Modified cuckoo search algorithm for short-term hydrothermalscheduling Thang Trung Nguyena, Dieu Ngoc Vob,⇑ a Faculty of Electrical and Electronics Engineering, Ton Duc Thang University,
Trang 1Modified cuckoo search algorithm for short-term hydrothermal
scheduling
Thang Trung Nguyena, Dieu Ngoc Vob,⇑
a
Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, 19 Nguyen Huu Tho str., 7th dist., Ho Chi Minh city, Viet Nam
b
Department of Power Systems, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet str., 10th dist., Ho Chi Minh city, Viet Nam
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 modifica-tions 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 pro-posed method can obtain higher quality solutions than many other methods Therefore, the propro-posed 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
alloca-tion of hydro energy so that the annual operating cost of a mixed
methods have been implemented for solving the hydrothermal
lambda-gamma iteration method, dynamic programming (DP)
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, com-putational 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 sub-problems and they are solved by network flow programming and
http://dx.doi.org/10.1016/j.ijepes.2014.10.004
0142-0615/Ó 2014 Elsevier Ltd All rights reserved.
⇑ 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: vndieu@gmail.com (D.N Vo).
Electrical Power and Energy Systems
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / i j e p e s
Trang 2priority 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
non-linear equations Therefore, it has a high potential for
implementa-tion on optimizaimplementa-tion problems such as economic load dispatch in
hydrothermal power systems However, a drawback of the
New-ton’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
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
sev-eral 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’’
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
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
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
detri-mental to actually reaching an optimal solution In this regard,
the EP technique is more likely better when overcoming these
dis-advantages where the mutation is a key search operator which
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
prob-lems 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
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 meta-heuristic search methods The HNN method is an efficient neural network for dealing with optimization problems However, it encounters a difficulty of predetermining the synaptic intercon-nections 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
techniques are efficient for finding near optimum solution for com-plex problems but they also usually suffer slow convergence, espe-cially for large-scale problems
Cuckoo search algorithm (CSA) is a new metaheuristic algo-rithm for solving optimization problems developed by Yang and
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
has been also successfully applied for solving non-convex
slow convergence for complex and large-scale problems Therefore,
a new modified CSA (MCSA) has been proposed by Walton et al
effi-ciency of the MCSA method over other methods such as
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 genera-tors with valve point loading effects satisfying power balance con-straint, water availability, and generator operating limits The MCSA method is based on the conventional CSA method with mod-ifications 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 clo-ser to the optimal solution On the other hand, there will be an information exchange between two eggs in the top egg group to
Nomenclature
ahj, bhj, chj water discharge coefficients of hydro plant j
asik, bsi, csi fuel cost coefficients of thermal plant i
valve-point effects
Bij, B0i, B00 B-matrix coefficients for transmission power loss
Phj,max maximum power output of hydro plant j
Phj,min minimum power output of hydro plant j
Psi,max maximum power output of thermal plant i
Psi,min minimum power output of thermal plant i
scheduled period
Trang 3speed 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
The remaining organization of the paper is as follows The
prob-lem formulation is given in Section ‘Probprob-lem formulation’ The
