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Accepted Manuscript
Solution of combined economic and emission dispatch problem using a novel
chaotic improved harmony search algorithm
H. Rezaie, M.H. Kazemi-Rahbar, B. Vahidi, H. Rastegar
PII:
DOI:
Reference:
S2288-4300(18)30125-8
https://doi.org/10.1016/j.jcde.2018.08.001
JCDE 160
To appear in:
Journal of Computational Design and Engineering
Received Date:
Revised Date:
Accepted Date:
23 May 2018
9 July 2018
16 August 2018
Please cite this article as: H. Rezaie, M.H. Kazemi-Rahbar, B. Vahidi, H. Rastegar, Solution of combined economic
and emission dispatch problem using a novel chaotic improved harmony search algorithm, Journal of Computational
Design and Engineering (2018), doi: https://doi.org/10.1016/j.jcde.2018.08.001
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Solution of combined economic and emission dispatch problem using a novel chaotic
improved harmony search algorithm
H. Rezaie1, M. H. Kazemi-Rahbar2, B. Vahidi1,*, H. Rastegar1
1
Department of Electrical Engineering, Amirkabir University of Technology (AUT), 424 Hafez Ave, Tehran, Iran
2
Department of Electrical Engineering, Shahed University, Persian Gulf Freeway, Tehran, Iran
Emails: h.rezaie@aut.ac.ir, mhkazemirahbar@gmail.com, vahidi@aut.ac.ir, rastegar@aut.ac.ir
*Corresponding author
Abstract
This paper presents a new optimization technique developed based on harmony search algorithm (HSA), called
chaotic improved harmony search algorithm (CIHSA). In the proposed algorithm, the original HSA is improved
using several innovative modifications in the optimization procedure such as using chaotic patterns instead of
uniform distribution to generate random numbers, dynamically tuning the algorithm parameters, and employing
virtual harmony memories. Also, a novel type of local optimization is introduced and employed in the algorithm
procedure. Applying these modifications to HSA has resulted in enhancing the robustness, accuracy and search
efficiency of the algorithm, and significantly reducing the iterations number required to achieve the optimal solution.
To validate the effectiveness of CIHSA, it is used to solve the combined economic emission dispatch (CEED)
problem, which practically is a complex high-dimensional non-convex optimization task with several equality and
inequality constraints. Six test systems having 6, 10, 13, 14, 40, and 140 generators are investigated in this study,
and the valve-point loading effects, ramp rate limits and power transmission losses are also taken into account. The
results obtained by CIHSA are compared with the results reported in a large number of other research works.
Furthermore, the statistical data regarding the CIHSA performance in all test systems is presented. The numerical
and statistical results confirm the high quality of the solutions found by CIHSA and its superiority compared to other
existing techniques employed in solving CEED problems.
Keywords: Economic emission dispatch, environmental/economic dispatch, improved harmony search algorithm,
chaotic harmony search algorithm, evolutionary algorithm, valve-point loading effect
1
Introduction
Optimization has always been an integral part of engineering design and decision making procedures to
attain the maximum benefit at the minimum cost. Over the last few decades, a large number of
metaheuristic algorithms have been introduced to remedy the computational drawbacks of traditional
optimization techniques which predominantly are based on numerical linear and nonlinear programming
methods. Some of the most popular and well-known metaheuristic algorithms are genetic algorithm (GA),
particle swarm optimization (PSO), differential evolution (DE), harmony search algorithm (HSA), tabu
search (TS), simulated annealing (SA), ant colony optimization (ACO), artificial bee colony (ABC),
cuckoo search (CS), and krill herd algorithm (KHA) which is younger than others but has attracted a lot
of attention over a few past years. Also, a lot of research works can be found in the literature, which have
proposed modified variants of these algorithms or have combined them with other algorithms to enhance
their search efficiency. For instance, some of the outstanding modified algorithms suggested based on
PSO, HSA, and KHA can be found in [1-8], [9-16], and [17-25], respectively.
This paper introduces a new HSA-based optimization method called chaotic improved harmony search
algorithm (CIHSA). This algorithm is actually formed by combining two algorithms proposed in this
paper, called chaotic harmony search algorithm (CHSA) and improved harmony search algorithm
(IHSA). In CHSA, to avoid being trapped into local optima and better covering the search space, the
chaotic pattern is used instead of uniform distribution to generate random numbers in the optimization
procedure, which leads to the improving the algorithm robustness. In IHSA, some innovative
modifications are incorporated into the original algorithm procedure that significantly enhances the
accuracy and efficiency of HSA. These modifications are explained in details in section 4.1. According to
the results and statistical data obtained in this work, CIHSA can be considered as one of the most
successful evolutions of the original HSA performance.
To validate the effectiveness of CIHSA, it is employed to solve the combined economic emission dispatch
(CEED) problem, which is one of the most important issues in optimizing the operation of electric power
systems. In practice, the CEED problem is a high-dimensional non-convex optimization problem with
several equality and inequality constraints, and it is one of the most complicated optimization problems in
electrical power engineering.
Economic load dispatch (ELD) problem is a computational process to determine the generation
contribution of each unit at minimum fuel cost while satisfying the total load demand and all operational
constraints [26], [27]. The electricity generation from fossil fuels is exacerbating the atmospheric
pollution which has become one of the most important concerns in recent years. So, besides moving
toward generating electricity from clean and renewable energies [28], pollutant emissions released by
traditional generation units should also be considered in the generation scheduling of the system.
Emission constrained dispatch (ECD) is similar to ELD, except that its goal is minimizing emissions
instead of fuel cost. However, due to the fact that fuel cost and emission are in conflict with each other
(minimizing one increases the other), system operation with either minimum fuel cost or minimum
emissions will be unworkable. To deal with this problem, a load dispatching technique to simultaneously
minimize both fuel cost and emissions, called combined economic emission dispatch (CEED), has been
developed. It is clear that the determination of a single optimal solution for the bi-objective CEED
problem requires that the weight of both objective functions be specified [11].
Generally, there are three main approaches to solve the CEED problems;
The first approach is to convert the bi-objective CEED problem into a single-objective optimization task
by defining a price penalty factor (pf) or normalizing the fuel cost and emissions. This approach has been
practiced in several previous research works using different methods such as particle swarm optimization
algorithm (PSO), gravitational search algorithm (GSA) and gravitational acceleration enhanced PSO
algorithm (GAEPSO) [29], moth swarm algorithm (MSA) [30], real coded chemical reaction algorithm
(RCCRO) [31], spiral optimization algorithm (SOA) [32], opposition-based harmony search algorithm
[11], biogeography based optimization algorithm (BBO) [33], and hybrid PSO and GSA (PSOGSA) [34].
The second method involves simultaneous minimization of fuel cost and emissions as a bi-objective
optimization problem. This approach has been practiced in several previous papers using different
methods such as multi-objective particle swarm optimization (MOPSO) [35], multi-objective differential
evolution (MODE) [36], multi-objective harmony search (MOHS) [37], tribe-modified differential
evolution (Tribe-MDE) [38], non-dominated sorting bacterial foraging (NSBF) and fuzzy dominance
sorting bacterial foraging (FSBF) [39], and elitist non-dominated sorting genetic algorithm (NSGA-II)
[40]. However, these methods are computationally involved and time-consuming and might produce only
sub-optimal solutions.
The third approach is to consider the emission as a constant within the permitted limits, in which the
problem can be solved through a single-objective optimization task. However, this approach is unable to
provide any information regarding trade-offs between fuel cost and emissions. For instance, DavidonFletcher-Powell’s method of optimization has been used in [41] for solving the CEED problem while the
emission amount has been considered as a constant in the allowed limits.
In this paper, to verify the effectiveness of the proposed CIHSA, it is applied to solve the CEED problem
in six test systems having 6, 10, 13, 14, 40, and 140 generators considering valve-point loadings effect.
The bi-objective CEED problem is converted into a single-objective optimization task by defining a
modified price penalty factor. In addition to solving the CEED problem, the ELD and ECD problem also
have been solved, and the obtained results are compared with a large number of other existing algorithms.
Also, it is taken into account that the selected papers for comparison, as much as possible, be among the
latest published articles in this field. Thereby, the section of numerical results in this paper is able to
clearly confirm the effectiveness and superiority of the proposed method. In addition, the statistical results
of the proposed algorithm are provided for all test systems which demonstrate that the proposed algorithm
not only has high ability to find the most appropriate solution, but also has high robustness and reliability
in its performance.
The rest of this paper is organized as follow: The ELD, ECD and CEED problems formulation is
expressed in section 2. A brief description of HSA is provided in section 3. The proposed CIHSA is
explained in details in section 4. Finally, the effectiveness of the proposed algorithm is verified by
applying CIHSA to six test systems, in section 5.
2
2.1
Problem formulation
Economic load dispatch (ELD)
The objective function of the ELD problem is minimizing the fuel cost for a specified load demand while
satisfying various system and unit constraints. The fuel cost of thermal power plants can be approximately
modeled as a quadratic function of the generators output power as given in (1).
FC    ai Pi 2  bi Pi  ci 
n
i 1
(1)
where FC is the total fuel cost of generations ($/hr), ai , bi and ci are the fuel cost coefficients of the i-th
unit, Pi is the output power of the i-th unit, and n is the number of generation units.
To obtain a more practical model for the cost function of thermal power plants, the effect of valve-point
loading should be also considered. The fuel cost function considering valve-point loadings can be
expressed as (2) [42]:


