Cuckoo search algorithm for short-term hydrothermal scheduling Thang Trung Nguyen a , Dieu Ngoc Vo b,⇑ , Anh Viet Truong c 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 c Faculty of Electrical and Electronics Engineering, University of Technical Education Ho Chi Minh City, 1 Vo Van Ngan Str., Thu Duc Dist., Ho Chi Minh City, Viet Nam highlights A new cuckoo search method is proposed for solving hydrothermal scheduling problem. There are few control parameters for the proposed method. The proposed method can properly deal with nonconvex short-term hydrothermal scheduling problem. The robustness and effectiveness of the proposed method have been validated for different test systems. graphical abstract Hydro Plants Minimize fuel cost ~ ~ Thermal Plants Electrical Load Cuckoo Search Algorithm Calculate slack thermal and hydro units Evaluate fitness funcon Generate new eggs via Lévy Flights Calculate slack thermal and hydro units Evaluate fitness funcon Discover alien egg And randomize Generate new eggs Randomize a number of nests, Nd Iteraon=1 Iteraon = Iteraon + 1 No Iteraon=Max STOP Yes 1010103.75 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 x 10Number of iterations = 1000 Fitness Function ($) Calculate slack thermal and hydro units Evaluate fitness funcon Host bird’s nest Alien bird’s egg Host bird’s egg X1 X2 … XNd-1 XNd Net solution article info Article history: Received 19 February 2014 Received in revised form 4 July 2014 Accepted 7 July 2014 Keywords: Cuckoo search algorithm Short-term hydrothermal scheduling Convex fuel cost function Nonconvex fuel cost function Lévy flights abstract This paper proposes a cuckoo search algorithm (CSA) for solving short-term fixed-head hydrothermal scheduling (HTS) problem considering power losses in transmission systems and valve point loading effects in fuel cost function of thermal units. The CSA method is a new meta-heuristic algorithm inspired from the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds of other species for solving optimization problems. The advantages of the CSA method are few con- trol parameters and effective for optimization problems with complicated constraints. The effectiveness of the proposed CSA has been tested on different hydrothermal systems and the obtained test results have been compared to those from other methods in the literature. The result comparison has shown that the CSA can obtain higher quality solutions than many other methods. Therefore, the proposed CSA can be an efficient method for solving short-term fixed head hydrothermal scheduling problems. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction The short term hydro-thermal scheduling (HTS) problem is to determine the power generation among the available thermal and hydro power plants so that the total fuel cost of thermal units is minimized over a schedule time of a single day or a week satis- fying both hydraulic and electrical operational constraints such as the quantity of available water, limits on generation, and power balance [1]. Several conventional methods have been implemented for solving the hydrothermal scheduling problem such as effective conventional method (ECM) based on Lagrange multiplier theory http://dx.doi.org/10.1016/j.apenergy.2014.07.017 0306-2619/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +84 88 657 296x5730; fax: +84 88 645 796. E-mail address: [email protected](D.N. Vo). Applied Energy 132 (2014) 276–287 Contents lists available at ScienceDirect Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Thang Trung Nguyen a, Dieu Ngoc Vo b,⇑, Anh Viet Truong c
a Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, 19 Nguyen Huu Tho Str., 7th Dist., Ho Chi Minh City, Viet Namb Department of Power Systems, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet Str., 10th Dist., Ho Chi Minh City, Viet Namc Faculty of Electrical and Electronics Engineering, University of Technical Education Ho Chi Minh City, 1 Vo Van Ngan Str., Thu Duc Dist., Ho Chi Minh City, Viet Nam
h i g h l i g h t s
� A new cuckoo search method isproposed for solving hydrothermalscheduling problem.� There are few control parameters for
the proposed method.� The proposed method can properly
deal with nonconvex short-termhydrothermal scheduling problem.� The robustness and effectiveness of
the proposed method have beenvalidated for different test systems.
g r a p h i c a l a b s t r a c t
Hydro Plants
Minimize fuel cost
~ ~
Thermal Plants
Electrical Load
Cuckoo Search Algorithm
Calculate slack thermal and hydro units
Evaluate fitness func�on
Generate new eggs via Lévy Flights
Calculate slack thermal and hydro units Evaluate fitness func�on
Discover alien egg
And randomizeGenerate new eggs
Randomize a number of nests, Nd
Itera�on=1
Itera�on = Itera�on + 1No
Itera�on=Max
STOP
Yes
101
102
103
3.75
3.8
3.85
3.9
3.95
4
4.05
4.1
4.15x 10
5
Number of iterations = 1000
Fitn
ess
Fun
ctio
n ($
)
Calculate slack thermal and hydro units Evaluate fitness func�on
Host bird’s nest
Alien bird’s egg
Host bird’s egg
X1 X2 … XNd-1 XNdNet solution
a r t i c l e i n f o
Article history:Received 19 February 2014Received in revised form 4 July 2014Accepted 7 July 2014
This paper proposes a cuckoo search algorithm (CSA) for solving short-term fixed-head hydrothermalscheduling (HTS) problem considering power losses in transmission systems and valve point loadingeffects in fuel cost function of thermal units. The CSA method is a new meta-heuristic algorithm inspiredfrom the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other hostbirds of other species for solving optimization problems. The advantages of the CSA method are few con-trol parameters and effective for optimization problems with complicated constraints. The effectivenessof the proposed CSA has been tested on different hydrothermal systems and the obtained test resultshave been compared to those from other methods in the literature. The result comparison has shown thatthe CSA can obtain higher quality solutions than many other methods. Therefore, the proposed CSA canbe an efficient method for solving short-term fixed head hydrothermal scheduling problems.
