3) and the respective capacity con- straints (5). Assuming the power loading of first (N − 1) generators are known, the power level of the Nth generator (i.e. the slack gen- erator) is given by PN = PD + PL − N−1 i=1 Pi (6) The transmission loss PL is a function of all generator outputs including the slack generator and it is given by PL = N−1 i=1 N−1 j=1 PiBijPj + 2PN N−1 i=1 BNiPi + BNNP2 N + N−1 i=1 B0iPi + B0NPN + B00 (7) Expanding and rearranging, Eq. (6) becomes BNNP2 N + 2 N−1 i=1 BNiPi + B0N − 1 PN + ⎛ ⎝PD + N−1 i=1 N−1 j=1 PiBijPj + N−1 i=1 B0iPi − N−1 i=1 Pi + B00 = 0 (8) The loading of the slack generator (i.e. Nth) can then be found by solving Eq. (8) using standard algebraic method. 5. Principle of multiobjective optimization Most of the real-world problems involve simultaneous opti- mization of several objective functions. These functions are non- commensurable and often competing and conflicting objectives. Multi-objective optimization having such conflicting objective functions gives rise to a set of optimal solutions, instead of one optimal solution because no solution can be considered to be bet- ter than any other with respect to all objective functions. These optimal solutions are known as pareto-optimal solutions. Generally, multi-objective optimization problem consisting of a number of objectives and several equality and inequality con-
- 3. M. Basu / Applied Soft Computing 11 (2011) 2845–2853 2847 straints can be formulated as follows: Minimize fi(x), i = 1, . . . , Nobj (9) Subject to gk(x) = 0 k = 1, . . . , K hl(x) ≤ 0 l = 1, . . . , L (10) where fi is the ith objective function, x is a decision vector. 6. Multi-objective differential evolution Differential evolution (DE) is a type of evolutionary algorithm [21,22] for optimization problems over a continuous domain. DE is exceptionally simple, significantly faster and robust. The basic idea of DE is to adapt the search during the evolutionary process. At the start of the evolution, the perturbations are large since parent populations are far away from each other. As the evolutionary pro- cess matures, the population converges to a small region and the perturbations adaptively become small. As a result, the evolution- ary algorithm performs a global exploratory search during the early stages of the evolutionary process and local exploitation during the mature stage of the search. In DE the fittest of an offspring competes one-to-one with that of corresponding parent which is different from other evolutionary algorithms. This one-to-one competition gives rise to faster convergence rate. In multi-objective differential evolution (MODE) [19,20], a pareto-based approach is introduced to implement the selection of the best individuals. Firstly, a pop- ulation of size, NP, is generated randomly and objective functions are evaluated. At a given generation of the evolutionary search, the population is sorted into several ranks based on non-domination. Secondly, DE operations are carried out over the individuals of the population. Trial vectors of size NP are generated and objective functions are evaluated. Both the parent vectors and trial vectors are combined to form a population of size 2NP. Then, the ranking of the combined population is carried out followed by the crowd- ing distance calculation. The best NP individuals are selected based on its ranking and crowding distance. These individuals