DSpace at VNU: Modified cuckoo search algorithm for short-term hydrothermal scheduling

Electrical Power and Energy Systems 65 (2015) 271–281

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Electrical Power and Energy Systems
journal homepage: www.elsevier.com/locate/ijepes

Modified cuckoo search algorithm for short-term hydrothermal
scheduling
Thang Trung Nguyen a, Dieu Ngoc Vo b,⇑
a
b

Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, 19 Nguyen Huu Tho str., 7th dist., Ho Chi Minh city, Viet Nam
Department of Power Systems, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet str., 10th dist., Ho Chi Minh city, Viet Nam

a r t i c l e

i n f o

Article history:
Received 8 March 2014
Received in revised form 11 June 2014
Accepted 10 October 2014

Keywords:
Lévy flight
Modified cuckoo search algorithm
Non-convex fuel cost function
Short-term hydrothermal scheduling
Water availability constraint

a b s t r a c t
This paper proposes a modified cuckoo search algorithm (MCSA) for solving short-term hydrothermal
scheduling (HTS) problem. The considered HTS problem in this paper is to minimize total cost of thermal
generators with valve point loading effects satisfying power balance constraint, water availability, and
generator operating limits. The MCSA method is based on the conventional CSA method with modifications to enhance its search ability. In the MCSA, the eggs are first sorted in the descending order of their
fitness function value and then classified in two groups where the eggs with low fitness function value
are put in the top egg group and the other ones are put in the abandoned one. The abandoned group,
the step size of the Lévy flight in CSA will change with the number of iterations to promote more localized
searching when the eggs are getting closer to the optimal solution. On the other hand, there will be an
information exchange between two eggs in the top egg group to speed up the search process of the eggs.
The proposed MCSA method has been tested on different systems and the obtained results are compared
to those from other methods available in the literature. The result comparison has indicated that the proposed method can obtain higher quality solutions than many other methods. Therefore, the proposed
MCSA can be a new efficient method for solving short-term fixed-head hydrothermal scheduling
problems.
Ó 2014 Elsevier Ltd. All rights reserved.

Introduction
A modern power system consists of a large number of thermal
and hydro plants connected at various load centers through a
transmission network. An important objective in the operation of
such a power system is to generate and transmit power to meet
the system load demand at minimum fuel cost by an optimal
mix of various types of plants. However, the hydro resources being
limited, thus the worth of water is greatly increased [1]. Therefore,
an optimal operation of a hydrothermal system will lead to a huge
saving in fuel cost of thermal power plants. The objective of the
hydrothermal scheduling problem is to find the optimum allocation of hydro energy so that the annual operating cost of a mixed
hydrothermal system is minimized [1]. Several conventional
methods have been implemented for solving the hydrothermal

scheduling problem such as gradient search techniques (GS) [2],
lambda-gamma iteration method, dynamic programming (DP)
⇑ Corresponding author at: Department of Power Systems, Ho Chi Minh City
University of Technology, 268 Ly Thuong Kiet str., 10th dist., Ho Chi Minh city, Viet
Nam. Tel.: +84 8 3864 7256x5730.
E-mail address: (D.N. Vo).
/>0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

[2], Lagrange relaxation (LR) [3], decomposition and coordination
method [4], mixed integer programming (MIP) [5], and Newton’s
method [6]. The GS method has been applied to the problem where
the hydro generation models were represented as piecewise linear
functions or polynomial approximation with a monotonically
increasing nature. However, such an approximation may be too
rough and seems impractical. In the lambda-gamma method, the
gamma values associated with different hydro plants are initially
chosen and then the lambda iterations are invoked for the given
power demand at each interval of the schedule time horizon. The
DP method is another popular optimization method implemented
for solving the hydrothermal scheduling problems. However, computational and dimensional requirements in the DP method will
drastically increase for large-scale systems [7]. On the contrary to
the DP method, the LR method is more reliable and efficient for
dealing with large-scale problems. However, the LR method may
suffer to the duality gap oscillation during the convergence process
due to the dual problem formulation, leading to divergence for
some problems with non-convexity of incremental heat rate curves
of thermal generators. In the decomposition and coordination
method, the problem is decomposed into thermal and hydro subproblems and they are solved by network flow programming and