implementation of MCSA for the problem is presented in
Sec-tion ‘ImplementaSec-tion of MCSA for HTS Problem’ The numerical
results are provided in Section ‘Numerical results’ Finally, the
con-clusion 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
con-siderable and it is negligible In this paper, the short-term
fixed-head 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
sched-uled in M sub-intervals with t hours for each is formulated as
fol-lows[20,21]
The objective is to minimize the total cost of thermal generators
considering valve loading effects:
Min CT¼XM
m¼1
XN1
i¼1
tmhasiþ bsiPsi;mþ csiP2si;mþ dj si
sin esi Pminsi Psi;m
ð1Þ
subject to:
– Power balance constraint: The total power generation from
ther-mal and hydro plants must satisfy the total load demand and
power loss in each subinterval:
XN 1
i¼1
Psi;mþXN 2
j¼1
Phj;m PL;m PD;m¼ 0; m ¼ 1; ; M ð2Þ
where the power losses in transmission lines are calculated using
Kron’s formula[2]:
PL;m¼NX1þN 2
i¼1
X
N1þN 2
j¼1
Pi;mBijPj;mþNX1þN 2
i¼1
B0iPi;mþ B00 ð3Þ
– Water availability constraint: The total available water discharge
of each hydro plant for the whole scheduled time horizon is
lim-ited by:
XM
m¼1
tmqj;m¼ Wj; j ¼ 1; ; N2 ð4Þ
where the rate of water flow from hydro plant j in subinterval m is
determined by:
qj;m¼ ahjþ bhjPhj;mþ chjP2hj;m ð5Þ
– Generator operating limits: Each thermal and hydro units have
their upper and lower generation limits:
Psi;min6Psi;m6Psi;max; i ¼ 1; ; N1; m ¼ 1; ; M ð6Þ
Phj;min6Phj;m6Phj;max; j ¼ 1; ; N2; m ¼ 1; ; M ð7Þ
Implementation of MCSA for HTS Problem Cuckoo search algorithm
The CSA is a new and efficient population-based heuristic evo-lutionary 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 behav-ior of some cuckoo species combined with the Lévy flight behavbehav-ior
of some birds and fruit flies There are mainly three principal rules
1 Each cuckoo lays one egg (a design solution) at a time and dumps its egg in a randomly chosen nest among the fixed num-ber 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 pa2 [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
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 algo-rithms Generally, the CSA method consists of two important oper-ations 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]:
Xðtþ1Þ
i ¼ Xtiþa Le0VyðkÞ ð8Þ
wherea> 0 is the step size for updating new solution related to the
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
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 paalso 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
rate of the method is not considerable and it can be fixed at 0.25 Modified cuckoo search
Although the CSA method outperforms the PSO and GA meth-ods in terms of success rate and number of required objective func-tion evaluafunc-tions[23], it finds an optimal solution based entirely on random walks which cannot guarantee a fast convergence
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 modifica-tions are performed as follows
Trang 4(a) Modification of the abandoned eggs: The improvement is
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
search-ing as the eggs are gettsearch-ing closer to the optimal solution
In the MCSA method, an initial Lévy flight step size is set
toa= A = 1 and at each iteration the new value ofais
calcu-lated usinga¼ A= ffiffiffiffi
G p where G is the current iteration num-ber 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
conver-gence 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 is calculated using the inverse of the
5 p Þ=2 The new egg is then located clo-ser 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 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,
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
guar-antee that the equality constraints are always satisfied The power
output of the slack hydro unit is calculated based on the
availabil-ity water constraint while the power output of the slack thermal
unit is determined using the power balance constraint
satisfied, a slack thermal unit is arbitrarily selected and thus its
power output will be dependent on the power output of the
the first thermal unit as the slack unit is calculated by:
Ps1;m¼ PD;mþ PL;mXN1
i¼2
Psi;mXN2 j¼1
follows:
PL;m¼ BTT;11P2s1;mþ 2XN 1
i¼2
BTT;1iPsi;mþ 2XN 2
j¼1
BTH;1jPhj;mþ BT;01
!