FC    ai Pi 2  bi Pi  ci   ei  sin fi   Pi min  Pi  

i 1 
n
(2)
where FC is the total fuel cost ($/hr) considering valve-point loadings, ei and fi are the fuel cost
coefficients of the i-th unit that reflect the valve-point effect.
a) Power output constraints
There is a practical range for the minimum and maximum of the electrical output power of each unit as
shown in (3).
Pi min  Pi  Pi max
(3)
b) System power balance constraint
In load dispatching, the system power balance, which is presented in (4), should be satisfied.
n
 Pi  PD  PL (4)
i 1
where PD is the total load demand, and PL is the total power transmission losses which can be expressed as
a function of the units output power and B-loss coefficients as presented in (5).
n
n
n
PL    PB
i ij Pj   B0i Pi  B00
i 1 j 1
i 1
(5)
where Bij is the ij-th element of the loss coefficients square matrix, B0i is the i-th element of the loss
coefficients vector and B00 is the loss coefficient constant.
c) Ramp rate limits constraint
In practice, the output power of the units cannot vary instantaneously. So, the operational range of the
output power of each unit is limited by its ramp up-/down-rate limits as given in (6).
max Pi min ,  Pi 0 - DR i   Pi  min Pi max ,  Pi 0  UR i 
(6)
where Pi0 is the previous operating output power of i-th unit (MW); DRi and URi are the down-rate and
up-rate limits of i-th unit (MW/h), respectively.
2.2
Emission constrained dispatch (ECD)
Total emissions released by thermal power plants also can be approximated as a quadratic function of the
output active power of the units. The emission constrained dispatch (ECD) problem can be expressed as
an optimization task to minimize the total amount of emissions, which is defined by (7) [38]:
E   102 i Pi 2  i Pi   i    i exp  i Pi 
n
i 1
(7)
where E is the total amount of emissions (lb/hr), αi ,βi ,γi, ζi and λi are the emission coefficients of the i-th
unit.
2.3
Combined economic emission dispatch (CEED)
The bi-objective CEED problem can be converted into a single objective function using a modified price
penalty factor (pf) approach as (8):
PTC  FC  pf  E (8)
where PTC is the pure total cost of the system operation. For providing a tradeoff between minimization
of the fuel cost and emission, (8) can be rewritten as follow:
TC  w  FC  (1  w)  pf  E (9)
where w is weight factor and specifies the optimization type; if w=1, then the problem is ELD; if w=0,
then the problem is ECD; if w=0.5, then the problem is CEED.
By the following steps pf for a specified load demand can be calculated:
Step 1: Calculate hi for each unit according to (10):
FC  Pi max 
E  Pi max 
 hi
i  1, 2,..., n $ / kg
(10)
Step 2: Sort hi values in an ascending order.
Step 3: Add maximum output power of each unit one at a time starting from the unit with smallest hi until
 Pi
max
 PD
Step 4: hi associated with the last unit is pf for the given load demand [33].
3
Harmony Search Algorithm (HSA)
Harmony search algorithm (HSA) is a derivative-free meta-heuristic algorithm which is inspired by the
music improvisation process. In comparison to other algorithms, the desirable feature of HSA is that it
generates a new solution vector (Harmony) after considering all the existing solution vectors in the
harmony memory (HM) matrix. This feature leads to the increasing the exploration power of HSA to
achieve better solutions. Furthermore, some modified variants of HSA have been introduced that have
employed a stochastic derivative in the optimization procedure and are applicable to the problems in
which the differential derivative cannot be used due to the problem characteristics, for instance, design
problems with discrete ranges of the control variables [15], [16].
There are five steps in the optimization procedure of HSA, as follows [13]:
Step 1. Definition of the optimization problem and HSA parameters initialization
The optimization problem is defined as follow:
Minimize f(x) subject to xi  X i
, i  1,, N
where f(x) is the objective function, x is the set of each decision variable (xi), Xi is the set of the possible
values range for each design variable that xi.l ≤ Xi ≤ xi,u , where xi,l and xi,u are the lower and upper bounds
for each decision variable, and N is the number of design variables. Also, in this step, the required HSA
parameters are specified. The harmony memory size (HMS) which determines the number of solution
vectors in HM matrix, harmony memory considering rate (HMCR), pitch adjusting rate (PAR), and the
termination criteria are selected in this step. HMCR and PAR are the parameters required to improve the
solution vector and they are defined in Step 3.
Step 2. Harmony memory (HM) initialization
In this step, the harmony memory (HM) matrix, given in (11), is filled with randomly generated solution
vectors and is sorted based on the objective function f(x) values.
 x11
 2
x
HM   1

 HMS
 x1
x12
x22
x2HMS
x1N 

xN2 

HMS 
... xN 
...
...
(11)
Step 3. Improvising a new harmony from the HM
A new harmony vector, x´ = (x´1, x´2,…, x´N), is generated based on memory consideration, pitch
adjustment, and random selection. According to memory consideration, the value of the first decision
variable x´1 for the new vector is selected from any value in the specified HM range (x11 - xHMS1). Values
of the other design variables (x´2,…, x´N) can be chosen with the same method. The HMCR varies
between 0 to 1, and determines the probability of choosing one value from the historical values stored in
the HM, while (1-HMCR) is the probability of randomly selecting one value from the possible range of
values.
 x {x1i , x i2 ,..., x iHMS }
with probability HMCR
xi   i
with probability 1  HMCR 
 xi  X i
(12)
Then, every component obtained by the memory consideration is examined to determine whether it
should be pitch-adjusted. This operation uses the PAR parameter, which is the rate of pitch adjustment as
follows:
Yes
Pitch adjusting decision for xi  
 NO
with probability PAR
with probability 1  PAR 
(13)
The value of (1-PAR) determines the probability of doing nothing. If the pitch adjustment decision for x´i
is Yes, then x´i is replaced according to (14).
xi  xi  r  bw
(14)
where r is a random number generated using uniform distribution between 0 and 1, and BW is an arbitrary
distance bandwidth. All of the mentioned considerations are applied to each variable of the new harmony
vector.
Step 4. Updating the HM
According to objective function value, if the new generated harmony vector, x´ = (x´1, x´2,…, x´N), is better
than the worst harmony in the HM, the new harmony will replace it in the HM.
Step 5. Repeating Steps 3 and 4 and checking the termination criteria
Steps 3 and 4 will be repeated until the termination criterion has been satisfied.
4
The chaotic improved harmony search algorithm (CIHSA)
HSA has a high ability to find the high-performance regions of the solution space at a reasonable time,
but gets into trouble in performing a local search for numerical applications. So, adding an extra local
optimization method to the optimization procedure of HSA can improve its performance in optimization
tasks. Also, in order to improve the fine-tuning characteristic of HSA, some modifications can be applied
to set the PAR and BW parameters of HSA. Moreover, to enhance the convergence rate of HSA and
reduce the required number of iterations to achieve the optimum solution, HSA can use more than one
HM in each iteration. Applying the mentioned modifications to the classic HSA leads to the creating a
new optimization algorithm with a more suitable performance called improved harmony search algorithm
(IHSA). The IHSA will be explained in section 4.1 in details.
Furthermore, to enrich the searching behavior of HSA and also to avoid being trapped into local optimu m
solutions, a chaotic pattern can be used instead of uniform distribution for generating the random numbers
in steps of HM initialization and improvising new harmonies in the HSA procedure. Using chaotic pattern
in HSA leads to the better coverage of the search space and increases the robustness of HSA. HSA with a
chaotic pattern for generating the random numbers in the optimization procedure called chaotic harmony
search algorithm (CHSA) and will be explained in section 4.2.
The combination of IHSA and CHSA results in creating a fast, accurate and robust optimization algorithm
which has the advantages of both of them and called chaotic improved harmony search algorithm
(CIHSA). The optimization procedures of CIHSA is explained in section 4.3 in details.
4.1
The Improved Harmony Search Algorithm (IHSA)
The improved harmony search algorithm (IHSA) has three main differences with the classic HSA;
1- The PAR and BW parameters are changed dynamically according to the number of iteration,
2- Three virtual HMs are employed in the optimization procedure,
3- An additional local optimization step after each iteration is added to the optimization procedure.
In the classic HSA, PAR and BW parameters are adjusted in the initialization step as fixed values and
cannot be changed during iterations. While to make the HSA performance more suitable, these parameters
should be changed dynamically according to the number of iteration. For example, a small value of BW
in final stages, which harmonies are close to the optimal solution, increases the fine-tuning of solutions,
but in early stages, the value of BW should be relatively large to enforce the algorithm to increase the
diversity of solutions. A large value of PAR with a small value of BW usually leads to the improving the
best-found solutions in final stages. A small value of PAR with a large value of BW can cause a poor
performance of HSA and a significant increase in the required number of iterations to achieve the
optimum solution. So, the PAR and BW parameters should be adapted with the optimization procedure.
For suitable setting the values of PAR and BW parameters, they are updated in each iteration according to
(15) and (16). Notice that the used equations for updating the PAR and BW parameters have been
previously used in some other papers such as [12].
 PARmax  PARmin
PAR(it )  PARmin  
NI