act as the parent vectors for the next generation. The algorithm of MODE can be described in the following steps: Step 1. Generate box, R, of size NP. Parent vectors of size NP is ran- domly generated and kept in R. Step 2. Classify these vectors into fronts based on nondomination [20] as follows: (a) Create new empty box R/ of size NP. (b) Compare each vector with all other vectors in R. (c) Start with i = 1. (d) If ith vector is not dominated by any other vector in R, keep ith vector in R/ and go to (f). (e) If ith vector is dominated by any other vector in R, go to (f). (f) Increment i by one. If i ≤ NP, go to (d) otherwise go to (g). (g) R/ now contains a sub-box (of size ≤NP) of nondomi- nated vectors, referred to as the first front or sub-box. Assign it a rank number equal to one (Irank = 1). (h) Create subsequent fronts or sub-boxes of R/ with the vectors remaining in R and assign these Irank = 2, 3, . . .. Finally, all NP vectors are in R/ into one or more fronts. Step 3. To calculate the crowding distance, Ii,dist, for the ith vector in any front, F, of R/, sort all the vectors in front, F, according to each objective function value in ascending order of mag- nitude. The crowding distance of the ith vector in its front F is the average side-length of the cuboid formed by using the nearest neighbors as the vertices. Assign large values of crowding distance Idist to the boundary vectors (vectors with smallest and largest function values). The following procedure is adopted to identify the bet- ter of the two vectors. Vector i is better than vector j (i) if Ii,rank Ij,rank or (ii) if Ii,rank = Ij,rank and Ii,dist Ij,dist. Step 4. Take a new empty box R// of size NP. Perform DE operations over NP vectors in R/ to generate NP trial vectors and store these vectors in R//. (a) Select a target vector, i in R/. (b) Start with i = 1. (c) Choose two vectors, r1 and r2 at random from the NP vectors in R/. Find the vector difference between these two vectors and multiply this difference with the scaling factor Fs to get the weighted difference. (d) Choose a third random vector r3 from the NP vectors in R/ and add this vector to the weighted difference to obtain the noisy random vector. (e) Perform crossover between the target vector and noisy random vector to find the trial vector. This is carried out by generating a random number and if random number CR (crossover factor), copy the target vector into the trial vector else copy the noisy random vector into the trial vector and put it in box R//. (f) Increment i by one. If i ≤ NP, go to (c) otherwise go to Step 5. Step 5. Copy all NP parent vectors from R/ and all NP trial vectors from R// into box R///. Box R/// has 2NP vectors. (a) Classify these 2NP vectors into fronts based on non- domination and calculate the crowding distance of each vector. Take the best NP vectors from box R/// and put into box R////. This completes one generation. Stop if generation number is equal to maximum number of generations. Else copy NP vectors from box R//// to the starting box R and go to Step 2. 