Ps1;m
þXN 1
i¼2
XN 1
j¼2
Psi;mBTT;ijPsj;mþXN 2
i¼1
XN 2
j¼1
Phi;mBHH;ijPhj;mþ 2XN 1
i¼2
XN 2
j¼1
Psi;mBTH;ijPhj;m
þXN 1
i¼2
BT;0iPsi;mþXN 2
j¼1
BH;0jPhj;mþ B00
ð10Þ
where
Bij¼ BTT;ij BTH;ij
BHT;ij BHH;ij
; B0i¼ BT;0i
BH;0i
BTT,ij, BT,0iPower loss coefficients due to thermal units
BHH,ij, BH,0iPower loss coefficients due to hydro units
BTH,ij, BHT,ij Power loss coefficients due to thermal and hydro units, BTH,ij= BHT,ijT
A P2 s1;mþ B Ps1;mþ C ¼ 0 ð11Þ
where
B ¼ 2XN 1
i¼2
BTT;1iPsi;mþ 2XN 2
j¼1
BTH;1jPhj;mþ BT;01 1 ð13Þ
C ¼XN 1
i¼2
‘XN 1
j¼2
Psi;mBTT;ijPsj;mþXN 2
i¼1
XN 2
j¼1
Phi;mBHH;ijPhj;mþ 2XN 1
i¼2
XN 2
j¼1
Psi;mBTH;ijPhj;m
þXN 1
i¼2
BT;0iPsi;mþXN 2
j¼1
BH;0jPhj;mþ B00þ PD;mXN 1
i¼2
Psi;mXN 2
j¼1
Phj;m ð14Þ
Ps1;m¼B
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B2 4AC p
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
qj;M¼ WjXM1
m¼1
tmqj;m
!,
Therefore, the power output of hydro unit j at subinterval M
(5)as follows:
Phj;M¼bhj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2hj 4chjðahj qj;MÞ q
2chj
; j ¼ 1; 2; ; N2 ð17Þ
where b2hj 4 chj ðahj qj;MÞ P 0
Implementation of MCSA for HTS The proposed MCSA method is implemented for solving the short-term fixed-head HTS problem as follows
Initialization
A population of Nphost 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,dPhj,m,d], where Psi,m,dis the power out of thermal unit i at subinterval m corresponding to nest d and Phj,m,dis the power out of hydro unit j at subinterval m corre-sponding 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:
Psi;m;d¼ Psi;minþ rand1 ðPsi;max Psi;minÞ; i ¼ 2; ; N1;m ¼ 1; ;M ð18Þ
Trang 5Phj;m;d¼ Phj;minþ rand2 ðPhj;max Phj;minÞ; j ¼ 1; ; N2;m ¼ 1; ;M 1
ð19Þ
in [0,1]
Consider vector Xd¼ ½Ps2;m;d;Ps3;m;d; :::;PsN 1 ;m;d;Ph1;m;d;Ph2;m;d; ;
PhN2;m;d of nest d including the thermal units from 2 to N1for M
subin-tervals At subinterval M, nest d only contains thermal units from 2
nests are randomly chosen satisfying Psi,min6Psi,m,d6Psi,max and
Phj,min6Phj,m,d6Phj,max
Based on the initialized population of the nests, the fitness
func-tion to be minimized corresponding to each nest for the considered
problem is calculated as:
FTd¼XM
m¼1
XN 1
i¼1
FiðPsi;m;dÞ þ Ks
XM m¼1
ðPs1;m;d Plims1Þ2þ Kq
XN 2
j¼1
ðqj;M;d qlim
j Þ2 ð20Þ
where Ksand Kqare penalty factors for the slack thermal unit 1 and
power output of the slack thermal unit calculated from Section
‘Cal-culation of power output for slack thermal and hydro units’
corre-sponding to nest d in the population
The limits for the slack thermal unit 1 and the water discharge
Plim
s1 ¼
Ps1;max if Ps1;m;d>Ps1;max
Ps1;min if Ps1;md<Ps1;min
Ps1;m;d otherwise
8
>
qlim
j ¼
qj;max if qj;M;d>qj;max
qj;min if qj;M;d<qj;min
qj;M;d otherwise
8
>
outputs of slack thermal unit 1, respectively; qj,maxand qj,minare
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
Xbest_nodiscardd nests is called Xbest_nodiscardr The nest
Gbest among all nests in the population
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:
Xdiscardnewd ¼ Xbest discarddþa rand3DXdiscardnewd ð23Þ
where rand3is the distributed random number in [0,1], the step size
a¼ A= ffiffiffiffi
G
p
is determined andDXdiscarddnewis obtained by:
DXdiscardnewd ¼vrxðbÞ
ryðbÞ ðXbest discardd GbestÞ; ð24Þ
where
m¼ randx
where randxand randyare two normally distributed stochastic vari-ables with standard deviationrx(b) andry(b) given by:
rxðbÞ ¼ Cð1 þ bÞ sinð
pb
2Þ
Cð1þb
2Þ b 2ð Þb12
2 4
3 5 1=b
ð26Þ
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:
Xnodiscardnewd ¼ Xbest nodiscarddþa rand4
DXnodiscardnewd ð28Þ
considered cases below:
– Case 1: The same egg is picked twice
DXnodiscardnewd ¼vrxðbÞ
ryðbÞ ðXbest nodiscardd GbestÞ
ð29Þ
wherea= A/G2,mandrx ðbÞ
r y ðbÞare calculated as in Section ‘Generation of new solution for the abandoned group’
– Case 2: Both eggs have the same fitness value function
DXnodiscardnewd ¼ ðXbest nodiscardr Xbest nodiscarddÞ=2
ð30Þ
– Case 3: The random egg has lower fitness than the egg d
DXnodiscardnewd ¼ ðXbest nodiscardr Xbest nodiscarddÞ=u;
ð31Þ
5 p Þ=2
– Case 4: The random egg has higher fitness than the egg d
Xnodiscardnewd ¼ Xbest nodiscardrþa rand5DXnodiscardnewd
ð32Þ
where
DXnodiscardnewd ¼ Xbest nodiscardð d Xbest nodiscardrÞ=u
ð33Þ
5
p Þ=2
For the newly obtained solution, its lower and upper limits should be satisfied according to the unit’s limits:
Psi;m;d¼
Psi;max if Psi;m;d>Psi;max
Psi;min if Psi;m;d<Psi;min
Psi;m;d otherwise
8
>
> ;i ¼ 2; ; N1;m ¼ 1; ;M ð34Þ
Phj;m;d¼
Phj;max if Phj;m;d>Phj;max
Phj;min if Phj;m;d<Phj;min
Phj;m;d otherwise
8
>
> ;j ¼ 1; ;N2;m ¼ 1; ;M 1
ð35Þ
Trang 6The power outputs of the slack hydro unit j at subinterval M,
PhjMdand the slack thermal unit 1 at each subinterval, Ps1mdare
cal-culated from section ‘Calculation of power output for slack thermal
and hydro units’ The fitness function value of the new egg is
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 paalso creates a new solution for the
prob-lem similar to the Lévy flights The new solution due to this action
can be found as follows:
Xdis
d ¼ Xbestdþ K DXdis
where K is the updated coefficient determined based on the
proba-bility of a host bird to discover an alien egg in its nest:
K ¼ 1 if rand6<pa
0 otherwise
ð37Þ
and the increased valueDXddisis determined by:
DXdisd ¼ rand7 randp½ 1ðXbestdÞ randp2ðXbestdÞ ð38Þ
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
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
parame-ters to be handled including number of nests, maximum number of
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
con-straints, 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
num-ber of iterations can be higher However, the most important
has a great effect on the final solution This parameter should be
tuned since it is a random number and there are no criteria for a
by the MCSA method for each test system will be analyzed with
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 coun-ter Icoun-ter = 1
Table 1
Results by MCSA for the system with quadratic fuel cost of thermal units with
different values of p a
p a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s)
0.1 376185.4434 377005.9876 377665.8782 344.6804 7.8
0.2 376135.9905 376489.2502 376753.4272 160.7845 8.1
0.3 376231.9095 376326.1712 376498.3798 84.9542 7.8
0.4 376071.9779 376182.8084 376301.0430 71.6546 7.9
0.5 376045.6635 376143.7147 376273.6160 61.7110 8.2
0.6 376044.8389 376104.4795 376218.8801 43.6353 8.1
0.7 375990.4608 376060.4277 376167.6549 40.9971 7.9
0.8 376005.7887 376046.4663 376101.4427 27.1796 8.2
0.9 376013.6162 376055.7229 376177.1540 38.8076 8.1
Table 2 The sensitivity analysis with respect to the stopping criterion for the system with quadratic fuel cost function.
