RECENT ADVANCES in ENERGY & ENVIRONMENT

Solving Economic Dispatch Problems with Improved Harmony Search

T. RATNIYOMCHAI, A. OONSIVILAI, P. PAO-LA-OR, and T. KULWORAWANICHPONG
Power System Research Unit, School of Electrical Engineering
Suranaree University of Technology
111 University Avenue, Suranaree District, Nakhon Ratchasima
THAILAND
thanatchai@gmail.com

Abstract: - This paper presents the use of the improved harmony search method for solving economic load dispatch
problems. The harmony search method mimics a jazz improvisation process by musicians in order to seek a fantastic
state of harmony. To assess the searching performance of the proposed method, a six-unit thermal generating system
acquired from the standard IEEE 30-bus test system was challenged. Satisfactory results obtained from the proposed
method were compared to those obtained by genetic algorithms, evolutionary programming, adaptive tabu search and
particle swarm. Also, effects of valve-point loading units were included and discussed.

Key-Words: - Economic dispatch, genetic algorithms, evolutionary programming, adaptive tabu search, particle swarm
optimization

1 Introduction due to having no restrictions on the shape of the cost

Engineering optimization problems contain many curves. Although it does not guarantee the globally
practical complex constraints. They can be formulated optimal solution in limited time, it does normally
and therefore solved as nonlinear programming models. provide good solutions with computational cost [5].
The methods for solving this kind of problems include In the past decades, many optimization algorithms
traditional mathematical programming (such as linear are tried with different kinds of constraints. Several
programming, quadratic programming, dynamic mathematical programming and modern heuristic search
programming, gradient methods and Lagrangian can be found extensively [5,6]. Evolutionary search
relaxation approaches [1]) and modern meta-heuristic methods have becomes more popular to solve any
methods (such as simulated annealing, genetic mathematical functions [7]. The natural selection and
algorithms, evolutionary algorithms, adaptive tabu meta-heuristic methods are useful for finding the global
search, particle swarm optimization, etc [2]). Some of optimum solution, since they all are maintaining
these methods are successful in locating the optimal population of solutions to the considered problem.
solution, but they are usually slow in convergence and Harmony search method has been developed by Geem et
require very expensive computational cost. Some other al [8]. It imitates the improvisation process of musicians
methods may risk being trapped to a local optimum, to find the perfect state of harmony. It has been
which is the problem of premature convergence. successfully applied to various mathematical
Economic load dispatch is one of well-known optimization problems in the application field of civil
problems in a field of power system optimization [3]. and mechanical engineering. However, its first version
The problem of dividing the total load demand among was invented as a combinatorial optimization where
available online generators economically and also decision variables are discrete. To apply the harmony
satisfying various system constraints simultaneously is search method to the real world engineering in which
called economic load dispatch. This is an important task many search spaces are continuous, some procedure of
in power system for allocating power generations among the harmony search method must be modified to be able
the committed units such that the constraints imposed to handle continuous search variables. Hence, it is an
are satisfied, the energy demands are met, and the improved version of the harmony search method which
corresponding cost is minimized. Improvements in is called as the improved harmony search method.
scheduling of the unit generations can lead to significant This paper solves an economic load dispatch problem
cost savings. In view of the nonlinear characteristics of using the improved harmony search method. The test
this problem, there is a demand for the optimization considers a six-unit generating system acquired from the
methods that do not have restrictions on the shape of the standard IEEE 30-bus test system [9]. The results
fuel-cost curves [4]. As some stochastic search obtained by the improved harmony search method are
algorithms as mentioned above may prove to be very compared with those of other promising methods. The
effective in nonlinear economic load dispatch problems

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 RECENT ADVANCES in ENERGY & ENVIRONMENT

proposed method proves to be a robust optimization Pi min ≤ Pi ≤ Pi max , i = 1, 2," , N G (3)