7. Parameter selection In differential evolution, scaling factor (SF), crossover constant (CR), and population size (NP) are the three control parameters. Proper selection of control parameters is important for the perfor- mance of the algorithm. The scaling factor controls the amount of perturbation in the mutation process. Its value lies in the range of [0, 1.2]. Lower value of SF results premature convergence while higher value of SF tends to slow down convergence speed. The crossover constant whose value is in the range of [0, 1], controls the diversity of the population. The diversity for searching the solution space depends on the population size. However, large population slows down convergence speed. The optimal control parameters are problem dependent. Gener- ally, parameter tuning is used to select control parameters. Through testing, parameter tuning adjusts the control parameters until the best settings are determined. 8. Implementation of MODE for EED problem In this section, an algorithm based on MODE for solving EED problem is described below. 8.1. Initialization Let pk = [P1, P2, . . ., Pi, . . ., PN]/ be the kth parent population to be evolved and k = 1, 2, . . ., NP. The elements of pk are real power out- puts of the committed N generating units. The real power output of the ith unit is determined by setting Pi∼U(Pmin i , Pmax i ), where i = 1, 2, . . ., N. U(Pmin i , Pmax i ) denotes a uniform random variable ranging over [Pmin i , Pmax i ]. Each population should satisfy the con-
- 4. 2848 M. Basu / Applied Soft Computing 11 (2011) 2845–2853 Table 1 Generation, cost, emission and CPU time of six-unit system for PD = 1200 MW. DE MODE PDE NSGA-II SPEA 2 Economic dispatch Emission dispatch EED EED EED EED P1 (MW) 84.4354 125.0000 108.6284 107.3965 113.1259 104.1573 P2 (MW) 93.3638 150.0000 115.9456 122.1418 116.4488 122.9807 P3 (MW) 225.0000 201.1816 206.7969 206.7536 217.4191 214.9553 P4 (MW) 209.9995 199.5454 210.0000 203.7047 207.9492 203.1387 P5 (MW) 325.0000 287.6191 301.8884 308.1045 304.6641 316.0302 P6 (MW) 314.9998 286.8137 308.4127 303.3797 291.5969 289.9396 Cost ($) 64,083 65,991 64,843 64,920 64,962 64,884 Emission (lb) 1345.6 1240.7 1286.0 1281.0 1281.0 1285 CPU time (s) 8.32 8.56 3.09 3.52 5.42 7.05 straints given by (3), and (5). Evaluate the value of F and E of each population. 8.2. Classification of parent population Classify these NP populations into fronts based on nondomi- nation and calculate the crowding distance of each population as described in Section 6. Sort these NP populations according to front level and crowding distance. 8.3. Mutation For each target population, a noisy population is generated by adding the weighted difference vector between two parent popu- lation members to a third member. For each target population p g k at gth generation the noisy population p /g k is obtained by p /g k = p g r1 + SF (p g r2 − p g r3 ), k ∈ NP (11) where p g r1 , p g r2 and p g r3 are selected randomly from NP populations at gth generation and r1 / = r2 / = r3 / = k. The noisy population should satisfy the constraints given by (3), and (5). Calculate the value of F and E of each noisy population. 8.4. Crossover Perform crossover for each target population p g k with its noisy population p /g k and create a trial population p //g k such that p //g k = p /g k , if ≤ CR p g i , otherwise , i ∈ NP (12) where is an uniformly distributed random number within [0, 1]. 8.5. Classification of combined populations Now, there are NP parent population and NP trial population. Total populations are 2NP. Classify these 2NP populations into fronts based on nondomination and calculate the crowding distance of each population as described in Section 5. Select the best NP popu- lation. This completes one generation. Stop if generation number is equal to maximum number of generations. Else copy best NP pop- ulation and go to 8.3. 