N max Min cost Avg cost Max cost Std dev CPU
1000 376127.9181 376298.291 376647.4134 102.3012 2.9
2000 376006.7362 376112.782 376224.7237 61.3486 5.9
3000 375990.4608 376060.4277 376167.6549 40.9971 7.9
4000 375989.8019 376036.395 376116.4892 29.3800 12.4
5000 375967.4969 376018.292 376069.1492 24.8528 14.2
6000 375977.1535 376004.682 376042.5783 18.0857 17.1
7000 375976.594 375994.014 376015.4418 11.9547 18.2
8000 375960.9742 375983.677 376018.3829 15.8972 22.1
9000 375956.0646 375981.686 376026.0262 17.5634 26.2 10,000 375951.3744 375974.696 376005.5239 14.5745 28.2
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 376114.734 376547.498 377211.261 274.463 30.29 MCSA 375990.461 376060.428 376167.655 40.997 7.90
3.76 3.78 3.8 3.82 3.84 3.86 3.88
3.9x 10
5
Number of iterations = 5000
MCSA CSA
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
Trang 7Step 2: Initialize nests for power output of all hydro and thermal
units using(18) and (19)
the value of their fitness function
Step 4: Generate new solutions for abandoned eggs via Lévy
flights as in Section ‘Generation of new solution for the
abandoned group’
(35)
– Calculate all hydro and thermal generation outputs from section ‘Calculation of power output for slack thermal and hydro units’
– Calculate the fitness function in(20) Step 6: Generate new solution for top eggs via Lévy flights as in Section ‘Generation of new solution for the top egg group’
(35) – Calculate all hydro and thermal generation outputs from section ‘Calculation of power output for slack thermal and hydro units’
– Calculate the fitness function in(20) Step 8: Put new eggs generated in step 4 and 6 in a group of egg Step 9: Discover alien egg and randomize as in Section ‘Alien egg discovery and randomization’
Step 10: Check for limit violations and repairing using(34) and (35) – Calculate all hydro and thermal generation outputs from section ‘Calculation of power output for slack thermal and hydro units’
– Evaluate fitness function to choose new Gbest
Otherwise, stop
Numerical results The proposed MCSA has been tested on three systems, in which one system with quadratic fuel cost function of thermal units and two other systems with non-convex fuel cost function of thermal units In addition, the conventional CSA method is also imple-mented for solving these systems for result comparison The both algorithms are coded in Matlab platform and run on a 2 GHz PC with 2 GB of RAM
System with quadratic fuel cost function of thermal units The test system is consisting of two thermal plants and two hydro plants scheduled in four subintervals with twelve hours
For implementation in the MCSA method, the number of nests and maximum number of iterations are set to 12 and 3000, respec-tively The effect of the probability pa on the optimal solution by the MCSA method for this system is analyzed and the obtained
analysis of the effect of the stopping criterion on the final solution
by MCSA with the fixed number of nests and probability pa is given
in Table 2 As observed from the table, the optimal solution is improved a little as the number of iterations is increased but the computational time is longer compared to the case with selected parameters as above On the contrary, the optimal solution is worse as the number of iterations is decreased but the computa-tional time is faster Therefore, the best parameters of MCSA for this system are Np= 12, Nmax= 3000, and pa= 0.7
num-ber of nests, and maximum numnum-ber of iterations are set to 0.9, 50, and 5000, respectively Both the MCSA and CSA methods are run 20 independent trials and the obtained results including minimal total cost, average total cost, maximal total cost, standard
optimal solution for the system by the proposed MCSA and CSA
characteris-tic of MCSA and CSA for the system
total costs than CSA for minimum cost, average cost, maximum cost, and standard deviation with a faster manner Therefore, it is indicated that the MCSA method is more robust and can obtain
Table 5
Results by MCSA for the first system with non-convex fuel cost of thermal units with
different values of p a
p a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s)
0.1 66118.2739 66148.3443 66180.9419 14.7858 6.5
0.2 66116.7871 66137.1950 66159.2199 14.0093 6.7
0.3 66115.7551 66124.9015 66150.5171 12.3232 6.7
0.4 66115.6562 66124.9541 66153.4599 12.6732 6.9
0.5 66115.5023 66116.5400 66122.6741 1.5408 6.8
0.6 66115.6378 66127.1959 66150.7313 13.8924 6.7
0.7 66115.5692 66119.4853 66147.1090 9.0145 6.8
0.8 66115.5053 66116.3965 66121.9564 1.5833 6.8
0.9 66115.4459 66116.8292 66128.1060 2.9602 6.6
Table 6
Results by MCSA for the second system with non-convex fuel cost of thermal units
with different values of p a
p a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s)
0.1 92825.8057 93178.3231 93760.1479 263.4581 13.6
0.2 92801.8044 92925.6080 93217.5736 104.6282 14.1
0.3 92755.2380 92860.9817 93055.7946 70.6202 14.0
0.4 92757.2273 92841.1892 92915.7016 48.8641 13.8
0.5 92770.2015 92835.5498 92961.4572 47.4677 13.7
0.6 92781.4987 92855.6608 92929.1243 48.5824 14.2
0.7 92741.9020 92836.6646 92938.5369 47.7365 13.3
0.8 92805.6040 92943.7870 93321.5465 113.1874 13.4
0.9 92863.1251 93039.8177 93477.5816 140.1228 13.7
Table 7
The sensitivity analysis with respect to the stopping criterion for the first system with
non-convex fuel cost of thermal units.