  it

 it
 BWmin  
BW (it )  BWmax  exp 
 Ln 

 BWmax  
 NI
(15)
(16)
where PARmax , PARmin , BWmax and BWmin are the maximum pitch adjusting rate, minimum pitch
adjusting rate, maximum bandwidth and minimum bandwidth, respectively. it and NI are the iteration
number and the specified maximum number of iterations, respectively.
To enhance the convergence rate and reduce the required number of iterations, in each iteration, based on
existing HM, four HMs are generated and the “improvising new harmonies” procedure is performed on
all of them. In other words, IHSA uses from three virtual HMs. After updating the HMs, their harmonies
are merged and sorted, and a new HM is provided with the best solutions found among all solutions, and
the extra solutions are discarded. The four HMs in the next iteration, are generated according to this new
HM.
After providing new HM in each iteration, a local optimization procedure is performed to increase the
quality of the solutions. In the local optimization step, two decision variables are selected randomly and
their values are replaced with the best possible values. This procedure is repeated N times for each
harmony. The local optimization can significantly decrease the required number of iterations to achieve
the optimal solution. The details of the local optimization step are explained in section 4.3.
4.2
The Chaotic Harmony Search Algorithm (CHSA)
Recently, the idea of using chaotic patterns instead of random sequences is taken into consideration in
several fields and suitable results have been shown in many applications such as optimization tasks.
Chaos is mathematically defined as generation randomly by simple deterministic systems. Generally, the
sensitive dependence on initial conditions, the semi-stochastic property and ergodicity are three main
dynamic properties of the chaos. The investigations show that often using chaotic patterns in heuristic
optimization techniques leads to the enhancing the searching behavior and it can avoid being trapped into
local optimum solutions [14], [43].
There are several one-dimensional chaotic maps such as Logistic map, Tent map, Bernoulli shift map,
Liebovitch map, Intermittency map and so on. Among these maps, the logistic map is relatively the
simplest one and its average computational time is less than others [43]. In the proposed chaotic
algorithms, the logistic map is used to generate the initial population and to improvise new harmonies.
The mathematical expression of the logistic map is given in (17) [43].
Yit 1  p  Yit  1  Yit 
for
0 p4
(17)
where ‘it’ is the number of iteration and ‘Y’ is the chaotic variable. In the chaotic pattern used in the
proposed method, the value of ‘p’ is considered 4, and by defining the initial condition as Yn ∈ (0,1) and
Yn ∉ {0,0.25,0.5,0.75,1}, the value of chaotic variables Yn will be distributed between [0, 1]. Figure 1
shows the variation of chaotic variables for 150 iterations with starting value of 0.91.
Equation (17) is used for HM initialization, but in the procedure of improvising new harmonies, random
numbers between -1 to +1 are needed. So, (17) is modified as (18) to obtain chaotic random numbers
between -1 to +1.
Yn1  2   4  Yn  1  Yn   0.5
(18)
Figure 2 shows the variation of chaotic variables according to Eq. (18) for 150 iterations with starting
value of 0.91.
4.3
Combination of IHSA and CHSA
The combination of IHSA and CHSA leads to the forming of an optimization technique which benefits
from the advantages of both of them and called chaotic improved harmony search algorithm (CIHSA).
For example, IHSA has a high accuracy and a very suitable convergence rate. On the other hand, CHSA
has a high robustness in the solutions found in different runs. So, CIHSA can provide high-quality
solutions with a high accuracy and robustness. In the following, the computational steps of the proposed
CIHSA are explained and the flowchart of the CIHSA optimization procedure is presented in Fig. 3.
Step 1: Read the system data.
In this step, the system data including fuel cost and emission coefficients, generation limits, and total load
demand, and also the algorithm parameters including HMS, HMCR, PAR min , PARmax , BWmin , BWmax ,
NI (specified maximum number of Iterations) and the number of decision variables N (number of
generating units) should be read and initialized.
Step 2: Initialize the HM according to chaotic selection between P min and Pmax
In this step, each harmony of HM should be initialized according to the following equation:
xi  Li  Yn 1  Ui  Li 
(19)
where xi is i-th decision variable (generator) of the harmony, and Yn+1 is a chaotic number distributed
between 0 and 1 using (17), Ui and Li denote the upper and lower limit of the i-th decision variable,
respectively. Since the total generation should be equal to total load demand, after HM initialization, this
condition should be checked and satisfied by modifying the decision variables value while the generation
limits are not violated. The strategy employed to meet the system power balance constraint is shown in
Fig. 4. In this figure, Δ determines our precision in satisfying this constraint.
Step 3: Improvise new harmonies
In this step, first the PAR and BW value should be updated according to (15) and (16), and four HMs
should be generated based on the existing HM. Then, each harmony of any of four HMs should be
updated according to the procedure shown in Fig. 3.
In the procedure of “improvise new harmonies” shown in Fig. 3, r1 and r2 are two 1×N matrices with
random numbers between 0 to 1. If r1(i) (i ϵ {1,2,..,N}) is less than HMCR, then one of the other
harmonies will be randomly selected and the value of its i-th decision variable will be considered as the
new value of x(i) (x'(i)). If r1(i) is more than HMCR, then x'(i) will be calculated according to (19). If r2(i)
(i ϵ {1,2,..,N}) is less than PAR, then the new value of x'(i) (x"(i)) will be calculated according to (20). If
r2(i) is more than PAR, then the value of x'(i) will be remained unchanged (x'(i) = x"(i)).
x ''i  x 'i  Yn1  BW
(20)
where the chaotic variable Yn+1 is obtained from (18). Note that the chaotic patterns used in updating of
each HM are different in initial value. After updating each decision variable in each harmony, this
variable should be checked to have a value between its minimum and maximum limit. If the decision
variable is out of limit, its value will be fixed with the limit. Also, after updating all decision variables in
each harmony, the system power balance constraint should be checked and met according to the flowchart
given in Fig. 4. After updating the HMs, their harmonies are merged and sorted, and a new HM is
provided with the best solutions found among all solutions, and the extra solutions are discarded. The four
HMs in the next iteration are generated according to this new HM.
Step 4: Local optimization
The following procedure will be repeated for N times for each harmony, where N is considered equal to
the number of the decision variables (generation units):
The r-th and t-th decision variables of the harmony will be selected randomly and by using a local
optimization, their value will be replaced with the best possible values according to the procedure
represented in Fig. 3. In this figure, ɛ determines the resolution of the search space. Two variables A and
B are defined for two reasons; first, to avoid infringement of the generation limits, second, to determine
the shortest possible period of search space and reduce the process time consequently. TCba denotes the
total cost calculated by fuel cost and emission coefficients related to the a-th generator with the output
power of ‘b’ MW. After local optimization in each harmony, the system power balance constraint will be
checked again, and the harmony will be modified if it has violated from this constraint.
Step 5: Check stopping criteria
The above procedure will be repeated from step3 until one of the termination criteria is reached.
Termination criteria are: the maximum number of iterations (NI) or no improvement in TC for a specified
number of iterations (SNI).
5
Numerical results and discussions
In order to assess the effectiveness of CIHSA, it is applied to solve the CEED problem on six test systems
having 6, 10, 13, 14, 40, and 140 generators considering valve-point loading effect. The algorithm has
been implemented on a PC with the detailed settings as follow:
Hardware: CPU: Intel® Core™ i5-4690, Frequency: 3.50 GHz, RAM: 4.0 GB, Hard drive: 500 GB
Software: Operating system: Windows 7 Ultimate, Language: MATLAB 8.1 (R2013a)
The results obtained by CIHSA are compared with the classic HSA, CHSA, IHSA and a large number of
other optimization algorithms which have been practiced in previous works. In addition, the statistical
results regarding the proposed algorithm performance are provided for each test system. The provided
comparisons and the presented statistical data clearly indicate the high efficiency, robustness, and
accuracy of the proposed algorithm. In the tables provided in this section, fuel cost, emission, total
generation, total cost, pure total cost, power transmission losses, and standard deviation have been
represented by FC, E, TG, TC, PTC, PL, and SD abbreviations, respectively.
5.1
Test system 1: 6-generator
i) In this case, the test system is the standard IEEE 30-bus six-generator system. This power system
consists of 21 load buses with 283.40 MW (2.834 per-unit) total load demand and 41 interconnected
transmission lines. The generation limits of the units and the fuel cost and the emission coefficients are
given in the appendix. To compare the results obtained by the proposed CIHSA with the results reported
in previous works, the system is considered lossless.
Table 1 presents the best solutions found by CIHSA, including generation output of each unit for ELD,
ECD and CEED problem. In this case, the achieved results using four algorithms HSA, CHSA, IHSA,
and CIHSA are identical. Also, to investigate whether w=0.5 leads to the obtaining the minimum PTC,
the PTC value obtained by CIHSA for different values of w for 20 trial runs is presented in Table 2.
According to this table, the minimum PTC is achieved in w=0.5 which has been considered as the CEED
case in this work. A comparison between the achieved solutions by CIHSA and by several recently
published methods for minimum FC in ELD, and minimum E in ECD problem, is performed in Table 3.
According to this table, the results achieved by CIHSA for the minimum fuel cost is less than many of
those algorithms and is similar to some of them. Only the Opposition-based Harmony Search algorithm
(OHS) [11] has achieved a lower fuel cost. But, according to the reported generation outputs in [11] and