9. Simulation results The proposed method has been applied to three test systems. In order to show the effectiveness of the proposed MODE approach, PDE, NSGA-II and SPEA 2 have been applied to solve the prob- lem. The proposed MODE, DE, PDE, NSGA-II and SPEA 2 algorithms have been implemented in MATLAB 7 on a PC (Pentium-IV, 80 GB, 3.0 GHz). Fuel cost and emission objectives are minimized individually by using differential evolution (DE) in order to explore the extreme points of the trade-off surface. In DE, population size, maximum number of generation, scaling factor and crossover factor have been selected as 200, 100, 0.75 and 1.0 respectively for all three test systems under consideration. MODE has been applied to optimize both cost and emission objectives simultaneously. For comparison, PDE, NSGA-II and SPEA 2 have been applied to solve EED problem. The population size, maximum number of generation, scaling factor and crossover fac- tor have been selected as 20, 30, 0.75 and 1.0 in the proposed MODE and PDE for all three test systems. In case of NSGA-II and SPEA 2, the population size, maximum number of generations, crossover and mutation probabilities have been selected as 20, 30, 0.9 and 0.2 for all the three test systems. 9.1. Test System 1 This test system consists of six generating units with quadratic cost and emission level functions. Unit data and loss coefficients have been found in Appendix A. It is seen that under the cost minimization criterion, fuel cost is 64,083 $ and emission is 1345.6 lb. But cost increases to 65,991 $ and emission decreases to 1240.7 lb for the case of emission mini- mization. MODE has been applied to solve EED problem. In case of MODE, fuel cost is 64,843 $ which is more than 64,083 $ and less than 65,991 $ and emission is 1286.0 lb which is less than 1345.6 lb and more than 1240.7 lb. Same problem has been solved by using PDE, NSGA-II and SPEA 2. Results obtained from DE, proposed MODE and best compromise solution of last generation obtained from PDE, NSGA-II and SPEA 2 are shown in Table 1. Fig. 1 depicts cost and emission convergence for six-unit system. The distribution of 20 nondominated solutions obtained in the last generation of proposed MODE, PDE, NSGA-II and SPEA2 for six-unit system is shown in Fig. 2. It is seen from Fig. 2 that no objective (i.e. cost or emission) can be further improved without degrading the other. 9.2. Test System 2 This test system consists of ten generating units with nons- mooth fuel cost and emission level functions. Unit data and loss coefficients have been given in Appendix A. During cost minimization, fuel cost is 1.11500 × 105 $ and emission is 4581.00 lb. But cost increases to 1.16400 × 105 $ and emission decreases to 3923.40 lb in case of emission minimiza- tion. In case of EED by using MODE, fuel cost is 1.1348 × 105 $ which is more than 1.11500 × 105 $ and less than 1.16400 × 105 $ and emission is 4124.90 lb which is less than 4581.00 lb and more than 3923.40 lb. Results obtained from DE, proposed MODE and