N max Min cost Avg cost Max cost Std dev CPU
1000 66116.2441 66130.5065 66154.8832 13.1013 2.0
2000 66115.6030 66123.4250 66147.2104 12.5686 4.5
3000 66115.4459 66116.8292 66128.1060 2.9602 6.6
4000 66115.4379 66121.315 66146.6594 12.1452 8.8
5000 66115.4379 66118.7912 66153.3468 10.2527 10.5
6000 66115.4379 66119.8527 66145.6194 10.7322 13.8
7000 66115.4379 66118.3408 66145.6559 8.9965 16.8
8000 66115.4379 66118.3637 66145.6107 9.0359 17.9
9000 66115.4379 66118.3103 66145.3794 8.9579 20.3
10,000 66115.4379 66117.2630 66153.3462 8.2578 23.3
Table 8
The sensitivity analysis with respect to the stopping criterion for the second system
with non-convex fuel cost of thermal units.
N max Min cost Avg cost Max cost Std dev CPU
3000 92813.5998 92991.4466 93358.3227 139.9290 8.8
4000 92808.9373 92969.6709 93499.2851 152.7867 10.5
5000 92741.9020 92836.6646 92938.5369 47.7365 13.3
6000 92763.7014 92900.4971 93231.8716 124.4226 16.2
7000 92749.8239 92812.5064 92914.4207 43.6531 20.1
8000 92738.4444 92797.726 92885.7751 31.2707 23.6
9000 92754.0452 92802.9407 92924.8274 41.9571 27.6
10,000 92752.6756 92798.1902 92899.0808 37.8289 28.9
Trang 8better solution quality than the conventional CSA The best total
cost for the system obtained by the proposed MCSA and CSA are
total cost than the others Therefore, MCSA is very effective for solving the short-term fixed-head HTS with quadratic fuel cost function of thermal units
Systems with valve point effects on fuel cost function of thermal units The proposed MCSA and CSA methods are tested on two
and two thermal plants scheduled in three subintervals with eight hours for each and the second system consists of two hydro plants and four thermal plants scheduled in four subintervals with twelve hours for each The data for the two test systems is given in Appen-dix A
For the MCSA method, the number of nests is set to 12 for the both systems, and the maximum number of iterations is set to
on the optimal solution of MCSA for the two systems is analyzed
inTables 5 and 6 Based on the analyses, the best values of pafor the two systems are 0.9 and 0.7, respectively The analysis of the sensitivity of the optimal solution by MCSA with respect to the
for the system with quadratic fuel cost function of thermal units, the selected parameters above are the best of MCSA for the two test system in this case
are respectively fixed at 0.9 and 50 for both systems while the maximum number of iterations is respectively set to 5000 and
7000 for the two systems Both MCSA and CSA are run 20 indepen-dent trials for the two systems and the obtained results including minimal total cost, average total cost, maximal total cost, standard
The optimal solutions obtained from the MCSA and CSA methods for the systems are given in Appendix B The convergence
and 3, respectively
Table 9
Results obtained from MCSA and CSA of two systems with non-convex fuel cost of thermal units.
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
6.61
6.615
6.62
6.625
6.63
6.635
6.64
6.645
6.65
x 104
Number of iterations = 5000
CSA MCSA
Fig 2 Convergence characteristic of MCSA and CSA for the first system with
non-convex fuel cost of thermal units.
1000 2000 3000 4000 5000 6000 7000
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
x 105
Number of iterations = 7000
CSA MCSA
Fig 3 Convergence characteristic of MCSA and CSA for the second system with
non-convex fuel cost of thermal units.
Table 10
Result comparison for two test systems with non-convex fuel cost of thermal units.