the fuel cost coefficients, the fuel cost is calculated as 600.0703416. Also, the reported total generation in
[11] for this case is 2.8338 per-unit which is less than the total load demand (2.834 per-unit) and has led
to the obtaining a lower fuel cost compared to other algorithms. Also, the achieved results using CIHSA
for the minimum emission is the least of all methods with the exception of SOA [32]. However, according
to the reported generation outputs in [32] and the emission coefficients, the emission is calculated as
0.195978 which is sufficiently more than the reported value in [32].
It should be noted that the minimum, average and maximum value of the objective functions obtained by
the four algorithms for 20 trial runs are equal to each other. It shows that the algorithms have suitable
performance and they are absolutely robust in this case. In Table 4, a statistical comparison between the
results obtained by CIHSA and by some other algorithms implemented in [29] and [38] is provided.
Figure 5 shows the convergence rate of the four algorithms for the fuel cost minimization. According to
this figure, the suggested modifications greatly affect the convergence rate of the classic HSA.
Figure 6 presents the optimal generation outputs found by CIHSA for ELD, ECD, and CEED, which
provides useful information about the generation units. For instance, the units with higher generation in
ELD rather than ECD can be considered by approximation as units with low generation cost and high
pollution. Those units with lower generation in ELD than ECD can be approximately considered as units
with high generation cost and low pollution.
ii) In this case, the ELD problem is only considered. The generation limits of the units and the coefficients
of the fuel cost and the transmission loss matrices are given in the appendix. Total load demand for this
test system is 700 MW. Table 5 gives the optimal solutions found by CIHSA and some other algorithms.
It is clear that CIHSA is superior to the algorithms presented in [49], [50] and [51] as the minimum fuel
cost achieved by CIHSA is significantly less than the values obtained by those algorithms. It is worth
mentioning that the CIHSA performance in this test system is absolutely robust and the minimum fuel
cost achieved for 20 trial runs are the same as what is given in Table 5.
5.2
Test system 2: 10-generator
This case study consists of ten generation units considering the valve-point effect. The fuel cost and
emission coefficients, the generation limit constraints and the transmission loss coefficients matrix are
given in the appendix. The generation outputs of the most appropriate solutions for ELD, ECD and CEED
problem for 2000 MW load demand are listed in Table 6. A comparison between the solutions found by
CIHSA and the results obtained by other algorithms for ELD and ECD is provided in Table 7. According
to this table, in ELD, the minimum fuel cost obtained by CIHSA is the least of all other methods, and in
ECD, the achieved minimum emission is more than DE [52] and is similar to TLBO [53], QOTLBO [53]
and RCCRO [31]. But, it should be mentioned that according to the generation outputs reported in [52]
and the emission coefficients, the emission is calculated as 3932.417288 which is more than the value
reported in [52] and the minimum emission obtained by CIHSA.
As a comparison regarding the CEED solution found by CIHSA, the minimum total cost obtained by FPA
[51] is 321,322.571, while the total cost of the optimal solution found by CIHSA is 320,948.532910. For
this case, the minimum, maximum, and average total cost achieved by CIHSA for 20 trail runs are
320,948.532910, 320,975.241456, and 320,953.658421, respectively. Thus, even the worth solution found
by CIHSA is significantly better than the best solution achieved by FPA [51].
5.3
Test system 3: 13-generator
In this case, the test system includes thirteen thermal power plants considering valve-point loading effect.
The system data is given in the appendix. For this test system, in the first case, the total load demand is
assumed 1800 MW, for which the best solutions found by CIHSA for ELD, ECD, and CEED problem are
listed in Table 8. In the second case, the total load demand is assumed 2520 MW, and the power
transmission losses are also considered. The transmission loss matrices are presented in appendix too. The
best solution found by CIHSA for ELD problem in this case is also included in Table 8. According to the
results, for 2520 MW load demand, the best FC found by CIHSA is 24,512.432933 $/h, which is lower
than the values found by other algorithms reported in the previous works such as OIWO (24,514.83 $/h)
[44], ORCCRO (24,513.91 $/h) [45], and SDE (24,514.88 $/h) [46]. The PTC value obtained by CIHSA
for different values of w for 20 trial runs is given in Table 9, which shows that w=0.5 leads to the
obtaining the minimum PTC value.
Table 10 shows the solutions of ELD problem for 1800 MW load demand using CIHSA and other
algorithms. According to this table, the best fuel cost found by CIHSA is the least among all others.
Furthermore, the CIHSA performance is absolutely robust in this case and the worth solution found by
CIHSA is better than the best solutions found by other algorithms. Also, as a comparison between the
ECD solutions obtained by CIHSA and other algorithms, the minimum emission value found by BBO
[33] and RCCRO [31] is reported as 58.241 and 58.2407 ton/h, respectively.
In addition, Table 11 presents the statistical data with respect to the results obtained by the four
algorithms for 20 trial runs. As it is clear from this table, all the algorithms have suitable performance
especially CIHSA which is completely robust in this case. Furthermore, a statistical comparison between
the results obtained by CIHSA and by several other algorithms in ELD for this case study has been
provided in Table 12, which clearly confirms the superb performance of CIHSA.
Figure 7 shows the convergence rate of the four algorithms in fuel cost minimization. In this figure, the
suitable effect of the suggested modifications on the convergence rate of the classic HSA is obvious.
Figure 8 shows the distribution of the fuel cost achieved by IHSA and CIHSA for this test system for 20
trial runs. According to Fig. 8, employing the chaotic pattern in the optimization procedure of IHSA leads
to the enhancing the performance and robustness of the algorithm.
5.4
Test system 4: 14-generator
This test system includes fourteen generation units considering valve-point loadings effect and power
transmission losses. The system data can be found in the appendix, total load demand is 2000 MW, and
only the CEED problem is solved for this case.
Table 13 presents the optimal solution found by CIHSA and by some other algorithms. According to this
table, the minimum total cost obtained by CIHSA is significantly lower than the others. Also, it is worth
mentioning that while in the solutions found by other algorithms, the sum of total load demand and power
transmission losses is not exactly equal to the total generation, the system power balance constraint has
been met precisely in the solution found by CIHSA that demonstrate the effective constraint handling
strategy employed in this algorithm, which leads to the obtaining high quality solutions.
For this case, the minimum, maximum, and average total cost achieved by CIHSA for 20 trail runs are
14,454.202397, 14462.352354, and 14456.759621, respectively. These results show the robustness of the
proposed algorithm as all solutions found are so close to each other. In this case, the worth solution found
by CIHSA is better than the best solutions found by GA, PSO, BBO [63], and DE [64].
5.5
Test system 5: 40-generator
In this test system, the four algorithms are applied to minimize the total cost in a power system with forty
thermal power plants considering valve-point loading effects. The system data including the generation
limits and the fuel cost and the emission coefficients is given in the appendix, and the total load demand is
assumed 10500 MW. Since this is a larger system with more nonlinear elements, it has more local
minima, thereby it will be more difficult to achieve the global solution. Table 14 presents the most
optimal solutions found by CIHSA for ELD, ECD and CEED problems. The minimum fuel cost in ELD
and the minimum emission in ECD is 121,412.536561 $/hr and 176,682.264680 ton/h, respectively. The
fuel cost and emission obtained in CEED are 128,726.248081 $/h and 178,577.661404 ton/h,
respectively. It is seen that according to the explanations mentioned in section 2, the CEED provides a
trade-off between minimum fuel cost and emission. Note that all of the units satisfy the generation limit
constraints. Also, the PTC value obtained by CIHSA for different values of w for 20 trial runs is
presented in Table 15, which again shows that the minimum PTC value is achieved in w=0.5.
The solution found by CIHSA for ELD problem is compared with the results obtained by several other
algorithms in Table 16. According to this table, the best fuel cost found by CIHSA is the least among all
of the other methods. Also, as a comparison between the ECD solutions obtained by CIHSA and other
algorithms, the minimum emission value found by MDE, Tribe-DE, Tribe-MDE [38], NSGA-II and
MODE [75] is reported as 176,719.22153, 176,825.6902, 176,682.2646796, 176,691.9677 and
176,683.2718 ton/h, respectively.
The statistical data of the results obtained by the four algorithms for 20 trial runs is given in Table 17, and
Table 18 presents a statistical comparison between the results obtained by CIHSA and by several other
optimization techniques in ELD for this test system, which demonstrates the superiority of the proposed
CIHSA compared to those optimization techniques.
Figure 9 presents the convergence rate of the four algorithms in ELD. According to Fig. 9, the proposed
modifications have great effects on the HSA performance and lead to the improving the efficiency,
accuracy and convergence rate of the algorithm. The distribution of the fuel cost achieved by IHSA and
CIHSA for this test system for 20 trial runs is shown in Fig. 10. As it is clear in this figure, due to the use
of chaotic pattern, CIHSA has more efficiency and robustness in comparison to IHSA.
5.6
Test system 6: 140-generator
To better show the effectiveness of CIHSA in large-scale test systems, it is tested for solving the ELD
problem in Korean power system which consists of 140 generators. The system data can be found in the
appendix. The minimum, maximum, and average value of the solutions found for 20 trial runs are equal to