- 5. M. Basu / Applied Soft Computing 11 (2011) 2845–2853 2849 6.4 6.45 6.5 x 10 4 Cost ($) Generations 0 5 10 15 20 25 30 35 40 45 50 1240 1245 1250 Emission (lb) Fig. 1. Cost and emission convergence for six-unit system. 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 6.4 6.42 6.44 6.46 6.48 6.5 6.52 6.54 6.56 6.58 6.6 x 10 4 Cost ($) Emission (lb) MODE PDE NSGA-II SPEA2 Fig. 2. Pareto-optimal front of the last generation for six-unit system. best compromise solution of last generation obtained from PDE, NSGA-II and SPEA 2 are summarized in Table 2. Cost and emission convergence for ten-unit system are shown in Fig. 3. The distribution of 20 nondominated solutions obtained in the last generation of proposed MODE, PDE, NSGA-II and SPEA2 for ten-unit system is shown in Fig. 4. 1.11 1.12 1.13 1.14 1.15 1.16 x 10 5 Cost ($) Generations 0 10 20 30 40 50 60 70 80 90 100 3900 4000 4100 4200 4300 4400 Emission (lb) Fig. 3. Cost and emission convergence for ten-unit system. 3950 4000 4050 4100 4150 4200 4250 4300 4350 1.12 1.125 1.13 1.135 1.14 1.145 1.15 1.155 x 10 5 Cost ($) Emission (lb) MODE PDE NSGA-II SPEA2 Fig. 4. Pareto-optimal front of the last generation for ten-unit system. 9.3. Test System 3 This test system consists of forty generating units with nons- mooth fuel cost and emission level functions. Unit data has been presented in Table A3 in Appendix A. It is seen that under the cost minimization criterion, fuel cost is 1.2184 × 105 $ and emission is 3.7479 × 105 ton during cost min- imization. But during emission minimization, cost increases to Table 2 Generation, cost, emission and CPU time of ten-unit system for PD = 2000 MW. DE MODE PDE NSGA-II SPEA 2 Economic dispatch Emission dispatch EED EED EED EED P1 (MW) 55.0000 55.0000 54.9487 54.9853 51.9515 52.9761 P2 (MW) 79.8063 80.0000 74.5821 79.3803 67.2584 72.8130 P3 (MW) 106.8253 80.5924 79.4294 83.9842 73.6879 78.1128 P4 (MW) 102.8307 81.0233 80.6875 86.5942 91.3554 83.6088 P5 (MW) 82.2418 160.0000 136.8551 144.4386 134.0522 137.2432 P6 (MW) 80.4352 240.0000 172.6393 165.7756 174.9504 172.9188 P7 (MW) 300.0000 292.7434 283.8233 283.2122 289.4350 287.2023 P8 (MW) 340.0000 299.1214 316.3407 312.7709 314.0556 326.4023 P9 (MW) 470.0000 394.5147 448.5923 440.1135 455.6978 448.8814 P10 (MW) 469.8975 398.6383 436.4287 432.6783 431.8054 423.9025 Cost (×105 $) 1.11500 1.16400 1.1348 1.1351 1.1354 1.1352 Emission (lb) 4581.00 3923.40 4124.90 4111.40 4130.20 4109.10 CPU time (s) 9.42 8.56 3.82 4.23 6.02 7.53
- 6. 2850 M. Basu / Applied Soft Computing 11 (2011) 2845–2853 Table 3 Generation (MW), cost (×105 $), emission (×105 ton) and CPU time (s) of forty-unit system for PD = 10,500 MW. Unit DE MODE PDE NSGA-II SPEA 2 Economic dispatch Emission dispatch EED EED EED EED 1 110.9515 114.0000 113.5295 112.1549 113.8685 113.9694 2 113.2997 114.0000 114.0000 113.9431 113.6381 114.0000 3 98.6155 120.0000 120.0000 120.0000 120.0000 119.8719 4 184.1487 169.2933 179.8015 180.2647 180.7887 179.9284 5 86.4013 97.0000 96.7716 97.0000 97.0000 97.0000 6 140.0000 124.2828 139.2760 140.0000 140.0000 139.2721 7 300.0000 299.4564 300.0000 299.8829 300.0000 300.0000 8 285.4556 297.8554 298.9193 300.0000 299.0084 298.2706 9 297.5110 297.1332 290.7737 289.8915 288.8890 290.5228 10 130.0000 130.0000 130.9025 130.5725 131.6132 131.4832 11 168.7482 298.5980 244.7349 244.1003 246.5128 244.6704 12 95.6950 297.7226 317.8218 318.2840 318.8748 317.2003 13 125.0000 433.7471 395.3846 394.7833 395.7224 394.7357 14 394.3545 421.9529 394.4692 394.2187 394.1369 394.6223 15 305.5234 422.6280 305.8104 305.9616 305.5781 304.7271 16 394.7147 422.9508 394.8229 