technique for solving economic load dispatch problems.
This paper organizes a total of five sections. Next Where
section, Section 2 illustrates economic load dispatch PD is the total power demand of the plant
problems with corresponding mathematical expressions PLoss is the total power losses of the plant
of its objective and various practical constraints. Section
Pi min is the minimum output of generating unit i
3 gives the brief of some meta-heuristic search methods
used for comparative purpose. It also provides the Pi max is the maximum output of generating unit i
algorithm procedure, described step-by-step. Section 4 is
the simulation results and discussion. Conclusion remark 2.2 Economic load dispatch problem
is in Section 5. The solution of economic dispatch problem will give the
amount of active power to be generated by different
units at the minimum production cost for a particular
2 Problem Formulation demand while keep operating the system within all the
Real power generation can be allocated to available constraint limits. This is the economic load dispatch
generating units in many different ways [6]. In this problems being as a constrained nonlinear optimization
paper, the economic objective and some practical problem [3,5,6] as follows.
constraints of the economic load dispatch problems are
illustrated as follows. Minimize FT
Subject to g i ( x ) = 0, ∀i (4)
2.1 Economic objective function
The economic dispatch problem is to find the optimal h j (x ) ≤ 0, ∀j
combination of power generation in such a way that the
total production cost of the entire system is minimized Where
while satisfying the total power demand and some key x is a vector of decision variables
power system constraints. The fuel cost for each power gi(x) is an equality constraint i
generation unit is defined. Hence, the total production hj(x) is an inequality constraint I
cost function of economic dispatch problem is defined as
the total sum of the fuel costs of all generating plant To solve this constrained optimization with some
units as described follows. efficient mathematical programming and modern meta-
heuristic methods [1,5], penalty method is used to
{ }
NG
FT = ∑ ai Pi 2 + bi Pi + ci + d i sin ei ( Pi min − Pi )
convert a constrained optimization problem to an
(1) unconstrained optimization problem. Therefore,
i =1
problems of a single objective function are formulated
Where and can be solved accordingly. The penalty function can
NG is the total number of generating units be expressed as follows.
FT is the total production cost
⎛ 2⎞
Pi is the power output of generating unit i
Pi min is the minimum output of generating unit i
[ ]
P( x ) = FT + ρ ⎜⎜ ∑ g i2 ( x ) + ∑ max{h j ( x ),0} ⎟⎟ (5)
⎝ i j ⎠
ai, bi, ci, di, ei are fuel cost coefficients of unit i
Where
It should note that (1) describes the fuel cost function in ρ is the penalty factor
which valve-point loading effect [4,10] is included.

2.2 Problem constraints

There are equality and inequality constraints in this kind
3 Meta-Heuristic Methods for Solving
of problems. A power balance equation (2) is set as an Optimization Problems
equality constraint whereas the limits of power
generation output (3) are inequality constraints. 3.1 Genetic algorithms (GAs)
There exist many different approaches to adjust the
NG motor parameters. The GAs is well-known [11,12], there
PD + PLoss − ∑ Pi = 0 (2) exist a hundred of works employing the GAs technique
i =1
to identify system parameters in various forms. The GAs

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 RECENT ADVANCES in ENERGY & ENVIRONMENT