each other and less than the reported results in the previous works, which clearly demonstrates the highquality performance and high robustness of the proposed CIHSA in large-scale test systems. The results
obtained by CIHSA and some other algorithms for this test system are given in Table 19. In addition,
Table 20 presents the scheduled output power for each generation unit in the optimal solution found by
CIHSA for this test system.
6
Conclusion
This paper presented a novel optimization technique for solving the CEED problems with several
operational considerations of the power system. In the proposed algorithm, the classic HSA has been
modified by several innovative modifications that result in enhancing the efficiency, accuracy, and
robustness of the algorithm and decreasing the required number of iterations to achieve the optimal
solution. The proposed CIHSA was employed in solving generation scheduling problem in six test
systems to minimize the system fuel cost and pollutant emissions. The solutions found by CIHSA were
compared with the results reported in several other research works. The numerical and statistical results
clearly demonstrated the high efficiency of CIHSA and its superiority compared to other existing
optimization techniques in solving CEED problems.
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Table1. Best achieved solutions for test system 1-i
ELD
ECD
CEED
P1
0.10971930
0.40607387
0.26832394
P2
0.29976607
0.45906893
0.37942508
P3
0.52429825
0.53793856
0.53956257
P4
1.01619883
0.38295303
0.67117460
P5
0.52429826
0.53793855
0.53956256
P6
0.35971928
0.51002706
0.43595125
FC
600.111408
638.273440
611.130692
E
0.222145
0.194203
0.199906
TC
600.111408
317.940559
469.204431
Table 2. The PTC values obtained by CIHSA for different values of W for test system 1-i
W
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Min PTC
956.213999
952.941891
949.986200
947.350846
945.040522
943.060847
941.418539
Max PTC
956.213999
952.941893
949.986204
947.350848
945.040523
943.060848
941.418540
Avg PTC
956.213999
952.941892
949.986202
947.350847
945.040522
943.060847
941.418539
W
0.35
0.40
0.45
0.50
0.55
0.60
0.65
Min PTC
940.121623
939.179684
938.604188
938.408863
938.610208
939.228140
940.286850
Max PTC
940.121623
939.179685
938.604188
938.408863
938.610208
939.228141
940.286850
Avg PTC
940.121623
939.179685
938.604188
938.408863
938.610208
939.228140
940.286850
W
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Min PTC
941.815950
943.852064
946.441037
949.641147
953.527864
958.201170
963.797487
Max PTC
941.815952
943.852065
946.441039
949.641149
953.527865
958.201171
963.797487
Avg PTC
941.815951
943.852065
946.441038
949.641148
953.527864
958.201170
963.797487
Table 3. Comparison of the minimum FC in ELD, and minimum E in ECD, obtained by different algorithms for test system 1-i
MOHS
Method
GSA [29]
DE [38]
SOA [32]
NPGA
GAEPSO
NSBF
SPEA
MDE
[36]
[29]
[39]
[36]
[38]
NSGA [36]
[37]
FCELD
602.2311
601.3428
600.986
600.6909
600.34
600.31
600.2978
600.2704
600.22
600.173
EECD
0.1954
0.194217
0.18729*
0.1947
0.1946
0.1943
0.1942
0.1944
0.1942
0.194208
NSGA-II
MOPSO
PSOGSA
FSBF
Tribe-
[34]
[39]
MDE [38]
MOMethod
DE/PSO
[40]
MSA [30]
[35]
OHS [11]
CIHSA
[48]
FCELD
600.155
600.12
600.115
600.11141
600.11141
600.1141
600.1114
600*
600.111408
EECD
0.1942
0.1942
0.194203
0.19420
0.194203
0.1942
0.194202
0.1942
0.194203
* The calculated fuel cost according to the reported generation outputs in [11] and the fuel cost coefficients is equal to 600.0703416.
* The calculated emission according to the reported generation outputs in [32] and the emission coefficients is equal to 0.195978.
Table 4. Statistical comparison between the results obtained by different algorithms in ELD and ECD problem for test system 1-i
ELD
ECD
Method
Min FC
Avg FC
Max FC
SD
Min E
Avg E
Max E
SD
CIHSA
600.111408
600.111408
600.111408
0
0.194203
0.194203
0.194203
0
Tribe-MDE [38]
600.1114
600.1114
600.1114
0
0.194202
0.194202
0.194202
0
MDE [38]
600.173
600.186
600.2214
2.3481
0.194206
0.194212
0.194219
4.25
GAEPSO [29]
600.2978
600.2997
600.3090
2.5601e-3
0.1942
0.1942
0.1942
4.0891e-9
PSO [29]
600.5692
603.1347
607.1275
1.5618
0.1947
0.1964
0.1990
1.1037e-3
DE [38]
601.3428
601.7951
602.1126
6.8624
0.194217
0.194225
0.194241
9.781
GSA [29]
602.2311
609.0541
615.2996
3.4189
0.1954
0.2004
0.2057
2.6205e-3
Table 5. Best solutions found for test system 1-ii in ELD by different algorithms
Unit
P1
P2
P3
P4
P5
P6
PL
FC
BFO
[50]
222.26
58.777
150.395
106.963
101.601
72.559
11.73
8428.69
SFL
[50]
287.392
67.637
140.933
98.357
64.052
53.15
11.59
8419.78
PSO
[50]
288.653
82.753
132.988
50
99.565
57.768
11.73
8401.45
HS [49]
FA [49]
NA
NA
NA
NA
NA
NA
NA
8398.06
293.312
79.546
123.334
69.7
79.546
63.778
11.44
8388.45
ABC
[49]
323.043
54.965
147.354
50
85.815
50.233
11.4
8372.27
CS [49]
324.113
76.859
158.094
50
51.963
50
11.03
8356.06
FPA
[51]
323.995
76.846
158.2
50
51.983
50
11.024
8356.05
CIHSA
303.521476
113.817531
143.442499
50.000296
50.004526
50.000000
10.786326
8313.222466
Table 6. Best solutions found by CIHSA for test system 2
Unit
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
TL
TG
FC
E
TC
ELD
55.000000
80.000000
106.934727
100.600317
81.476793
83.026871
300.000000
340.000000
470.000000
470.000000
87.038709
2087.038709
111,497.630981
4572.276303
-
ECD
55.000000
80.000000
81.149904
81.359769
160.000000
240.000000
294.507931
297.268922
396.720288
395.587840
81.594656
2081.594656
116,412.565528
3932.243301
-
CEED
55.000000
80.000000
81.081501
80.930292
160.000000
240.000000
290.800949
296.689692
398.842744
398.331226
81.676404
2081.676404
116,390.278321
3932.447341
320,948.532910
Table 7. Comparison of the solutions found by different algorithms for ELD problem
Method
DE [52]
TLBO [53]
FCELD
EECD
111,500
3923.4*
111,500
3932.2
QOTLBO
[53]
111,498
3932.2
RCCRO
[31]
111,497.6319
3932.2433
CIHSA
111497.630981
3932.243301
* The calculated emission according to the reported generation outputs in [52] and the emission coefficients is equal to 3932.417288.
Table 8. Best solutions found by CIHSA for test system 3
Demand 2520MW
Demand 1800MW
ELD
ELD
ECD
CEED
P1
628.318526
628.318531
80.640745
89.759790
P2
299.199291
149.599650
166.328710
149.599650
P3
297.367318
222.749069
166.328709
158.133171
P4
159.733097
109.866550
154.733174
159.733100
P5
159.733094
109.866550
154.733172
159.733100
P6
159.733094
60.000000
154.733174
159.733100
P7
159.733099
109.866550
154.733174
159.733100
P8
159.733095
109.866550
154.733174
159.733100
P9
159.733094
109.866550
154.733173
159.733100
P10
77.399902
40.000000
119.963737
114.799825
P11
114.799811
40.000000
119.963737
114.799825
P12
92.399910
55.000000
109.187661
94.509138
P13
92.399902
55.000000
109.187660
120.000000
PL
40.283232
0.000000
0.000000
0.000000
FC
24512.432933
17960.366122
19113.256777
18376.521665
E
526.302229
461.480560
58.240712
58.737659
TC
24512.432933
17960.366122
16779.772577
17649.734958
Table 9. The PTC values obtained by CIHSA for different values of W for test system 3
w
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Min PTC
35893.029326
35763.443948
35613.244855
35459.469547
35388.708514
35350.704459
35348.552378
Max PTC
35893.029408
35763.444003
35613.244896
35459.469587
35388.708531
35350.704462
35348.552380
Avg PTC
35893.029386
35763.443974
35613.244874
35459.469571
35388.708520
35350.704461
35348.552379
W
0.35
0.40
0.45
0.50
0.55
0.60
0.65
Min PTC
35344.922472
35300.685247
35299.884610
35299.469975
35300.193306
35300.358787
35300.358836
Max PTC
35344.922482
35300.688353
35299.899802
35299.469980
35300.193306
35300.358845
35300.358845
Avg PTC
35344.922477
35300.686124
35299.887725
35299.469977
35300.193306
35300.358810
35300.358843
w
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Min PTC
35300.358845
35300.358845
35819.693512
35819.693514
35819.693819
35819.694193
150917.855090
Max PTC
35300.358845
35300.358902
35819.693512
35819.694166
35819.694190
35819.694193
150917.855137
Avg PTC
35300.358845
35300.358866
35819.693512
35819.693805
35819.694047
35819.694193
150917.855099
Table 10. The minimum fuel costs obtained by different algorithms in ELD problem for test system 3 with 1800 load demand
PSO-TVAC
Method
MSL [56]
HMAPSO[57]
PSM [58]
QPSO [59]
SHDE [60]
HGA [62]
SDE [46]
17,963.83
17,963.83
[61]
FC
18,158.68
17,969.31
RCCRO
FAPSO-VDE
Method
FC
[31]
[47]
17963.8292
17,963.82
17,969.17
17,969.01
DE [54]
CRO [54]
17,961.6108
17,961.0703
17,963.89
17,963.879
HCRO-DE
IPSO-
[54]
TVAC [55]
17,960.3820
17,960.3703
CIHSA
17960.366122
Table 11. Statistical data regarding the solutions found by the four algorithms for test system 3
HSA
CHSA
IHSA
CIHSA
Min FC
17960.510699
17960.366729
17960.366148
17960.366122
Avg FC
17976.978559
17972.844383
17960.366153
17960.366122
Max FC
17986.249765
17985.848755
17960.366159
17960.366122
SD
11.248992
9.172839
0.000003
0.000000
Min E
58.240712
58.240712
58.240712
58.240712
Avg E
58.240712
58.240712
58.240712
58.240712
Max E
58.240712
58.240712
58.240712
58.240712
SD
0.000000
0.000000
0.000000
0.000000
Min TC
17649.734948
17649.734944
17649.735008
17649.734958
Avg TC
17649.760961
17649.756915
17649.735015
17649.734983
Max TC
17650.015798
17649.916931
17649.735024
17649.734990
SD
0.061504
0.041899
0.000009
0.000004
ELD
ECD
CEED
Table 12. Statistical comparison between the results obtained by different algorithms in ELD problem for test system 3
Method
MSL [56]
HMAPSO [57]
PSM [58]
QPSO [59]
SHDE [60]
PSO-TVAC [61]
HGA [62]
Min FC
18,158.68
17,969.31
17,969.17
17,969.01
17,963.89
17,963.88
17,963.83
Avg FC
NA
17,969.31
18,088.84
18,075.11
18,046.38
18,154.56
17,988.04
Max FC
NA
17,969.31
18,233.52
NA
NA
18,358.31
NA
NA
NA
SD
NA
NA
NA
NA
NA
Method
FAPSO-VDE [47]
DE [54]
CRO [54]
HCRO-DE [54]
IPSO-TVAC [55]
CIHSA
Min FC
17,963.82
17,961.61
17,961.07
17,960.38
17,960.37
17,960.366122
Avg FC
17,963.83
17,963.83
17,962.77
17,960.59
17,960.64
17,960.366122
Max FC
17,963.83
17,980.35
17,974.82
17,961.04
17,961.27
17,960.366122
SD
NA
1.44
1.18
0.069
NA
0
Table 13. Best solutions found by different algorithms for test system 4
Method
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
TG
TL
FC
E
TC
GA [63]
264.0103
150.0000
102.7986
119.6997
200.0000
277.2228
238.0072
157.7048
124.4686
136.7911
65.0000
79.7628
62.5000
52.3999
2030.4
30.3658
9889.3
3351.8
15017.3
PSO [63]
239.7598
150.0000
129.9988
119.7331
150.0000
281.3344
184.8666
159.7331
161.9994
160.0000
79.9986
80.0000
85.0000
52.3999
2034.8
34.8236
10166.0
2990.6
14941.0
DE [64]
239.76
150.01
95.21
119.75
199.86
284.59
234.86
159.73
124.89
137.32
66.65
79.95
84.97
52.43
2030.0
29.98
9592.4
3232.2
14537.0
BBO [63]
239.7594
150.0000
126.7956
119.7331
150.0000
284.5995
184.8665
159.7331
161.8835
160.0000
80.0000
80.0000