394.1321 394.6968 394.7289 17 489.7972 439.2581 487.9872 489.3040 489.4234 487.9857 18 489.3620 439.4411 489.1751 489.6419 488.2701 488.5321 19 520.9024 439.4908 500.5265 499.9835 500.8000 501.1683 20 510.6407 439.6189 457.0072 455.4160 455.2006 456.4324 21 524.5336 439.2250 434.6068 435.2845 434.6639 434.7887 22 526.6981 439.6821 434.5310 433.7311 434.1500 434.3937 23 530.7467 439.8757 444.6732 446.2496 445.8385 445.0772 24 526.3270 439.8937 452.0332 451.8828 450.7509 451.8970 25 525.6537 440.4401 492.7831 493.2259 491.2745 492.3946 26 522.9497 439.8408 436.3347 434.7492 436.3418 436.9926 27 10.0000 28.7758 10.0000 11.8064 11.2457 10.7784 28 11.5522 29.0747 10.3901 10.7536 10.0000 10.2955 29 10.0000 28.9047 12.3149 10.3053 12.0714 13.7018 30 89.9076 97.0000 96.9050 97.0000 97.0000 96.2431 31 190.0000 172.4036 189.7727 190.0000 189.4826 190.0000 32 190.0000 172.3956 174.2324 175.3065 174.7971 174.2163 33 190.0000 172.3144 190.0000 190.0000 189.2845 190.0000 34 198.8403 200.0000 199.6506 200.0000 200.0000 200.0000 35 174.1783 200.0000 199.8662 200.0000 199.9138 200.0000 36 197.1598 200.0000 200.0000 200.0000 199.5066 200.0000 37 110.0000 100.8765 110.0000 109.9412 108.3061 110.0000 38 109.3565 100.9000 109.9454 109.8823 110.0000 109.6912 39 110.0000 100.7784 108.1786 108.9686 109.7899 108.5560 40 510.9752 439.1894 422.0682 421.3778 421.5609 421.8521 Cost 1.2184 1.2996 1.2579 1.2573 1.2583 1.2581 Emission 3.7479 1.7668 2.1119 2.1177 2.1095 2.1110 CPU time 13.25 14.09 5.39 6.15 7.32 8.57 1.2996 × 105 $ and emission decreases to 1.7668 × 105 ton. MODE has been applied to solve EED problem. In case of EED by using MODE fuel cost is 1.2579 × 105 $ which is more than 1.2184 × 105 $ and less than 1.2996 × 105 $ and emission is 2.1119 × 105 ton 1.2 1.25 1.3 x 10 5 Cost ($) Generations 0 10 20 30 40 50 60 70 80 90 100 1 2 3 x 10 5 Emission (ton) Fig. 5. Cost and emission convergence for forty-unit system. which is less than 3.7479 × 105 ton and more than 1.7668 × 105 ton. Results obtained from DE, proposed MODE and best compromise solution of last generation obtained from PDE, NSGA-II and SPEA 2 are shown in Table 3. 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 x 10 5 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 x 10 5 Cost ($) Emission (ton) MODE PDE NSGA-II SPEA2 Fig. 6. Pareto-optimal front of the last generation for forty-unit system.
- 7. M. Basu / Applied Soft Computing 11 (2011) 2845–2853 2851 Cost and emission convergence of this system are shown in Fig. 5. The distribution of 20 nondominated solutions obtained in the last generation of proposed MODE, PDE, NSGA-II and SPEA2 for forty-unit system is shown in Fig. 6. It is seen from Fig. 6 that cost or emission cannot be further improved without degrading the other. 10. Conclusion In this paper, multi-objective differential evolution has been presented for solving economic environmental dispatch problem. The problem has been formulated as multi-objective optimiza- tion problem with competing fuel cost and emission objectives. Results obtained from the proposed approach have been compared with those obtained from pareto differential evolution, strength pareto evolutionary algorithm 2 and nondominated sorting genetic algorithm-II. It is seen from the comparison that the proposed approach provides a competitive performance in terms of solution as well as computation time. The proposed multi-objective