is a stochastic search technique that leads a set of payoff for the sequence of symbols (e.g., average payoff
population in solution space evolved using the principles per symbol) indicates the fitness of the machine or
of genetic evolution and natural selection, called genetic program. Offspring machines are created by randomly
operators e.g. crossover, mutation, etc. With successive mutating the parents and are scored in a similar manner.
updating new generation, a set of updated solutions Those machines that provide the greatest payoff are
gradually converges to the real solution. The GAs is retained to become parents of the next generation, and
very popular and widely used in most research areas the process iterates. When new symbols are to be
where an intelligent search technique is applied. predicted, the best available machine serves as the basis
In this paper, the GAs is selected to build up an for making such a prediction and the new observation is
algorithm to solve economic dispatch problems (all added to the available database. Fogel described this
generation from available generating units). To reduce process as “evolutionary programming” in contrast to
programming complication, the Genetic Algorithms “heuristic programming” [13].
(GADS TOOLBOX in MATLAB [12]) is employed to
generate a set of initial random parameters. With the 3.3 Adaptive tabu search (ATS)
searching process, the parameters are adjusted to give The tabu search method [10, 14] is an iterative process
the best result. that searches for the best solution by moving from a
current solution to find a better solution repeatedly. One
3.2 Evolutionary programming (EP) of the important features of the TS method is its tabu list
Evolutionary programming was invented by Lawrence that keeps the history of search paths. The information in
J. Fogel [7] in 1960. At the time, artificial intelligence the list is used for finding a new direction of search
was limited to two main avenues of investigation: movement. Every new is expected to search a better
modeling the human brain or neural networks, and solution and ultimately the optimum one. Another
modeling the problem solving behavior of human feature of the tabu search method is its aspiration
experts or heuristic programming. Both focused on criterion. The aspiration criterion provides preferable
emulating humans as the most advanced intelligent characteristics of any possible solutions. It is particularly
organism produced by evolution. The alternative, useful for the selection of a proper solution from a set of
envisioned by Fogel, was to refrain from modeling the satisfied solutions.
end product of evolution but rather to model the process In order to improve the performance of the tabu
of evolution itself as a vehicle for producing intelligent search method, we have proposed two additional
behavior. Fogel viewed intelligence as a composite mechanisms namely back-tracking and adaptive search
ability to make predictions in an environment coupled radius. The enhanced version of the tabu search method
with the translation of each prediction into a suitable has been named the adaptive tabu search [15].
response in light of a given goal (e.g. to maximize a Regarding to the intensification mechanism, the back-
payoff function). Thus, the viewed prediction is a tracking mechanism allows the search to look backward
prerequisite for intelligent behavior. The modeling of to some previous solutions stored in the tabu list. This
evolution as an optimization process was a consequence mechanism may become necessary when the search
of Fogel’s expertise in the emerging fields of encounters an entrapment caused by a local solution. An
biotechnology (at the time defined as the utilization of alternative solution is then chosen from the current and
mathematics to describe the functioning of a human the previous solutions. With the back-tracked solution, a
operator), cybernetics, and engineering. new search space is created. Given this new search space
Fogel crafted a series of experiments in which finite to explore, the search moves in a new direction away
state machines represented individual organisms in a from that approaching the local solution. Note that the
population of problem solvers. These graphical models new solution chosen here is not necessary to be the best
are used to describe the behavior or computer software solution within the current search space but it helps the
and hardware, which is why he termed his approach search to escape from an entrapment.
"Evolutionary Programming". The experimental
procedure was as follows. A population of finite state 3.4 Particle swarm optimization (PSO)
machine is exposed to the environment – that is, the Kennedy and Eberhart developed a particle swarm
sequence of symbols that has been observed up to the optimization algorithm based on the behavior of
current time. For each parent machine, as each input individuals (i.e., particles or agents) of a swarm [16-18].
symbol is presented to the machine, the corresponding Its roots are in zoologist’s modeling of the movement of
output symbol is compared with the next input symbol. individuals (i.e., fish, birds, and insects) within a group.
The worth of this prediction is then measured with It has been noticed that members of the group seem to
respect to the payoff function (e.g., all-none, squared share information among them to lead to increased
error). After the last prediction is made, a function of the

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 RECENT ADVANCES in ENERGY & ENVIRONMENT

efficiency of the group. The particle swarm optimization their notes make a new harmony (Sol, Ti, Do)
algorithm searches in parallel using a group of which is musically the chord C7. If this new
individuals similar to other AI-based heuristic harmony is better than the existing worst harmony
optimization techniques. Each individual corresponds to in their memories, the new harmony is included in
a candidate solution to the problem. Individuals in a
their memories and the worst harmony is excluded
swarm approach to the optimum through its present
from their memories. This procedure is repeated
velocity, previous experience, and the experience of its
until a fantastic harmony is found.
neighbors. In a physical n-dimensional search space, the
However, its first version was invented as a
position and velocity of individual i are represented as
combinatorial optimization where decision variables are
the velocity vectors. Using these information individual i
discrete. To apply the harmony search method to the real
and its updated velocity can be modified under the
world engineering in which many search spaces are
following equations in the particle swarm optimization
continuous, some procedure of the harmony search
algorithm.
method must be modified to be able to handle
continuous search variables. Together, the parameter
xi( ) = xi( ) + vi( )
k +1 k k +1
(7) called bandwidth is used and adaptively changed by

( )
variance of population. Hence, it is an improved version
vi( ) = vi( ) + αi xilbest − xi( ) +
k +1 k k
of the harmony search method which is called as the