272

T.T. Nguyen, D.N. Vo / Electrical Power and Energy Systems 65 (2015) 271–281

Nomenclature
ahj, bhj, chj water discharge coefficients of hydro plant j
asik, bsi, csi fuel cost coefficients of thermal plant i
dsi esi
fuel cost coefficients of thermal plant i with valvepoint effects
Bij, B0i, B00 B-matrix coefficients for transmission power loss
PD,m
total system load demand at subinterval m
Phj,m
power output of hydro plant j in subinterval m
Phj,max
maximum power output of hydro plant j

priority list based dynamic programming methods. In order to
solve the HTS problem, the MIP method requires a linearization
of equations whereas the decomposition and coordination method
may encounter the difficulties when dealing with the non-linearity
of objective function and/or constraints. The Newton’s method is
computationally stable, effective, and fast for solving a set of nonlinear equations. Therefore, it has a high potential for implementation on optimization problems such as economic load dispatch in
hydrothermal power systems. However, a drawback of the Newton’s method is the dependence on the formulation and inversion
of Jacobian matrix, leading to its restriction of applicability on
large-scale problems. Generally, these conventional methods can
be efficiently applicable for the HTS problems with differentiable
fuel cost function and constraints. A multistage Benders decomposition method has been presented in [8,9] for solving a short-term
hydrothermal scheduling problem. In this method, an alternative
strategy is proposed to decompose the HTS problem into many

stages with each stage comprising variables and constraints of several time-steps. The advantage of this approach is that it allows
exploring the best trade-off between solving a ‘‘larger number of
shorter stages’’ and solving a ‘‘shorter number of larger stages’’.
As a result, the multistage Benders decomposition method in [8]
can reduce the number of iterations for convergence. However,
the computational time for the subproblem in each stage is
increased. For enhancing the efficiency of the method, there is an
additionally optimal aggregation factor suggested in [9] that yields
the least computational time for the overall problem.
Recently, several novel methods based on artificial intelligence
techniques have been implemented for solving the HTS problems
such as simulated annealing approach (SA) [10], evolutionary programming (EP) [11–14], genetic algorithm (GA) [15–18], differential evolution (DE) [19], artificial immune system (AIS) [20], and
Hopfield neural network (HNN) [21]. In the SA technique, the
appropriate setting of the relevant control parameters is a difficult
task and it usually suffers slow speed of convergence when dealing
with practical sized power systems. Both the GA and EP algorithms
are evolutionary based methods for solving optimization problems.
However, the essential encoding and decoding schemes in the both
methods are different. In the GA method, the required crossover
and mutation operations to diversify the offspring may be detrimental to actually reaching an optimal solution. In this regard,
the EP technique is more likely better when overcoming these disadvantages where the mutation is a key search operator which
generates new solutions from the current ones [17]. However,
one disadvantage of the EP method in solving some multimodal
optimization problems is its slow convergence to a near optimum.
Another evolutionary based method for solving optimization problems is DE method which has the ability to search in very large
spaces of candidate solutions with few or no assumptions about
the considered problem. However, the DE method is slow or no
convergence to the near optimum solution when dealing with
large-scale problems. The AIS method is one of the efficient

Phj,min
PL,m
Psi,m
Psi,max
Psi,min
qj,m
Wj

minimum power output of hydro plant j
total transmission loss at subinterval m
power output of thermal plant i in subinterval m
maximum power output of thermal plant i
minimum power output of thermal plant i
rate of water flow from hydro plant j in subinterval m
volume of water available for hydro unit j during the
scheduled period

metaheuristic search methods for solving optimization problems.
In the AIS method, the most important step is the application of
the aging operator to eliminate the old antibodies to maintain
the diversity of the population and avoid a premature convergence.
The advantages of the AIS method are few control parameters and
small number of iterations. However, the AIS method also suffers a
difficulty when dealing with large-scale problems like other metaheuristic search methods. The HNN method is an efficient neural
network for dealing with optimization problems. However, it
encounters a difficulty of predetermining the synaptic interconnections among neurons which may lead to constraint mismatch
if the weighting coefficients associated with constraints in its
energy function are not carefully selected. In addition, the HNN
method also suffers slow convergence to an optimal solution and
the constraints of the problems must be linearized when applying