differ- ential evolution is simple, robust and efficient. It does not impose any limitation on the number of objectives and can be extended to include more objectives. Appendix A. See Tables A1–A3. Table A1 Six-unit generator characteristics. Unit Pmax i (MW) Pmin i (MW) ai ($/h) bi ($/MWh) ci ($/(MW)2 h) ˛i (lb/h) ˇi (lb/MWh) i (lb/(MW)2 h) 1 10 125 756.7988 38.5390 0.15247 13.8593 0.32767 0.00419 2 10 150 451.3251 46.1591 0.10587 13.8593 0.32767 0.00419 3 35 210 1243.5311 38.3055 0.03546 40.2669 −0.54551 0.00683 4 35 225 1049.9977 40.3965 0.02803 40.2669 −0.54551 0.00683 5 125 315 1356.6592 38.2704 0.01799 42.8955 −0.51116 0.00461 6 130 325 1658.5696 36.3278 0.02111 42.8955 −0.51116 0.00461 Table A2 Ten-unit generator characteristics. Unit Pmax i (MW) Pmin i (MW) ai ($/h) bi ($/MWh) ci ($/(MW)2 h) di ($/h) ei (rad/MW) ˛i (lb/h) ˇi (lb/MWh) i (lb/(MW)2 h) i (lb/h) ıi (1/MW) 1 10 55 1000.403 40.5407 0.12951 33 0.0174 360.0012 −3.9864 0.04702 0.25475 0.01234 2 20 80 950.606 39.5804 0.10908 25 0.0178 350.0056 −3.9524 0.04652 0.25475 0.01234 3 47 120 900.705 36.5104 0.12511 32 0.0162 330.0056 −3.9023 0.04652 0.25163 0.01215 4 20 130 800.705 39.5104 0.12111 30 0.0168 330.0056 −3.9023 0.04652 0.25163 0.01215 5 50 160 756.799 38.5390 0.15247 30 0.0148 13.8593 0.3277 0.00420 0.24970 0.01200 6 70 240 451.325 46.1592 0.10587 20 0.0163 13.8593 0.3277 0.00420 0.24970 0.01200 7 60 300 1243.531 38.3055 0.03546 20 0.0152 40.2669 −0.5455 0.00680 0.24800 0.01290 8 70 340 1049.998 40.3965 0.02803 30 0.0128 40.2669 −0.5455 0.00680 0.24990 0.01203 9 135 470 1658.569 36.3278 0.02111 60 0.0136 42.8955 −0.5112 0.00460 0.25470 0.01234 10 150 470 1356.659 38.2704 0.01799 40 0.0141 42.8955 −0.5112 0.00460 0.25470 0.01234 Table A3 Forty-unit generator characteristics. Unit Pmax i (MW) Pmin i (MW) ai ($/h) bi ($/MWh) ci ($/(MW)2 h) di ($/h) ei (rad/MW) ˛i (ton/h) ˇi (ton/MWh) i (ton/(MW)2 h) i (ton/h) ıi (1/MW) 1 36 114 94.705 6.73 0.00690 100 0.084 60 −2.22 0.0480 1.3100 0.05690 2 36 114 94.705 6.73 0.00690 100 0.084 60 −2.22 0.0480 1.3100 0.05690 3 60 120 309.540 7.07 0.02028 100 0.084 100 −2.36 0.0762 1.3100 0.05690 4 80 190 369.030 8.18 0.00942 150 0.063 120 −3.14 0.0540 0.9142 0.04540 5 47 97 148.890 5.35 0.01140 120 0.077 50 −1.89 0.0850 0.9936 0.04060 6 68 140 222.330 8.05 0.01142 100 0.084 80 −3.08 0.0854 1.3100 0.05690 7 110 300 287.710 8.03 0.00357 200 0.042 100 −3.06 0.0242 0.6550 0.02846 8 135 300 391.980 6.99 0.00492 200 0.042 130 −2.32 0.0310 0.6550 0.02846 9 135 300 455.760 6.60 0.00573 200 0.042 150 −2.11 0.0335 0.6550 0.02846 10 130 300 722.820 12.9 0.00605 200 0.042 280 −4.34 0.4250 0.6550 0.02846 11 94 375 635.200 12.9 0.00515 200 0.042 220 −4.34 0.0322 0.6550 0.02846 12 94 375 654.690 12.8 0.00569 200 0.042 225 −4.28 0.0338 0.6550 0.02846 13 125 500 913.400 12.5 0.00421 300 0.035 300 −4.18 0.0296 0.5035 0.02075 14 125 500 1760.400 8.84 0.00752 300 0.035 520 −3.34 0.0512 0.5035 0.02075 15 125 500 1760.400 8.84 0.00752 300 0.035 510 −3.55 0.0496 0.5035 0.02075 16 125 500 1760.400 8.84 0.00752 300 0.035 510 −3.55 0.0496 0.5035 0.02075 17 220 500 647.850 7.97 0.00313 300 0.035 220 −2.68 0.0151 0.5035 0.02075 18 220 500 649.690 7.95 0.00313 300 0.035 222 −2.66 0.0151 0.5035 0.02075 19 242 550 647.830 7.97 0.00313 300 0.035 220 −2.68 0.0151 0.5035 0.02075 20 242 550 647.810 7.97 0.00313 300 0.035 220 −2.68 0.0151 0.5035 0.02075 21 254 550 785.960 6.63 0.00298 300 0.035 290 −2.22 0.0145 0.5035 0.02075 22 254 550 785.960 6.63 0.00298 300 0.035 285 −2.22 0.0145 0.5035 0.02075 23 254 550 794.530 6.66 0.00284 300 0.035 295 −2.26 0.0138 0.5035 0.02075 24 254 550 794.530 6.66 0.00284 300 0.035 295 −2.26 0.0138 0.5035 0.02075 25 254 550 801.320 7.10 0.00277 300 0.035 310 −2.42 0.0132 0.5035 0.02075 26 254 550 801.320 7.10 0.00277 300 0.035 310 −2.42 0.0132 0.5035 0.02075 27 10 150 1055.100 3.33 0.52124 120 0.077 360 −1.11 1.8420 0.9936 0.04060