Table 11 Results by MCSA for the third system with non-convex fuel cost of thermal units with different values of p a
p a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s) 0.1 208762.7817 213515.558 218874.476 3026.5798 30.9 0.2 204341.9458 206511.495 208536.579 1081.6181 30.5 0.3 203707.1445 205084.626 205884.938 584.0704 30.6 0.4 203065.9561 204599.077 206267.853 738.2208 30.5 0.5 203726.0858 205179.783 206515.485 659.2901 30.2 0.6 203909.0909 205603.094 208225.794 1091.5761 30.6 0.7 204162.857 206661.253 209396.63 1350.5882 31.0 0.8 206244.1729 208659.567 213045.941 1960.5150 30.8 0.9 203876.9789 213134.302 227258.392 5582.4592 30.5
Trang 9As observed fromTable 3, the MCSA method can obtain better
minimum, average, and maximum costs and standard deviation
than the CSA method with faster computational times for both
sys-tems Therefore, the MCSA method is more effective and efficient
than the CSA method for more complex systems The minimum
total costs obtained from the MCSA and CSA methods are
com-pared to those from other methods including AIS, EP, PSO, and
has indicated that the proposed MCSA can obtain better solution
quality in terms of total cost and computational time than the
on a Pentium-IV 3.0 GHz PC Therefore, the proposed MCSA is a
very effective method for solving the short-term fixed-head HTS
problem with non-convex fuel cost function of thermal units
For the practical applicability demonstration of the proposed
method, a larger scale system including eight thermal plants with
non-convex fuel cost function and four hydro plants scheduled in
four subintervals with twelve hours for each is considered In this
system, the power loss in transmission system is included The
data of the test system is given in Appendix A
For implementation in the MCSA method, the number of nests
and maximum number of iterations are set to 12 and 10,000,
respectively A sensitivity analysis of the effect of paon the final
observed from the table, the proposed MCSA method can reach
the best optimal solution for the system at pa= 0.4
maximum number of iterations are set to 0.75, 50, and 12,000,
respectively Both the MCSA and CSA methods are run 20
independent trials and the obtained results including minimal total cost, average total cost, maximal total cost, standard
are no results from other methods for comparison in this case As observed from the table, the proposed MCSA method outperforms the conventional CSA method in total cost and computational time The optimal solutions by MCSA and CSA for the system are given in Appendix B and the convergence characteristic of the two methods for the system is shown inFig 4
Conclusions
In this paper, the MCSA method has been successfully applied for solving short-term fixed-head HTS problem with both smooth and nonsmooth fuel cost curves of thermal units The main modi-fications of the MCSA method based on the conventional CSA method are classification of nest in two groups based on their fit-ness function value and information exchange in each groups to enhance its search ability and speed up convergence process The proposed MCSA has been tested on three hydrothermal systems with different fuel cost functions of thermal units The result com-parisons with the conventional CSA and other methods in the liter-ature have indicated that the proposed method is better than the compared methods in terms of total cost and computational time Therefore, the proposed MCSA can be a very favorable method for solving the short-term fixed-head HTS problems, especially for nonsmooth fuel cost function of thermal units
Appendix A Data of test systems SeeTables A1–A8
The transmission loss coefficients for the third system with non-convex fuel cost of thermal units are as follows:
B ¼ 104
0:39 0:1 0:12 0:12 0:15 0:16 0:39 0:1 0:12 0:12 0:15 0:16 0:1 0:4 0:14 0:1 0:15 0:2 0:1 0:4 0:14 0:1 0:15 0:2 0:12 0:14 0:35 0:11 0:2 0:18 0:12 0:14 0:35 0:11 0:2 0:18 0:12 0:1 0:11 0:36 0:17 0:15 0:12 0:1 0:11 0:36 0:17 0:15 0:15 0:15 0:2 0:17 0:49 0:14 0:15 0:15 0:2 0:17 0:49 0:14 0:16 0:2 0:18 0:15 0:14 0:45 0:16 0:2 0:18 0:15 0:14 0:45 0:39 0:1 0:12 0:12 0:15 0:16 0:39 0:1 0:12 0:12 0:15 0:16 0:1 0:4 0:14 0:1 0:15 0:2 0:1 0:4 0:14 0:1 0:15 0:2 0:12 0:14 0:35 0:11 0:2 0:18 0:12 0:14 0:35 0:11 0:2 0:18 0:12 0:1 0:11 0:36 0:17 0:15 0:12 0:1 0:11 0:36 0:17 0:15 0:15 0:15 0:2 0:17 0:49 0:14 0:15 0:15 0:2 0:17 0:49 0:14 0:16 0:2 0:18 0:15 0:14 0:45 0:16 0:2 0:18 0:15 0:14 0:45
Appendix B Optimal solutions for test systems SeeTables B1–B4
Table 12
The obtained result from MCSA and CSA for the third system with non-convex fuel cost of thermal units.