85.0000
52.3999
2034.8
34.7706
9874.3
2999.0
14462.5
CIHSA
239.751073
150.000001
128.900000
119.733137
199.800031
234.730936
184.866516
159.726441
162.000000
160.000000
80.000000
80.000000
85.000000
52.400000
2036.908134
36.908134
9809.081794
3036.134581
14454.202397
Table 14. Best solutions found by CIHSA for test system 5
Unit
ELD
ECD
CEED
Unit
ELD
ECD
CEED
P1
110.799824
114.000000
114.000000
P21
523.279369
439.446403
437.467746
P2
110.799821
114.000000
114.000000
P22
523.279368
439.446404
437.467748
P3
97.399908
120.000000
120.000000
P23
523.279370
439.772065
437.976522
P4
179.733098
169.368008
178.225888
P24
523.279370
439.772067
437.976521
P5
87.799902
97.000000
97.000000
P25
523.279368
440.111764
437.759425
P6
140.000000
124.257413
129.443556
P26
523.279366
440.111764
437.759419
P7
259.599646
299.711393
300.000000
P27
10.000000
28.993702
19.531198
P8
284.599639
297.914857
299.540696
P28
10.000000
28.993704
19.531198
P9
284.599647
297.260102
298.627148
P29
10.000000
28.993703
19.531201
P10
130.000000
130.000000
130.000000
P30
87.799899
97.000000
97.000000
P11
94.000000
298.410143
307.573033
P31
190.000000
172.331904
175.807155
P12
94.000000
298.026013
307.001570
P32
190.000000
172.331904
175.807156
P13
214.759789
433.557638
433.807799
P33
190.000000
172.331903
175.807158
P14
394.279365
421.728406
408.955374
P34
164.799820
200.000000
200.000000
P15
394.279361
422.779651
411.450301
P35
194.397895
200.000000
200.000000
P16
394.279360
422.779650
411.450298
P36
199.999989
200.000000
200.000000
P17
489.279365
439.412855
452.156805
P37
109.999993
100.838377
104.255068
P18
489.279366
439.402887
452.179572
P38
110.000000
100.838377
104.255066
P19
511.279371
439.412855
437.466768
P39
109.999999
100.838377
104.255067
P20
511.279370
439.412854
437.466768
P40
511.279366
439.412857
437.466772
ELD
ECD
CEED
FC
121,412.536561
129,995.271365
128,726.248081
E
359,901.367106
176,682.264680
178,577.661404
TC
-
-
95,790.897555
Table 15. The PTC values obtained by CIHSA for different values of W for test system 5
w
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Min PTC
192183.670224
192121.792709
192059.937736
191995.546995
191928.509235
191859.977989
191790.486531
Max PTC
192183.929224
192121.996468
192060.278195
191995.624038
191928.739017
191860.062080
191790.682177
Avg PTC
192183.781361
192121.914493
192060.128025
191995.584579
191928.623352
191860.029794
191790.573188
W
0.35
0.40
0.45
0.50
0.55
0.60
0.65
Min PTC
191722.075166
191658.196138
191605.964745
191581.795179
191624.175129
191798.972333
192141.399370
Max PTC
191722.151665
191658.263990
191605.995518
191581.795266
191624.233307
191799.061231
192141.650122
Avg PTC
191722.108907
191658.222692
191605.973925
191581.795234
191624.199095
191799.009589
192141.563575
w
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Min PTC
192989.513462
193985.622434
194468.657787
195083.252153
201072.283619
243632.698809
248090.164140
Max PTC
192989.982299
193986.387450
194468.740024
195083.254160
201072.292447
243632.715432
248090.172770
Avg PTC
192989.772353
193985.858881
194468.717847
195083.253166
201072.288762
243632.707784
248090.169535
Table 16. The minimum fuel costs obtained by different algorithms in ELD problem for test system 5
Method
SCA [65]
EP-SQP [66]
PSO-SQP [67]
PSO [68]
AA (Dist.) [69]
FAPSO [4]
Catfish PSO [68]
FC
122,713.6828
122,323.97
122,094.67
121,818.04
121,788.7
121,712.4
121,683.70
Method
NPSO-LRS [5]
SOH-PSO [1]
CSO [65]
QPSO [3]
BBO [70]
FAPSO-NM [4]
MSSA [71]
FC
121,664.4308
121,501.14
121,461.6707
121,448.21
121,426.953
121,418.3
121,413.4686
Method
CE-SQP [67]
DE [72]
Tribe-MDE [38]
SQPSO [3]
CQGSO [73]
CBA [74]
CIHSA
FC
12,1412.88
121,412.68
121,412.5704
121,412.57
121,412.5512
121,412.5468
121,412.536561
Table 17. Statistical data regarding the solutions found by the four algorithms for test system 5
HSA
CHSA
IHSA
CIHSA
Min FC
121,748.313700
121,747.486373
121,412.537474
121,412.536561
Avg FC
121,823.084480
121,782.368421
121,417.134018
121,413.373697
Max FC
122,192.942093
122,185.721380
121,420.895493
121,420.896252
SD
157.536309
97.133565
4.266395
2.572547
Min E
176,688.401224
176,684.339661
176,682.264787
176,682.264680
Avg E
176,697.526418
176,687.732416
176,682.264843
176,682.264680
Max E
176,710.689257
176,701.074282
176,682.264899
176,682.264680
SD
5.400849
3.613488
3.255279
0.000000
Min TC
95,791.739657
95,791.184559
95,790.897599
95,790.897555
Avg TC
95,794.250667
95,792.056144
95,790.897612
95,790.897555
Max TC
95,812.516821
95,793.345271
95,790.897623
95,790.897555
SD
4.447689
0.669487
0.000009
0.000000
ELD
ECD
CEED
Table 18. Statistical comparison between the results obtained by different algorithms in ELD problem for test system 5
Method
SCA [65]
EP-SQP [66]
PSO-SQP [67]
NPSO-LRS [5]
SOH-PSO [1]
CSO [65]
QPSO [3]
Min FC
122,713.6828
122,323.97
122,094.67
121,664.4308
121,501.14
121,461.6707
121,448.21
Avg FC
125,235.1288
122,379.63
122,245.25
122,209.3185
121,853.57
121,936.1926
122,225.07
Max FC
130,918.3914
NA
NA
122,981.5913
122,446.3
NA
121,994.0267
SD
NA
NA
NA
NA
NA
32
114.08
Method
BBO [70]
CE-SQP [67]
DE [72]
SQPSO [3]
CBA [74]
CIHSA
Min FC
121,426.953
12,1412.88
121,412.68
121,412.57
121,412.5468
121,412.536561
Avg FC
121,508.0325
121,423.65
121,439.89
121,455.7
121,418.9826
121,413.373697
Max FC
121,688.6634
NA
121,479.63
121,709.5582
121,436.15
121,420.896252
SD
NA
NA
NA
49.8076
NA
2.572547
Table 19. Statistical comparison between the results obtained by different algorithms in ELD problem for test system 6
CCPSO [76]
CTPSO [76]
GSO [73]
CQGSO [73]
CIHSA
Min FC
1,655,685
1,655,685
1,734,405.7923
1,655,679.426
1,655,679.425866
Max FC
1,655,685
1,655,685
1,745,315.0894
1,655,679.430
1,655,679.425866
Avg FC
1,655,685
1,655,685
1,739,400.0802
1,655,679.428
1,655,679.425866
Table 20. The optimal generation outputs determined by CIHSA in ELD problem for test system 6
Unit
Power
Unit
Power
Unit
Power
Unit
Power
Unit
Power
P1
119.000000
P29
501.000000
P57
103.000000
P85
115.000000
P113
94.000000
P2
164.000000
P30
499.000000
P58
198.000000
P86
207.000000
P114
94.000000
P3
190.000000
P31
506.000000
P59
312.000000
P87
207.000000
P115
244.000000
P4
190.000000
P32
506.000000
P60
308.589343
P88
175.000000
P116
244.000000
P5
190.000000
P33
506.000000
P61
163.000000
P89
175.000000
P117
244.000000
P6
190.000000
P34
506.000000
P62
95.000000
P90
180.423911
P118
95.000000
P7
490.000000
P35
500.000000
P63
511.000000
P91
175.000000
P119
95.000000
P8
490.000000
P36
500.000000
P64
511.000000
P92
575.400000
P120
116.000000
P9
496.000000
P37
241.000000
P65
490.000000
P93
547.500000
P121
175.000000
P10
496.000000
P38
241.000000
P66
256.825727
P94
836.800000
P122
2.000000
P11
496.000000
P39
774.000000
P67
490.000000
P95
837.500000
P123
4.000000
P12
496.000000
P40
769.000000
P68
490.000000
P96
682.000000
P124
15.000000
P13
506.000000
P41
3.000000
P69
130.000000
P97
720.000000
P125
9.000000
P14
509.000000
P42
3.000000
P70
294.561866
P98
718.000000
P126
12.000000
P15
506.000000
P43
250.000000
P71
141.585409
P99
720.000000
P127
10.000000
P16
505.000000
P44
250.000000
P72
365.907593
P100
964.000000
P128
112.000000
P17
506.000000
P45
250.000000
P73
195.000000
P101
958.000000
P129
4.000000
P18
506.000000
P46
250.000000
P74
217.548960
P102
947.900000
P130
5.000000
P19
505.000000
P47
250.000000
P75
217.549207
P103
934.000000
P131
5.000000
P20
505.000000
P48
250.000000
P76
258.662735
P104
935.000000
P132
50.000000
P21
505.000000
P49
250.000000
P77
403.245249
P105
876.500000
P133
5.000000
P22
505.000000
P50
250.000000
P78
330.000000
P106
880.900000
P134
42.000000
P23
505.000000
P51
165.000000
P79
531.000000
P107
873.700000
P135
42.000000
P24
505.000000
P52
165.000000
P80
531.000000
P108
877.400000
P136
41.000000
P25
537.000000
P53
165.000000
P81
542.000000
P109
871.700000
P137
17.000000
P26
537.000000
P54
165.000000
P82
56.000000
P110
864.800000
P138
7.000000
P27
549.000000
P55
180.000000
P83
115.000000
P111
882.000000
P139
7.000000
P28
549.000000
P56
180.000000
P84
115.000000
P112
94.000000
P140
26.000000
Fig. 1. Logistic map with the initial value of 0.91 for generating random numbers between 0 to 1.
Fig. 2. Modified logistic map with the initial value of 0.91 for generating random numbers between -1 to 1.
for it = 1 to NI
Generate four new
HMs according to
the existing HM
PAR = PAR(it)
BW = BW (it)
C == 0
No
for j = 1 to HMS
x i = xki
Yes
k is a random int eger
number between 1 to HMS
r1(i) < HMCR
Yes
r2(i) < PAR
Yes
C = C+1
for i = 1 to N
No
TC(it) == TC(it+1)
x i = Li + Yn+1×(Ui - Li)
x i = x i + Yn+1×BW
No
if TC (Xjnew) < TC (Xjworst )
then Xjworst = Xjnew
HM HMS×1 = {The best harmonies among all harmonies in the four HMs}
Run local optimization
for i = 1 to HMS
It == NI
No
Yes
for j = 1 to N
C == SNI
Yes
Stop
Select r-th and t-th decision variable randomly
V = xir + xit & TCold = TC (xir) + TC (xit)
No
max
min
min
max
max
min
min
max
A = max (V - Pr
B = max (V - Pt
, Pt
) to min (V - Pr
, Pr
) to min (V - Pt
, Pt
Length(A) > Length(B)
, Pr
H=A
d1 = t & d2 = r
)
TCold = TCnew
xid1 = k & xid2 = V-k
H=B
d1 = r & d2 = t
for k = min(H) : Ɛ : max(H)
No
Yes
TCnew = TC (k) + TC (V-k)
j == N
Yes
)
Yes
TCnew < TCold
No
Fig. 3. Flowchart of the CIHSA optimization procedure.
N
PM =  Pi - PD +PL 
i=1
PM < Δ
YES
NO
Select j-th decision variable randomly
Fix it with the limit
YES
Output power is
out of limits
NO
Fig. 4. The handling strategy for the system power balance constraint.
Fig. 5. Convergence characteristic for fuel cost minimization obtained by four algorithms for test system 1.
Fig. 6. Optimal generation outputs found by CIHSA for test system 1.
Fig. 7. Convergence characteristic for fuel cost minimization obtained by four algorithms for test system 3.
Fig. 8. Distribution of the fuel cost achieved by IHSA and CIHSA for test system 3 for 20 trial runs.
Fig. 9. Convergence characteristic for fuel cost minimization obtained by four algorithms for test system 5.
Fig. 10. Distribution of the fuel cost achieved by IHSA and CIHSA for test system 5 for 20 trial runs.
Appendix
The data of all test systems including the fuel cost and emission coefficients and transmission losses
matrices are given in this section. This data is collected from [32], [33], [44], [49], [52], [64], and [76].
Test system 1-i
Unit
Pmin
Pmax
a
b
c
α
β
γ
ξ
λ
1
5
150
100
200
10
6.490
−5.554
4.091
2e-4
2.857
2
5
150
120
150
10
5.638
−6.047
2.543
5e-4
3.333
3
5
150
40
180
20
4.586
−5.094
4.258
1e-6
8.000
4
5
150
60
100
10
3.380
−3.550
5.326
2e-3
2.000
5
5
150
40
180
20
4.586
−5.094
4.258
1e-6
8.000
6
5
150
100
150
10
5.151
−5.555
6.131
1e-5
6.667
Test system 1-ii
Unit
Pmin
Pmax
a
b
c
1
100
500
0.007
7
240
2
50
200
0.005
10
200
3
80
300
0.009
8.5
220
4
50
150
0.009
11
200
5
50
200
0.008
10.5
220
6
50
120
0.0075
12
120
0.0014
0.0017