- 8. 2852 M. Basu / Applied Soft Computing 11 (2011) 2845–2853 Table A3 (Continued ) Unit Pmax i (MW) Pmin i (MW) ai ($/h) bi ($/MWh) ci ($/(MW)2 h) di ($/h) ei (rad/MW) ˛i (ton/h) ˇi (ton/MWh) i (ton/(MW)2 h) i (ton/h) ıi (1/MW) 28 10 150 1055.100 3.33 0.52124 120 0.077 360 −1.11 1.8420 0.9936 0.04060 29 10 150 1055.100 3.33 0.52124 120 0.077 360 −1.11 1.8420 0.9936 0.04060 30 47 97 148.890 5.35 0.01140 120 0.077 50 −1.89 0.0850 0.9936 0.04060 31 60 190 222.920 6.43 0.00160 150 0.063 80 −2.08 0.0121 0.9142 0.04540 32 60 190 222.920 6.43 0.00160 150 0.063 80 −2.08 0.0121 0.9142 0.04540 33 60 190 222.920 6.43 0.00160 150 0.063 80 −2.08 0.0121 0.9142 0.04540 34 90 200 107.870 8.95 0.00010 200 0.042 65 −3.48 0.0012 0.6550 0.02846 35 90 200 116.580 8.62 0.00010 200 0.042 70 −3.24 0.0012 0.6550 0.02846 36 90 200 116.580 8.62 0.00010 200 0.042 70 −3.24 0.0012 0.6550 0.02846 37 25 110 307.450 5.88 0.01610 80 0.098 100 −1.98 0.0950 1.4200 0.06770 38 25 110 307.450 5.88 0.01610 80 0.098 100 −1.98 0.0950 1.4200 0.06770 39 25 110 307.450 5.88 0.01610 80 0.098 100 −1.98 0.0950 1.4200 0.06770 40 242 550 647.830 7.97 0.00313 300 0.035 220 −2.68 0.0151 0.5035 0.02075 The transmission loss formula coefficients of six-unit system are: B = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0.000140 0.000017 0.000015 0.000019 0.000026 0.000022 0.000017 0.000060 0.000013 0.000016 0.000015 0.000020 0.000015 0.000013 0.000065 0.000017 0.000024 0.000019 0.000019 0.000016 0.000017 0.000071 0.000030 0.000025 0.000026 0.000015 0.000024 0.000030 0.000069 0.000032 0.000022 0.000020 0.000019 0.000025 0.000032 0.000085 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ The transmission loss formula coefficients of ten-unit system are: B = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0.000049 0.000014 0.000015 0.000015 0.000016 0.000017 0.000017 0.000018 0.000019 0.000020 0.000014 0.000045 0.000016 0.000016 0.000017 0.000015 0.000015 0.000016 0.000018 0.000018 0.000015 0.000016 0.000039 0.000010 0.000012 0.000012 0.000014 0.000014 0.000016 0.000016 0.000015 0.000016 0.000010 0.000040 0.000014 0.000010 0.000011 0.000012 0.000014 0.000015 0.000016 0.000017 0.000012 0.000014 0.000035 0.000011 0.000013 0.000013 0.000015 0.000016 0.000017 0.000015 0.000012 0.000010 0.000011 0.000036 0.000012 0.000012 0.000014 0.000015 0.000017 0.000015 0.000014 0.000011 0.000013 0.000012 0.000038 0.000016 0.000016 0.000018 0.000018 0.000016 0.000014 0.000012 0.000013 0.000012 0.000016 0.000040 0.000015 0.000016 0.000019 0.000018 0.000016 0.000014 0.000015 0.000014 0.000016 0.000015 0.000042 0.000019 0.000020 0.000018 0.000016 0.000015 0.000016 0.000015 0.000018 0.000016 0.000019 0.000044 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ References [1] IEEE Current Operating Problems Working Group, in: Potential impacts of clean air regulations on system operations, IEEE Trans. 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