in HNN [22]. In general, most of the artificial intelligence based
techniques are efficient for finding near optimum solution for complex problems but they also usually suffer slow convergence, especially for large-scale problems.
Cuckoo search algorithm (CSA) is a new metaheuristic algorithm for solving optimization problems developed by Yang and
Deb in 2009 [23]. This algorithm is inspired from the reproduction
strategy of cuckoo species in the nature. At the most basic level,
cuckoos lay their eggs in the nests of other host birds which may
be of different species. The host bird may discover strange eggs
in its nest and it either destroys the eggs or abandons the nest to
build a new one. The effectiveness of the CSA method over other
methods such as GA and particle swarm optimization (PSO) has
been validated on benchmarked functions [23]. Moreover, CSA
has been also successfully applied for solving non-convex economic dispatch (ED) problems [24,25] and micro grid power dispatch problem [25]. However, the conventional CSA still suffers
slow convergence for complex and large-scale problems. Therefore,
a new modified CSA (MCSA) has been proposed by Walton et al.
[26] to speed up its convergence to the optimal solution. The efficiency of the MCSA method over other methods such as conventional CSA, DE and PSO has been given in [26].
This paper proposes MCSA method for solving short-term
hydrothermal scheduling (HTS) problem. The considered HTS
problem in this paper is to minimize total cost of thermal generators with valve point loading effects satisfying power balance constraint, water availability, and generator operating limits. The
MCSA method is based on the conventional CSA method with modifications to enhance its search ability. In the MCSA, the eggs are
first sorted in the descending order of their fitness function value
and then classified in two groups where the eggs with low fitness
function value are put in the top egg group and the other ones are
put in the abandoned one. The abandoned group, the step size of
the Lévy flight in CSA will change with the number of iterations
to promote more localized searching when the eggs are getting closer to the optimal solution. On the other hand, there will be an
information exchange between two eggs in the top egg group to


T.T. Nguyen, D.N. Vo / Electrical Power and Energy Systems 65 (2015) 271–281

speed up the search process of the eggs. The proposed MCSA
method has been tested on different hydrothermal systems and
the obtained results have been compared to those from other
methods available in the literature such as Newton’s method and
HNN in [21], and PSO, DE, EP and AIS in [20].
The remaining organization of the paper is as follows. The problem formulation is given in Section ‘Problem formulation’. The
implementation of MCSA for the problem is presented in Section ‘Implementation of MCSA for HTS Problem’. The numerical
results are provided in Section ‘Numerical results’. Finally, the conclusion is given.
Problem formulation
The objective of the HTS problem is to minimize the total fuel
cost of thermal generators while satisfying various hydraulic,
power balance, and generator operating limits constraints. The
objective function of the problem includes only total operation cost
of thermal units since the operation cost of hydro units is not considerable and it is negligible. In this paper, the short-term fixedhead HTS problems are considered where the effect of reservoir
head variation on the power output of hydro units is neglected.
The mathematical formulation of the short-term fixed-head HTS
problem consisting of N1 thermal units and N2 hydro units scheduled in M sub-intervals with t hours for each is formulated as follows [20,21].
The objective is to minimize the total cost of thermal generators
considering valve loading effects:

Min C T ¼

N1
M X
h
X
t m asi þ bsi Psi;m þ csi P2si;m þ jdsi
m¼1 i¼1





i


À Psi;m

 sin esi  P min
si

ð1Þ

subject to:
– Power balance constraint: The total power generation from thermal and hydro plants must satisfy the total load demand and
power loss in each subinterval:
N1
N2
X
X
Psi;m þ
Phj;m À P L;m À P D;m ¼ 0;
i¼1

m ¼ 1; . . . ; M

ð2Þ

j¼1

where the power losses in transmission lines are calculated using

Kron’s formula [2]:

PL;m ¼

NX
1 þN 2 NX
1 þN 2
i¼1

Pi;m Bij Pj;m þ

j¼1

NX
1 þN 2

B0i Pi;m þ B00

ð3Þ

i¼1

– Water availability constraint: The total available water discharge
of each hydro plant for the whole scheduled time horizon is limited by:
M
X