0.0015
Bij = 
0.0019
0.0026

0.0022
0.0017
0.0060
0.0013
0.0016
0.0015
0.0020
B0i = -0.0003908
0.0015
0.0013
0.0065
0.0017
0.0024
0.0019
-0.0001297
0.0019
0.0016
0.0017
0.0071
0.0030
0.0025
0.0022 
0.0020 
0.0019 

0.0025
0.0032 

0.0085
0.0026
0.0015
0.0024
0.0030
0.0069
0.0032
0.0007047
0.0000591
-0.0006635
0.0002161
B00 =0.00056
Test system 2
Unit
Pmin
Pmax
a
b
c
e
f
α
β
γ
ξ
λ
1
2
3
4
5
6
7
8
9
10
10
20
47
20
50
70
60
70
135
150
55
80
120
130
160
240
300
340
470
470
0.12951
0.10908
0.12511
0.12111
0.15247
0.10587
0.03546
0.02803
0.02111
0.01799
40.5407
39.5804
36.5104
39.5104
38.539
46.1592
38.3055
40.3965
36.3278
38.2704
1000.403
950.606
900.705
800.705
756.799
451.325
1243.531
1049.998
1658.569
1356.659
33
25
32
30
30
20
20
30
60
40
0.0174
0.0178
0.0162
0.0168
0.0148
0.0163
0.0152
0.0128
0.0136
0.0141
4.702
4.652
4.652
4.652
0.420
0.420
0.680
0.680
0.460
0.460
−398.64
−395.24
−390.23
−390.23
+032.77
+032.77
−054.55
−054.55
−051.12
−051.12
36000.12
35000.56
33000.56
33000.56
1385.93
1385.93
4026.69
4026.69
4289.55
4289.55
0.25475
0.25475
0.25163
0.25163
0.2497
0.2497
0.248
0.2499
0.2547
0.2547
0.01234
0.01234
0.01215
0.01215
0.012
0.012
0.0129
0.01203
0.01234
0.01234
0.0049
 0.0014

0.0015

 0.0015
0.0016
Bij = 
 0.0017
 0.0017

0.0018

0.0019
0.0020
0.0014
0.0045
0.0016
0.0016
0.0017
0.0015
0.0015
0.0016
0.0018
0.0018
0.0015
0.0016
0.0039
0.0010
0.0012
0.0012
0.0014
0.0014
0.0016
0.0016
0.0015
0.0016
0.0010
0.0040
0.0014
0.0010
0.0011
0.0012
0.0014
0.0015
0.0016
0.0017
0.0012
0.0014
0.0035
0.0011
0.0013
0.0013
0.0015
0.0016
0.0017
0.0015
0.0012
0.0010
0.0011
0.0036
0.0012
0.0012
0.0014
0.0015
0.0017
0.0015
0.0014
0.0011
0.0013
0.0012
0.0038
0.0016
0.0016
0.0018
0.0018
0.0016
0.0014
0.0012
0.0013
0.0012
0.0016
0.0040
0.0015
0.0016
0.0019
0.0018
0.0016
0.0014
0.0015
0.0014
0.0016
0.0015
0.0042
0.0019
0.0020
0.0018
0.0016

0.0015
0.0016

0.0015
0.0018

0.0016 

0.0019
0.0044 
Test system 3
Unit
Pmin
Pmax
a
b
c
e
f
α
β
γ
ξ
λ
1
0
680
0.00028
8.10
550
300
0.035
0.06320
−2.434
40
0.855
0.0087
2
0
360
0.00056
8.10
309
200
0.042
0.03480
−3.630
50
0.623
0.0068
3
0
360
0.00056
8.10
307
150
0.042
0.03480
−3.630
50
0.623
0.0068
4
60
180
0.00324
7.74
240
150
0.063
0.04376
−5.271
40
0.312
0.0085
5
60
180
0.00324
7.74
240
150
0.063
0.04376
−5.271
40
0.312
0.0085
6
60
180
0.00324
7.74
240
150
0.063
0.04376
−5.271
40
0.312
0.0085
7
60
180
0.00324
7.74
240
150
0.063
0.04376
−5.271
40
0.312
0.0085
8
60
180
0.00324
7.74
240
150
0.063
0.04376
−5.271
40
0.312
0.0085
9
60
180
0.00324
7.74
240
150
0.063
0.04376
−5.271
40
0.312
0.0085
10
40
120
0.00284
8.60
126
100
0.084
0.05710
−4.852
100
0.424
0.0052
11
40
120
0.00284
8.60
126
100
0.084
0.05710
−4.852
100
0.424
0.0052
12
55
120
0.00284
8.60
126
100
0.084
0.05710
−4.343
100
1.130
0.0055
13
55
120
0.00284
8.60
126
100
0.084
0.05710
−4.343
100
1.130
0.0055
 +0.14
 +0.12

 +0.07

 0.01
 0.03

 0.01
-2
Bij =10   0.01

 0.01

 0.03
 0.05

 0.03
 0.02

 +0.04
+0.12
+0.15
+0.13
0.000
0.05
0.02
0.000
+0.01
0.02
0.04
0.04
0.000
+0.04
+0.07
+0.13
+0.76
0.01
0.13
0.09
0.01
0.000
0.08
0.12
0.17
0.000
0.26
0.01
0.000
0.01
0.34
0.07
0.04
0.11
0.50
0.29
0.32
0.11
0.000
0.01
0.03
0.05
0.13
0.07
0.90
0.14
0.03
0.12
0.10
0.13
0.07
0.02
0.02
0.01
0.02
0.09
0.04
0.14
0.16
0.000
0.06
0.05
0.08
0.11
0.01
0.02
0.01
0.000
0.01
0.11
0.03
0.000
0.15
0.17
0.15
0.09
0.05
0.07
0.000
0.01
0.01
0.000
0.50
0.12
0.06
0.17
1.68
0.82
0.79
0.23
0.36
0.01
0.03
0.02
0.08
0.29
0.10
0.05
0.15
0.82
1.29
1.16
0.21
0.25
0.07
0.05
0.04
0.12
0.32
0.13
0.08
0.09
0.79
1.16
2.00
0.27
0.34
0.09
0.03
0.04
0.17
0.11
0.07
0.11
0.05
0.23
0.21
0.27
1.40
0.01
0.04
0.02
0.000
0.000
0.000
0.02
0.01
0.07
0.36
0.25
0.34
0.01
0.54
0.01
+0.04 
+0.04 
0.26 