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3 x 105
Number of iterations = 12000
CSA MCSA
Fig 4 Convergence characteristic of MCSA and CSA for the third system with
non-convex fuel cost of thermal units.
Table A1
Data of thermal units in the system with quadratic fuel cost function of thermal units.
Trang 10Table A2
Data of hydro units in the system with quadratic fuel cost function of thermal units.
Hydro plant a hj (acre-ft/h) b hj (acre-ft/MW h) c hj (acre-ft/MW 2
h) W j (acre-ft) P hj,min (MW) P hj,max (MW)
Table A3
Data of thermal units in the first system with non-convex fuel cost of thermal units.
Thermal plant a si ($/h) b si ($/MW h) c si ($/MW 2
h) d si ($/h) e si (1/MW) P si,min (MW) P si,max (MW)
Table A4
Data of hydro units in the first system with non-convex fuel cost of thermal units.
Hydro plant a hj (MCF/h) b hj (MCF/MW h) c hj (MCF /MW 2
h) W j (MCF) P hj,min (MW) P hj,max (MW)
Table A5
Data of thermal units in the second system with non-convex fuel cost of thermal units.
Thermal plant a si ($/h) b si ($/MW h) c si ($/MW 2
h) d si ($/h) e si (rad/MW) P si,min (MW) P si,max (MW)
Table A6
Data of hydro units in the second system with non-convex fuel cost of thermal units.
Hydro
plant
a hj
(acre-ft/
h)
b hj
(acre-ft/
MW h)
c hj
(acre-ft/
MW 2 h)
W j
(acre-ft)
P hj,min
(MW)
P hj,max
(MW)
Table A8
Data of hydro units in the third system with non-convex fuel cost of thermal units.
Hydro
plant
a hj
(acre-ft/h)
b hj
(acre-ft/MW h)
c hj
(acre-ft/MW 2 h)
W j
(acre-ft)
P hj,min
(MW)
P hj,max
(MW)
Table B1 Optimal solutions obtained by MCSA and CSA for the system with quadratic fuel cost function of thermal units.
MCSA P s1 (MW) 432.5053 449.9445 448.8348 449.9593
P s2 (MW) 326.2515 449.0193 402.4822 570.3089
P h1 (MW) 164.0322 240.6800 221.7214 494.0726
P h2 (MW) 308.6007 409.3062 369.5016 494.0726 CSA P s1 (MW) 437.9985 450.0000 450.0000 450.0000
P s2 (MW) 324.1876 445.9295 385.2570 584.6512
P h1 (MW) 164.4848 232.6844 228.6392 250.0000
P h2 (MW) 304.7345 420.5458 378.8693 478.5961
Table B2 Optimal solutions obtained by MCSA and CSA for the first system with non-convex fuel cost of thermal units.
MCSA P h1 (MW) 238.2119 323.9756 274.3932
P h2 (MW) 82.9216 183.8083 125.4489
P s1 (MW) 220.0598 221.3594 221.3566
P s2 (MW) 399.0655 538.6915 538.6903 CSA P h1 (MW) 237.7067 323.8045 275.0592
P h2 (MW) 83.254 183.966 124.9636
P s1 (MW) 220.2122 221.3596 221.3596
P s2 (MW) 399.0658 538.6922 538.5385
Table A7
Data of thermal units in the third system with non-convex fuel cost of thermal units.
Thermal
plant
a si
($/h)
b si
($/MW h)
c si
($/MW 2
h)
d si
($/h)
e si
(rad/MW)
P si,min
(MW)
P si,max
(MW)