273

Implementation of MCSA for HTS Problem

Cuckoo search algorithm
The CSA is a new and efficient population-based heuristic evolutionary algorithm for solving optimization problems with the
advantages of simple implement and few control parameters
[23]. This algorithm is based on the obligate brood parasitic behavior of some cuckoo species combined with the Lévy flight behavior
of some birds and fruit flies. There are mainly three principal rules
during the search process as follows [27].
1. Each cuckoo lays one egg (a design solution) at a time and
dumps its egg in a randomly chosen nest among the fixed number of available host nests.
2. The best nests with high quality of egg (better solution) will be
carried over to the next generation.
3. The number of available host nests is fixed, and a host bird can
discover an alien egg with a probability pa 2 [0,1]. In this case, it
can either throw the egg away or abandon the nest so as to
build a completely new nest in a new location.
As a further approximation, the last assumption can be approximated by a fraction pa of the n host nests are replaced by new
nests (with new random solutions). For maximization problems,
the quality or fitness of a solution can simply be proportional to
the value of the objective function. Other forms of fitness can be
defined in a similar way to the fitness function in genetic algorithms. Generally, the CSA method consists of two important operations including (1) laying egg and (2) destroying and rebuilding
nest [28].
In this paper, the optimal path for the Lévy flights of the CSA is
calculated using Mantegna’s algorithm. The new solution by each
nest is calculated as follows [23]:
ðtþ1Þ

Xi

¼ X ti þ a È Le0 VyðkÞ

ð8Þ

where a > 0 is the step size for updating new solution related to the
scales of the problem of interests. As reported in [23], the CSA
method can obtain high success rates if this value is set to one. In
most cases, the value can be set to one [23]. However, the small value
could be beneficial in problems of small domain [26]. Therefore, the
step size will be tuned and set to different values in the range of [0,1]
corresponding to different systems considered in the paper.
The action of discovery of an alien egg in a nest of the host bird
with the probability of pa also creates a new solution for the problem
similar to the Lévy flights. One of the advantages of the CSA over the
PSO method is that only one parameter, the fraction of nests to
abandon pa, needs to be adjusted in the range of [0,1]. Yang and
Deb [23] found that the effect of this parameter on the convergence
rate of the method is not considerable and it can be fixed at 0.25.
Modified cuckoo search

tm qj;m ¼ W j ;

j ¼ 1; . . . ; N2

ð4Þ

m¼1

where the rate of water flow from hydro plant j in subinterval m is
determined by:

qj;m ¼ ahj þ bhj Phj;m þ chj P2hj;m

ð5Þ

– Generator operating limits: Each thermal and hydro units have
their upper and lower generation limits:

Psi;min 6 Psi;m 6 Psi;max ;

Phj;min 6 Phj;m 6 P hj;max ;

i ¼ 1; . . . ; N 1 ;

j ¼ 1; . . . ; N2 ;

m ¼ 1; . . . ; M

m ¼ 1; . . . ; M

ð6Þ

ð7Þ

Although the CSA method outperforms the PSO and GA methods in terms of success rate and number of required objective function evaluations [23], it finds an optimal solution based entirely on
random walks which cannot guarantee a fast convergence. Therefore, Walton et al. [26] have proposed two modifications to the
CSA method to increase its convergence rate, making the method
more practical for a wider range of applications.
In the MCSA method, the eggs are first sorted in a descending
order based on their corresponding fitness function value in the
problem and then classified into two groups where the eggs with
high fitness function value are put in the abandoned egg group
and the other ones are put in the top egg group. The two modifications are performed as follows.

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T.T. Nguyen, D.N. Vo / Electrical Power and Energy Systems 65 (2015) 271–281

(a) Modification of the abandoned eggs: The improvement is
focused on the step size a which is used to update the new
solution via Lévy flight as in (8). Unlike the CSA method
where the step size value is constant, the step size value in
the MCSA method decreases as the number of iteration
increases. This modification is to enhance localized searching as the eggs are getting closer to the optimal solution.
In the MCSA method, an initial Lévy flight step size is set
to a = A = 1 and at p
each
ffiffiffiffi iteration the new value of a is calculated using a ¼ A= G where G is the current iteration number and A is an initial value of the Lévy flight step size.
(b) Modification of the top eggs and information exchange between
two eggs: This modification is focused on the top eggs with
information exchange between two eggs to speed up convergence to the optimal solution. In the CSA method, the search
for optimal solutions is independently performed due to no
information exchange among eggs. However, for each of
the top eggs in the MCSA method, a second egg in this group
is randomly selected and a new egg is then generated based
on the line connecting these two top eggs. Along this line, the
location of the new egg
pffiffiffiis calculated using the inverse of the
golden ratio u ¼ ð1 þ 5Þ=2. The new egg is then located closer to the egg with the best fitness function value. Note that
the new egg is generated at the midpoint if both eggs have
the same fitness function value. In case that the same egg is
picked twice, a local Lévy flight search is performed from