+0.01 
0.02 

0.02 
0.000 

+0.01 

+0.07 
+0.09 

+0.04 
0.01 

+1.03 
B0i =  0.0001 0.0002 0.0028 0.0001 0.0001 0.0003 0.0002 0.0002 0.0006 0.0039 0.0017 0.0000 0.0032
B00 = +0.0055
Test system 4
Unit
Pmin
Pmax
a
b
c
e
f
α
β
γ
1
150
455
0.0050
1.89
150
300
0.035
1.6
−150.0
2333.3
2
150
455
0.0055
2.00
115
200
0.042
3.1
−182.0
2102.2
3
20
130
0.0060
3.50
40
200
0.042
1.3
−124.9
2205.0
4
20
130
0.0050
3.15
122
150
0.063
1.2
−135.5
2298.3
5
150
470
0.0050
3.05
125
150
0.063
2.0
−190.0
2131.3
6
135
460
0.0070
2.75
120
150
0.063
0.7
+80.50
2190.0
7
135
465
0.0070
3.45
70
150
0.063
1.5
−140.0
2300.1
8
60
300
0.0070
3.45
70
150
0.063
1.8
−180.0
2400.3
9
25
162
0.0050
2.45
130
150
0.063
1.9
−200.0
2512.1
10
25
160
0.0050
2.45
130
100
0.084
1.2
−136.0
2299.0
11
20
80
0.0055
2.35
135
100
0.084
3.3
−210.0
2701.0
12
20
80
0.0045
1.60
200
100
0.084
1.8
−180.0
2510.1
13
25
85
0.0070
3.45
70
100
0.084
1.8
−181.0
2431.3
14
15
55
0.0060
3.89
45
100
0.084
3.0
−192.1
2711.9
Test system 5
Unit
Pmin
Pmax
a
b
c
d
e
α
β
γ
ξ
λ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
36
36
60
80
47
68
110
135
135
130
94
94
125
125
125
125
220
220
242
242
254
254
254
254
254
254
10
10
10
47
60
60
60
90
90
90
25
25
25
114
114
120
190
97
140
300
300
300
300
375
375
500
500
500
500
500
500
550
550
550
550
550
550
550
550
150
150
150
97
190
190
190
200
200
200
110
110
110
0.00690
0.00690
0.02028
0.00942
0.01140
0.01142
0.00357
0.00492
0.00573
0.00605
0.00515
0.00569
0.00421
0.00752
0.00708
0.00708
0.00313
0.00313
0.00313
0.00313
0.00298
0.00298
0.00284
0.00284
0.00277
0.00277
0.52124
0.52124
0.52124
0.01140
0.00160
0.00160
0.00160
0.00010
0.00010
0.00010
0.01610
0.01610
0.01610
6.73
6.73
7.07
8.18
5.35
8.05
8.03
6.99
6.60
12.9
12.9
12.8
12.5
8.84
9.15
9.15
7.97
7.95
7.97
7.97
6.63
6.63
6.66
6.66
7.10
7.10
3.33
3.33
3.33
5.35
6.43
6.43
6.43
8.95
8.62
8.62
5.88
5.88
5.88
94.705
94.705
309.54
369.03
148.89
222.33
287.71
391.98
455.76
722.82
635.20
654.69
913.40
1760.4
1728.3
1728.3
647.85
649.69
647.83
647.81
785.96
785.96
794.53
794.53
801.32
801.32
1055.1
1055.1
1055.1
148.89
222.92
222.92
222.92
107.87
116.58
116.58
307.45
307.45
307.45
100
100
100
150
120
100
200
200
200
200
200
200
300
300
300
300
300
300
300
300
300
300
300
300
300
300
120
120
120
120
150
150
150
200
200
200
80
80
80
0.084
0.084
0.084
0.063
0.077
0.084
0.042
0.042
0.042
0.042
0.042
0.042
0.035
0.035
0.035
0.035
0.035
0.035
0.035
0.035
0.035
0.035
0.035
0.035
0.035
0.035
0.077
0.077
0.077
0.077
0.063
0.063
0.063
0.042
0.042
0.042
0.098
0.098
0.098
4.800
4.800
7.620
5.400
8.500
8.540
2.420
3.100
3.350
42.50
3.220
3.380
2.960
5.120
4.960
4.960
1.510
1.510
1.510
1.510
1.450
1.450
1.380
1.380
1.320
1.320
184.2
184.2
184.2
8.500
1.210
1.210
1.210
0.120
0.120
0.120
9.500
9.500
9.500
−222
−222
−236
−314
−189
−308
−306
−232
−211
−434
−434
−428
−418
−334
−355
−355
−268
−266
−268
−268
−222
−222
−226
−226
−242
−242
−111
−111
−111
−189
−208
−208
−208
−348
−324
−324
−198
−198
−198
6000
6000
10000
12000
5000
8000
10000
13000
15000
28000
22000
22500
30000
52000
51000
51000
22000
22200
22000
22000
29000
28500
29500
29500
31000
31000
36000
36000
36000
5000
8000
8000
8000
6500
7000
7000
10000
10000
10000
1.3100
1.3100
1.3100
0.9142
0.9936
1.3100
0.6550
0.6550
0.6550
0.6550
0.6550
0.6550
0.5035
0.5035
0.5035
0.5035
0.5035
0.5035
0.5035
0.5035
0.5035
0.5035
0.5035
0.5035
0.5035
0.5035
0.9936
0.9936
0.9936
0.9936
0.9142
0.9142
0.9142
0.6550
0.6550
0.6550
1.4200
1.4200
1.4200
0.05690
0.05690
0.05690
0.04540
0.04060
0.05690
0.02846
0.02846
0.02846
0.02846
0.02846
0.02846
0.02075
0.02075
0.02075
0.02075
0.02075
0.02075
0.02075
0.02075
0.02075
0.02075
0.02075
0.02075
0.02075
0.02075
0.04060
0.04060
0.04060
0.04060
0.04540
0.04540
0.04540
0.02846
0.02846
0.02846
0.06770
0.06770
0.06770
40
242
550
0.00313
7.97
647.83
300
0.035
1.510
−268
22000
0.5035
0.02075
Test system 6
Unit
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
Pmin
71
120
125
125
90
90
280
280
260
260
260
260
260
260
260
260
260
260
260
260
260
260
260
260
280
280
280
280
260
260
260
260
260
260
260
260
120
120
423
423
3
3
160
160
160
160
160
160
160
160
165
165
165
165
180
180
103
198
100
153
163
95
160
160
196
196
196
196
130
130
Pmax
119
189
190
190
190
190
490
490
496
496
496
496
506
509
506
505
506
506
505
505
505
505
505
505
537
537
549
549
501
501
506
506
506
506
500
500
241
241
774
769
19
28
250
250
250
250
250
250
250
250
504
504
504
504
471
561
341
617
312
471
500
302
511
511
490
490
490
490
432
432
a
0.032888
0.008280
0.003849
0.003849
0.042468
0.014992
0.007039
0.003079
0.005063
0.005063
0.005063
0.003552
0.003901
0.003901
0.003901
0.003901
0.002393
0.002393
0.003684
0.003684
0.003684
0.003684
0.004004
0.003684
0.001619
0.005093
0.000993
0.000993
0.002473
0.002547
0.003542
0.003542
0.003542
0.003542
0.003132
0.001323
0.002950
0.002950
0.000991
0.001581
0.902360
0.110295
0.024493
0.029156
0.024667
0.016517
0.026584
0.007540
0.016430
0.045934
0.000044
0.000044
0.000044
0.000044
0.002528
0.000131
0.010372
0.007627
0.012464
0.039441
0.007278
0.000044
0.000044
0.000044
0.018827
0.010852
0.018827
0.018827
0.034560
0.081540
b
61.242
41.095
46.310
46.310
54.242
61.215
11.791
15.055
13.226
13.226
13.226
14.498
14.651
14.651
14.651
14.651
15.669
15.669
14.656
14.656
14.656
14.656
14.378
14.656
16.261
13.362
17.203
17.203
15.274
15.212
15.033
15.033
15.033
15.033
13.992
15.679
16.542
16.542
16.518
15.815
75.464
129.544
56.613
54.451
54.736
58.034
55.981
61.520
58.635
44.647
71.584
71.584
71.584
71.584
85.120
87.682
69.532
78.339
58.172
46.636
76.947
80.761
70.136
70.136
49.840
65.404
49.840
49.840
66.465
22.941
c
1220.645
1315.118
874.288
874.288
1976.469
1338.087
1818.299
1133.978
1320.636
1320.636
1320.636
1106.539
1176.504
1176.504
1176.504
1176.504
1017.406
1017.406
1229.131
1229.131
1229.131
1229.131
1267.894
1229.131
975.926
1532.093
641.989
641.989
911.533
910.533
1074.810
1074.810
1074.810
1074.810
1278.460
861.742
408.834
408.834
1288.815
1436.251
669.988
134.544
3427.912
3751.772
3918.780
3379.580
3345.296
3138.754
3453.050
5119.300
1898.415
1898.415
1898.415
1898.415
2473.390
2781.705
5515.508
3478.300
6240.909
9960.110
3671.997
1837.383
3108.395
3108.395
7095.484
3392.732
7095.484
7095.484
4288.320
13813.001
UR
30
30
60
60
150
150
180
180
300
300
300
300
600
600
600
600
600
600
600
600
600
600
600
600
300
300
360
360
180
180
600
600
600
600
660
900
180
180
600
600
210
366
702
702
702
702
702
702
702
702
1350
1350
1350
1350
1350
720
720
2700
1500
1656
2160
900
1200
1200
1014
1014
1014
1014
1350
1350
DR
120
120
60
60
150
150
300
300
510
510
510
510
600
600
600
600
600
600
600
600
600
600
600
600
300
300
360
360
180
180
600
600
600
600
660
900
180
180
600
600
210
366
702
702
702
702
702
702
702
702
1350
1350
1350
1350
1350
720
720
2700
1500
1656
2160
900
1200
1200
1014
1014
1014
1014
1350
1350
P0
98.4
134
141.5
183.3
125
91.3
401.1
329.5
386.1
427.3
412.2
370.1
301.8
368
301.9
476.4
283.1
414.1
328
389.4
354.7
262
461.5
371.6
462.6
379.2
530.8
391.9
480.1
319
329.5
333.8
390
432
402
428
178.4
194.1
474
609.8
17.8
6.9
224.3
210
212
200.8
220
232.9
168
208.4
443.9
426
434.1
402.5
357.4
423
220
369.4
273.5
336
432
220
410.6
422.7
351
296
411.1
263.2
370.3
418.7
Unit
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
Pmin
137
137
195
175
175
175
175
330
160
160
200
56
115
115
115
207
207
175
175
175
175
360
415
795
795
578
615
612
612
758
755
750
750
713
718
791
786
795
795
795
795
94
94
94
244
244
244
95
95
116
175
2
4
15
9
12
10
112
4
5
5
50
5
42
42
41
17
7
7
26
Pmax
455
455
541
536
540
538
540
574
531
531
542
132
245
245
245
307
307
345
345
345
345
580
645
984
978
682
720
718
720
964
958
1007
1006
1013
1020
954
952
1006
1013
1021
1015
203
203
203
379
379
379
190
189
194
321
19
59
83
53
37
34
373
20
38
19
98
10
74
74
105
51
19
19
40
a
0.023534
0.035475
0.000915
0.000044
0.000044
0.001307
0.000392
0.000087
0.000521
0.000498
0.001046
0.132050
0.096968
0.054868
0.054868
0.014382
0.013161
0.016033
0.013653
0.028148
0.013470
0.000064
0.000252
0.000022
0.000022
0.000203
0.000198
0.000215
0.000218
0.000193
0.000197
0.000324
0.000344
0.000690
0.000650
0.000233
0.000239
0.000261
0.000259
0.000707
0.000786
0.014355
0.014355
0.014355
0.030266
0.030266
0.030266
0.024027
0.001580
0.022095
0.076810
0.953443
0.000044
0.072468
0.000448
0.599112
0.244706
0.000042
0.085145
0.524718
0.176515
0.063414
2.740485
0.112438
0.041529
0.000911
0.005245
0.234787
0.234787
1.111878
b
64.314
45.017
70.644
70.959
70.959
70.302
70.662
71.101
37.854
37.768
67.983
77.838
63.671
79.458
79.458
93.966
94.723
66.919
68.185
60.821
68.551
2.842
2.946
3.096
3.040
1.709
1.668
1.789
1.815
2.726
2.732
2.651
2.798
1.595
1.503
2.425
2.499
2.674
2.692
1.633
1.816
89.830
89.830
89.830
64.125
64.125
64.125
76.129
81.805
81.140
46.665
78.412
112.088
90.871
97.116
83.244
95.665
91.202
104.501
83.015
127.795
77.929
92.779
80.950
89.073
161.288
161.829
84.972
84.972
16.087
c
4435.493
9750.750
1042.366
1159.895
1159.895
1303.990
1156.193
2118.968
779.519
829.888
2333.690
2028.954
4412.017
2982.219
2982.219
3174.939
3218.359
3723.822
3551.405
4322.615
3493.739
226.799
382.932
156.987
154.484
332.834
326.599
345.306
350.372
370.377
367.067
124.875
130.785
878.746
827.959
432.007
445.606
467.223
475.940
899.462
1000.367
1269.132
1269.132
1269.132
4965.124
4965.124
4965.124
2243.185
2290.381
1681.533
6743.302
394.398
1243.165
1454.740
1011.051
909.269
689.378
1443.792
535.553
617.734
90.966
974.447
263.810
1335.594
1033.871
1391.325
4477.110
57.794
57.794
1258.437
UR
1350
1350
780
1650
1650
1650
1650
1620
1482
1482
1668
120
180
120
120
120
120
318
318
318
318
18
18
36
36
138
144
144
144
48
48
36
36
30
30
30
30
36
36
36
36
120
120
120
480
480
480
240
240
120
180
90
90
300
162
114
120
1080
60
66
12
300
6
60
60
528
300
18
18
72
DR
1350
1350
780
1650
1650
1650
1650
1620
1482
1482
1668
120
180
180
180
180
180
318
318
318
318
18
18
36
36
204
216
216
216
48
48
54
54
30
30
30
30
36
36
36
36
120
120
120
480
480
480
240
240
120
180
90
90
300
162
114
120
1080
60
66
6
300
6
60
60
528
300
30
30
120
P0
409.6
412
423.2
428
436
428
425
497.2
510
470
464.1
118.1
141.3
132
135
252
221
245.9
247.9
183.6
288
557.4
529.5
800.8
801.5
582.7
680.7
670.7
651.7
921
916.8
911.9
898
905
846.5
850.9
843.7
841.4
835.7
828.8
846
179
120.8
121
317.4
318.4
335.8
151
129.5
130
218.8
5.4
45
20
16.3
20
22.1
125
10
13
7.5
53.2
6.4
69.1
49.9
91
41
13.7
7.4
28.6
Highlights:





An innovative and strong optimization technique based on harmony
search is proposed
The proposed algorithm is tested on solving economic emission
dispatch problem
It has the potential to be applied to many other engineering
optimization problems
Four test systems considering valve point effect and transmission loss
are studied
High quality solutions are obtained and compared with a large number
of other methods
Conflict of Interest:
Authors have no conflict of interest.
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