β (x − x( ) )
(8) improved harmony search method [19-23].
gbest k
i i
Where
5 Simulation Results
xi( ) is the individual i at iteration k
k
To verify the effectiveness of the proposed improved
harmony search method, a six-unit thermal power
vi( ) is the updated velocity of individual i at
k
generating plant acquired from the standard IEEE 30-
iteration k bus test system was tested. Fuel cost coefficients and
αi, βi are uniformly random numbers between [0,1] generation limits for each generating unit of the test
system were given in Table 1.
xilbest is the individual best of individual i
x gbest is the global best of the swarm Table 1: Fuel cost coefficients for each generating unit
i a b c D e min max
1 100 200 10 15 6.283 0.05 0.5
3.5 Improved harmony search (IHS)
2 120 150 10 10 8.976 0.05 0.6
The harmony search algorithm [8] was 3 40 180 20 10 14.784 0.05 1.0
conceptualized from the musical process of 4 60 100 10 5 20.944 0.05 1.2
searching for a ‘perfect state’ of harmony, such as 5 40 180 20 5 25.133 0.05 1.0
jazz improvisation. Jazz improvisation seeks a best 6 100 150 10 5 18.48 0.05 0.6
state (fantastic harmony) determined by aesthetic
estimation, just as the optimization algorithm seeks The simulations were performed using MATLAB
a best state (global optimum) determined by software. The test were carried out by solving economic
evaluating the objective function. Aesthetic load dispatch of a single power demand case, PD = 3.6
p.u.. For comparison purposes, some meta-heuristic
estimation is performed by the set of pitches played
search (GA, EP, ATS and PSO) were also applied to
by each instrument, just as the objective function solve this test case. The results of which are presented as
evaluation is performed by the set of values follows.
assigned by each decision variable. The harmony
quality is enhanced practice after practice, just as 5.1 Solution by genetic algorithms
the solution quality is enhanced iteration by In this case, some parameters must be assigned for the
iteration. Consider a jazz trio composed of a use of genetic algorithms to solve the economic dispatch
saxophone, double bass, and guitar. Assume there problems as follows:
exists a certain number of preferable pitches in each • Population size = 20
musician’s memory: saxophonist {Do, Mi, Sol}, • Maximum generation = 1000
double bassist {Ti, Sol, Re}, and guitarist {La, Fa, • Crossover rate = 0.8
Do}. If the saxophonist plays note Sol, the double • Mutation rate = 0.2
bassist plays Ti, and the guitarist plays Do, together

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 RECENT ADVANCES in ENERGY & ENVIRONMENT

The obtained results for the six-unit system using the 5.4 Solution by particle swarm optimization
genetic algorithms were given in Table 2. It showed that In this case, some parameters must be assigned for the
the genetic algorithms has succeeded in finding a global use of particle swarm optimization to solve the
optimal solution for this case. economic dispatch problems as follows:
• Number of particles = 20
Table 2: Optimal solution for GA case • Maximum generation = 1000
P1 0.4200 P4 1.0705 • Maximum velocity = 15
P2 0.3826 P5 0.6875
P3 0.7217 P6 0.3177 The obtained results for the six-unit system using the
FT = 1704.2 Baht/h particle swarm optimization were given in Table 5. It
showed that the particle swarm optimization has
5.2 Solution by evolutionary programming succeeded in finding a global optimal solution for this
In this case, some parameters must be assigned for the case.
use of evolutionary programming to solve the economic
dispatch problems as follows: Table 5: Optimal solution for PSO case
• Population size = 20 P1 0.0798 P4 1.1123
• Maximum generation = 1000 P2 0.4701 P5 0.5801
• Scaling factor β = 0.01 P3 0.9261 P6 0.4314
FT = 1702.0 Baht/h
The obtained results for the six-unit system using the
evolutionary programming were given in Table 3. It 5.5 Solution by improved harmony search
showed that the evolutionary programming has In this case, some parameters must be assigned for the
succeeded in finding a global optimal solution for this use of improved harmony search to solve the economic
case. dispatch problems as follows:
• Maximum generation = 5000
Table 3: Optimal solution for EP case • Harmony memory size = 20
P1 0.4189 P4 0.8581 • Maximum stalled generation = 250
P2 0.4946 P5 0.6354
P3 0.9611 P6 0.2320 2800

FT = 1678.7 Baht/h 2600

2400
Cost function

5.3 Solution by adaptive tabu search 2200

In this case, some parameters must be assigned for the 2000

use of adaptive tabu search to solve the economic 1800

dispatch problems as follows: 1600

0 100 200 300 400 500 600

• Neighborhood size = 30
Iteration

• Maximum generation = 1000 250

• Initial neighborhood radius = 0.05 200

Iteration stalled

150

The obtained results for the six-unit system using the 100

adaptive tabu search were given in Table 4. It showed

50
that the adaptive tabu search has succeeded in finding a
global optimal solution for this case. 0
0
Iteration

Table 4: Optimal solution for ATS case Fig. 1. Solution convergence by IHS
P1 0.4189 P4 0.9309
Table 6: Optimal solution for IHS case
P2 0.3298 P5 0.7081
P1 0.1899 P4 1.0449
P3 0.6448 P6 0.5779
P2 0.4679 P5 0.6330
FT = 1699.7 Baht/h
P3 0.9096 P6 0.3571
FT = 1696.0 Baht/h

The obtained results for the six-unit system using the

improved harmony search were given in Table 6. It

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 RECENT ADVANCES in ENERGY & ENVIRONMENT

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