the randomly picked nest with the step size a ¼ A=G2 . In
the MCSA method, there are two parameters need to be
tuned including the nest fraction to make up the top nests
and the fraction of nests to be abandoned. Based on the
experiments on the benchmark functions, the best values of
the two parameters are suggested to be 0.25 and 0.75,
respectively [26]. These two selected values are also used
to obtain the best solution as in [29].
Calculation of power output for slack thermal and hydro units
In this research, a thermal unit and a hydro unit are arbitrarily
selected based on the equality constraints in the problem to guarantee that the equality constraints are always satisfied. The power
output of the slack hydro unit is calculated based on the availability water constraint while the power output of the slack thermal
unit is determined using the power balance constraint.
To guarantee that the power balance constraint (2) is always
satisfied, a slack thermal unit is arbitrarily selected and thus its
power output will be dependent on the power output of the
remaining N1À1 thermal units and N2 hydro units in the system.
Suppose that the power outputs of last (N1À1) thermal unit and
N2 hydro units at subinterval m are known, the power output of
the first thermal unit as the slack unit is calculated by:
N1
X

N2
X

i¼2

j¼1

where




BTT;ij
Bij ¼


BHT;ij



BTH;ij


;


B
HH;ij






BT;0i

B0i ¼


BH;0i


BTT,ij, BT,0i Power loss coefficients due to thermal units
BHH,ij, BH,0i Power loss coefficients due to hydro units
BTH,ij, BHT,ij Power loss coefficients due to thermal and hydro
units, BTH,ij = BTHT,ij.
Substituting (10) into (9), a quadratic equation is obtained:

A Â P2s1;m þ B Â Ps1;m þ C ¼ 0

ð11Þ

where

A ¼ BTT;11
B¼2

ð12Þ

N1
N2
X
X
BTT;1i Psi;m þ 2 BTH;1j Phj;m þ BT;01 À 1
i¼2

C¼

ð13Þ

j¼1

N1 X
N1
N2 X
N2
N1 X
N2
X
X
X
‘ P si;m BTT;ij P sj;m þ
P hi;m BHH;ij P hj;m þ 2
P si;m BTH;ij P hj;m
i¼2 j¼2

i¼1 j¼1

i¼2 j¼1

N1
N2
N1
N2
X

X
X
X
þ BT;0i P si;m þ
BH;0j P hj;m þ B00 þ P D;m À
P si;m À
P hj;m
i¼2

j¼1

i¼2

j¼1

ð14Þ
The solution of the second order Eq. (11) is obtained by:

Ps1;m ¼

ÀB Æ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B2 À 4AC
2A

ð15Þ

where B2À4AC P 0.
Similarly, suppose that the power output of all hydro units in

the first M-1 subintervals is known. Therefore, the water discharge
of all hydro units in the first M-1 subintervals is then obtained
using (5). Then its water discharge at subinterval M is calculated
using the available water constraint (4) as follows:

qj;M ¼

Wj À

MÀ1
X

!,

t m qj;m

tM

ð16Þ

m¼1

Therefore, the power output of hydro unit j at subinterval M
corresponding to the obtained qj,M from (16) is determined using
(5) as follows:

Phj;M ¼

Àbhj Æ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
bhj À 4chj ðahj À qj;M Þ
2chj

;

j ¼ 1; 2; . . . ; N2

ð17Þ

2

where bhj À 4  chj  ðahj À qj;M Þ P 0.
Implementation of MCSA for HTS

ð9Þ

The proposed MCSA method is implemented for solving the
short-term fixed-head HTS problem as follows.

Eq. (3) is rewritten in terms of the slack thermal unit 1 as
follows:

Initialization
A population of Np host nests is represented by X = [X1, X2, . . .,
XNp]T, in which each Xd (d = 1, ..., Np) represents a solution vector
of variables given by Xd = [Psi,m,d Phj,m,d], where Psi,m,d is the power
out of thermal unit i at subinterval m corresponding to nest d
and Phj,m,d is the power out of hydro unit j at subinterval m corresponding to nest d.

In the MCSA, each egg can be regarded as a solution which is
randomly generated in the initialization. Therefore, each element
in nest d of the population is randomly initialized as follows:

Ps1;m ¼ PD;m þ PL;m À

PL;m ¼

BTT;11 P2s1;m

Psi;m À

Phj;m

!
N1
N2
X
X
þ 2 BTT;1i Psi;m þ 2 BTH;1j Phj;m þ BT;01 Ps1;m
i¼2

j¼1

N1 X
N1
N2 X
N2
N1 X
N2

X
X
X
þ
Psi;m BTT;ij Psj;m þ
Phi;m BHH;ij Phj;m þ 2
Psi;m BTH;ij Phj;m
i¼2 j¼2

i¼1 j¼1

i¼2 j¼1

N1
N2
X
X
þ BT;0i Psi;m þ
BH;0j P hj;m þ B00
i¼2

j¼1

ð10Þ

P si;m;d ¼ P si;min þ rand1 Ã ðP si;max À P si;min Þ; i ¼ 2; . .. ; N 1 ; m ¼ 1; . .. ; M

ð18Þ

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T.T. Nguyen, D.N. Vo / Electrical Power and Energy Systems 65 (2015) 271–281

P hj;m;d ¼ P hj;min þ rand2 Ã ðPhj;max À P hj;min Þ; j ¼ 1;...; N2 ;m ¼ 1;. ..;M À 1
ð19Þ
where rand1 and rand2 are uniformly distributed random numbers
in [0,1].
Consider vector X d ¼ ½P s2;m;d ; Ps3;m;d ; :::; P sN1 ;m;d ; P h1;m;d ; P h2;m;d ; . . . ;
P hN2 ;m;d Š of nest d including the thermal units from 2 to N1 for M
subintervals and hydro units from 1 to N2 for the first (M-1) subintervals. At subinterval M, nest d only contains thermal units from 2
to N1. The power output of the thermal and hydro units in the Np
nests are randomly chosen satisfying Psi,min 6 Psi,m,d 6 Psi,max and
Phj,min 6 Phj,m,d 6 Phj,max.
Based on the initialized population of the nests, the fitness function to be minimized corresponding to each nest for the considered
problem is calculated as:

FT d ¼

N1
N2
M X
M
X
X
X
2
2
F i ðPsi;m;d Þ þ K s ðPs1;m;d À Plim
ðqj;M;d À qlim

j Þ
s1 Þ þ K q
m¼1 i¼1

m¼1

j¼1

ð20Þ
where Ks and Kq are penalty factors for the slack thermal unit 1 and
the available water at subinterval M, respectively; Ps1,m,d is the
power output of the slack thermal unit calculated from Section ‘Calculation of power output for slack thermal and hydro units’ corresponding to nest d in the population; qj,M,d is the water discharge
of hydro plant j at the subinterval M calculated from Eq. (9) corresponding to nest d in the population.
The limits for the slack thermal unit 1 and the water discharge
at the subinterval M in (20) are determined as follows:

8
>
< P s1;max if Ps1;m;d > P s1;max
Plim
¼
P s1;min if Ps1;md < Ps1;min
s1
>
:
P s1;m;d otherwise
8
>
< qj;max if qj;M;d > qj;max
qj;min if qj;M;d < qj;min

qlim
¼
j
>
:
qj;M;d otherwise

where randx and randy are two normally distributed stochastic variables with standard deviation rx(b) and ry(b) given by:

2

rx ðbÞ ¼

31=b

pb
4 Cð1 þ bÞ Â sinð 2 Þ 5

ð26Þ

bÀ1
Cð1þb
Þ Â b  2ð 2 Þ
2

ry ðbÞ ¼ 1

ð27Þ

where b is the distribution factor (0.3 6 b 6 1.99) and C(.) is the

gamma distribution function.
Generation of new solution for the top egg group. The modification
applied to the eggs in the top group (d = 1, . . ., Notop) is described
in Section ‘Modified cuckoo search’. There are three cases for the
new generated eggs based on the information exchange among
the top eggs. The optimal path for the Lévy flights is calculated
using Mantegna’s algorithm as follows:
new

Xnodiscardd

¼ Xbest nodiscardd þ a  rand4
new

 DXnodiscardd

where rand4 is the distributed random numbers in [0,1]. The value
of a and DXnodiscardnew
will be selected depending on one of the
d
considered cases below:
– Case 1: The same egg is picked twice
new

DXnodiscardd

¼vÂ

rx ðbÞ
 ðXbest nodiscardd À GbestÞ

ry ðbÞ
ð29Þ

ð21Þ

rx ðbÞ
ry ðbÞ

2

where a = A/G , m and
are calculated as in Section ‘Generation of
new solution for the abandoned group’.
– Case 2: Both eggs have the same fitness value function
new

DXnodiscardd

ð22Þ

¼ ðXbest nodiscardr À Xbest nodiscardd Þ=2
ð30Þ

where Ps1,max and Ps1,min are the maximum and minimum power
outputs of slack thermal unit 1, respectively; qj,max and qj,min are
the maximum and minimum water discharges of hydro plant j.
The nests are first sorted in the descending order based on their
fitness function value and then classified into two groups. The
nests with high fitness function value are put in an abandoned
group and the other ones are put in a top group where each nest

in the abandoned group is named Xbest_discardd and the top group
is Xbest_nodiscardd. A nest which is randomly picked among the
Xbest_nodiscardd nests is called Xbest_nodiscardr. The nest corresponding to the best fitness function in (20) is set to the best nest
Gbest among all nests in the population.

where a = A = 1.
– Case 3: The random egg has lower fitness than the egg d

Generation of New Solution via Lévy Flights
Generation of new solution for the abandoned group. Based on the
modification applied to the abandoned eggs (d = Notop + 1,. . ., Nd),
the optimal path for the Lévy flights is calculated using Mantegna’s
algorithm as follows:

where

new

Xdiscardd

new

¼ Xbest discardd þ a  rand3  DXdiscardd

ð23Þ

where p
rand
ffiffiffiffi 3 is the distributed random number in [0,1], the step size
a ¼ A= G is determined and DXdiscardnew

is obtained by:
d

rx ðbÞ
new
DXdiscardd ¼ v Â
 ðXbest discardd À GbestÞ;
ry ðbÞ

ð24Þ

where

m¼

randx
jrandy j1=b

ð28Þ

ð25Þ

new

DXnodiscardd

¼ ðXbest nodiscardr À Xbest nodiscardd Þ=u;
ð31Þ

pffiffiffi

where a = A and u ¼ ð1 þ 5Þ=2.
– Case 4: The random egg has higher fitness than the egg d
By modifying Eq. (28), the new solution is obtained as below
new

Xnodiscardd

new

¼ Xbest nodiscardr þ a  rand5  DXnodiscardd

ð32Þ
new

DXnodiscardd

¼ ðXbest nodiscardd À Xbest nodiscardr Þ=u

ð33Þ
pffiffiffi
and a = A = 1 and u ¼ ð1 þ 5Þ=2.
For the newly obtained solution, its lower and upper limits
should be satisfied according to the unit’s limits:
8
>
< Psi;max if Psi;m;d > Psi;max
Psi;m;d ¼ Psi;min if Psi;m;d < Psi;min ; i ¼ 2;. .. ; N1 ; m ¼ 1; .. .; M
>
:
Psi;m;d otherwise

ð34Þ

8
>
< Phj;max if Phj;m;d > Phj;max
Phj;m;d ¼ Phj;min if Phj;m;d < Phj;min ;j ¼ 1; .. .; N2 ; m ¼ 1; .. .; M À 1
>
:
Phj;m;d otherwise
ð35Þ


276

T.T. Nguyen, D.N. Vo / Electrical Power and Energy Systems 65 (2015) 271–281

The power outputs of the slack hydro unit j at subinterval M,
PhjMd and the slack thermal unit 1 at each subinterval, Ps1md are calculated from section ‘Calculation of power output for slack thermal
and hydro units’. The fitness function value of the new egg is calculated using (20) and then compared to that from the old egg.
The egg with better fitness function value is considered as the
new solution.
Alien egg discovery and randomization
The action of discovery of an alien egg in a nest of a host bird
with the probability of pa also creates a new solution for the problem similar to the Lévy flights. The new solution due to this action
can be found as follows:
dis
X dis
d ¼ Xbest d þ K Â DX d

1 if rand6 < pa
0

DX dis
d

ð37Þ

otherwise

and the increased value

Min cost

Avg. cost

Max cost

Std. dev.

CPU

1000
2000
3000
4000
5000
6000
7000
8000

9000
10,000

376127.9181
376006.7362
375990.4608
375989.8019
375967.4969
375977.1535
375976.594
375960.9742
375956.0646
375951.3744

376298.291
376112.782
376060.4277
376036.395
376018.292
376004.682
375994.014
375983.677
375981.686
375974.696

376647.4134
376224.7237
376167.6549
376116.4892
376069.1492

376042.5783
376015.4418
376018.3829
376026.0262
376005.5239

102.3012
61.3486
40.9971
29.3800
24.8528
18.0857
11.9547
15.8972
17.5634
14.5745

2.9
5.9
7.9
12.4
14.2
17.1
18.2
22.1
26.2
28.2

DXdis
d

Table 3
The obtained result from MCSA and CSA for the system with quadratic fuel cost
function of thermal units.
Method

Min cost ($)

Avg. cost ($)

Max cost ($)

Std. dev. ($)

CPU (s)

CSA
MCSA

376114.734
375990.461

376547.498
376060.428

377211.261
376167.655

274.463
40.997

30.29
7.90

is determined by:

¼ rand7  ½randp1 ðXbestd Þ À randp2 ðXbestd ފ

ð38Þ

where rand6 and rand7 are the distributed random numbers in [0,1]
and randp1(Xbestd) and randp2(Xbestd) are the random perturbation
for positions of the nests in Xbestd.
For the newly obtained solution, its lower and upper limits
should be also satisfied using (34) and (35). The value of the fitness
function is recalculated using (20) and the nest corresponding to
the best fitness function is set to the best nest Gbest of the
population.
Stopping criterion
In the proposed MCSA method, the stopping criterion for the
algorithm is based on the maximum number of iterations. The
algorithm is terminated as the maximum number of iterations
reached.
Selection of parameters
In the proposed MCSA method, there are three control parameters to be handled including number of nests, maximum number of
iterations and probability of an alien egg to be discovered pa.
Among the parameters, the number of nest and maximum number
of iterations are easily to be fixed in advance depending on the
considered systems. For small-scale systems with simple constraints, the number of nest and maximum number of iterations
can be set to small values. On the contrary, for large-scale systems

with complex constraints, the number of nest and maximum number of iterations can be higher. However, the most important
parameter of the proposed method is the probability pa which
has a great effect on the final solution. This parameter should be

Table 1
Results by MCSA for the system with quadratic fuel cost of thermal units with
different values of pa.
pa

Min cost ($)

Avg. cost ($)

Max cost ($)

Std. dev. ($)

CPU (s)

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

376185.4434

376135.9905
376231.9095
376071.9779
376045.6635
376044.8389
375990.4608
376005.7887
376013.6162

377005.9876
376489.2502
376326.1712
376182.8084
376143.7147
376104.4795
376060.4277
376046.4663
376055.7229

377665.8782
376753.4272
376498.3798
376301.0430
376273.6160
376218.8801
376167.6549
376101.4427
376177.1540

344.6804

160.7845
84.9542
71.6546
61.7110
43.6353
40.9971
27.1796
38.8076

7.8
8.1
7.8
7.9
8.2
8.1
7.9
8.2
8.1

3.9

x 10

5

MCSA
CSA

3.88

Fitness Function ($)



Nmax

ð36Þ

where K is the updated coefficient determined based on the probability of a host bird to discover an alien egg in its nest:

K¼

Table 2
The sensitivity analysis with respect to the stopping criterion for the system with
quadratic fuel cost function.

3.86

3.84

3.82

3.8

3.78

3.76
2
10

3

4

10

10

Number of iterations = 5000
Fig. 1. Convergence characteristic of MCSA and CSA for the system with quadratic
fuel cost function of thermal unit.

Table 4
Result comparison for the system with quadratic fuel cost function of thermal units.
Method

Newton [21]

HNN [21]

CSA

MCSA

Cost ($)

377,374.67

377,554.94

376,114.734

375,990.461

tuned since it is a random number and there are no criteria for a
proper selection. Therefore, the effect of pa on the final solution
by the MCSA method for each test system will be analyzed with
the value of pa ranging from 0.1 to 0.9 with a step size of 0.1 to
obtain the suitable value of pa for each system.
Overall procedure
The overall procedure of the proposed MCSA for solving the
fixed head short-term HTS problem is described as follows.
Step 1: Select parameters for MCSA. Initialize population of host
nests as in Section ‘Initialization’. Set the iteration counter Iter = 1.