� 2014 Elsevier Ltd. All rights reserved.
1. Introduction
The short term hydro-thermal scheduling (HTS) problem is todetermine the power generation among the available thermal
and hydro power plants so that the total fuel cost of thermal unitsis minimized over a schedule time of a single day or a week satis-fying both hydraulic and electrical operational constraints such asthe quantity of available water, limits on generation, and powerbalance [1]. Several conventional methods have been implementedfor solving the hydrothermal scheduling problem such as effectiveconventional method (ECM) based on Lagrange multiplier theory
ahj, bhj, chj water discharge coefficients of hydro plant jasi, bsi, csi fuel cost coefficients of thermal plant idsi, esi fuel cost coefficients of thermal plant i reflecting
valve-point effectsBij, B0i, B00 B-matrix coefficients for transmission power lossPD,m total system load demand at subinterval mPhj,m power output of hydro plant j in subinterval mPhj,max maximum power output of hydro plant iPhj,min minimum power output of hydro plant i
PL,m total transmission loss at subinterval mPsi,m power output of thermal plant i in subinterval mPsi,max maximum power output of thermal plant iPsi,min minimum power output of thermal plant iqj,m rate of water flow from hydro plant j in subinterval mtm duration for subinterval mWj volume of water available for generation by hydro
unit j during the scheduling period
T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287 277
[1], k–c iteration method, dynamic programming (DP) [2],Lagrange relaxation (LR) [3], decomposition and coordinationmethod [4], and mixed integer programming (MIP) [5], Newton’smethod [6,7]. In the ECM method, the coordination equations arelinearized and solved for the water availability constraint sepa-rately from generating units, thus the Lagrangian multiplier associ-ated with water availability constraint is separately from theoutputs of generating units. Based on the obtained Lagrangian mul-tiplier of water constraint, the Lagrangian multiplier associatedwith power balance constraint is determined and the outputs ofthermal and hydro units are finally calculated. In the k–c method,the c values of the different hydro plants are initially chosen andthereafter the k iterations are invoked for the given power demandat each interval of the scheduling period. The DP method is apopular optimization method implemented for solving the hydro-thermal scheduling problem. However, computational and dimen-sional requirements in the DP method increase drastically withlarge-scale system planning horizon [8]. On the contrary to theDP method, the LR method is more efficient for dealing withlarge-scale problems. However, the LR method may suffer to dual-ity gap oscillation resulting from the dual problem formulation,leading to divergence for some problems with operation limitsand non-convexity of incremental heat rate curves of generators.In the decomposition and coordination method, the problem isdecomposed into thermal and hydro sub-problems and they aresolved by network flow programming and priority list baseddynamic programming methods. In order to solve the hydrother-mal scheduling problem, MIP requires linearization of equationswhereas the decomposition and coordination method mayencounter difficulties when dealing with the operation limits andnon-linearity of objective function and/or constraints. TheNewton’s method is computationally stable, effective, and fast forsolving a set of nonlinear equations. Therefore, it has a highpotential for implementation on optimization problems such eco-nomic load dispatch in hydrothermal power systems. However,the Newton’s method mainly depends on the formulation andinversion of Jacobian matrix, leading to restriction of applicabilityon large-scale problems. In general, these conventional methodscan be applicable for only the HTS problems with differentiablefuel cost function and constraints.
Recently, several artificial intelligence techniques have beenproposed for solving the hydrothermal scheduling problems suchas evolutionary programming (EP) [8], genetic algorithm (GA)[9–12], differential evolution (DE) [13], artificial immune system(AIS) [14], and Hopfield neural network (HNN) [7]. Both the GAand EP algorithms are evolutionary based method for solving opti-mization problems. However, the essential encoding and decodingschemes in the both methods are different. In the GA method, thecrossover and mutation operations required to diversify the off-spring may be detrimental to actually reaching an optimal solu-tion. In this regard, the EP is more likely better when overcomingthese disadvantages. In the EP method, the mutation is a key
search operator which generates new solutions from the currentones [15]. However, one disadvantage of the EP method in solvingsome of the multimodal optimization problems is its slow conver-gence to a near optimum. The DE method has the ability to searchin very large spaces of candidate solutions with few or no assump-tions about the considered problem. However, the DE method isdifficult to deal with large-scale problems with slow or no conver-gence to the near optimum solution. The AIS method is one of theefficient methods for solving the nonconvex short-term hydrother-mal scheduling. The most important step of the AIS method is theapplication of the aging operator to eliminate the old antibodies, tomaintain the diversity of the population, and to avoid the prema-ture convergence. The advantages of the AIS method are fewparameters and small maximum number of iterations. However,the AIS method is also difficult to deal with large-scale problemslike other meta-heuristic search methods. Optimal gamma basedgenetic algorithm (OGB-GA) [9] is an improvement of GA for effi-ciently solving the HTS problem. In the OGB-GA method, the c val-ues of the hydro plants are considered as the GA variables and the kiterations over the scheduling period can be called to find the ther-mal and hydro generations for each chromosome in the populationto calculate the value of the fitness function. Therefore, the numberof the GA variables is drastically reduced and does not even dependon the number of intervals in the scheduling period [9]. The HNNmethod is an efficient neural network for dealing with optimiza-tion problems. However, it encounters a difficulty of predetermin-ing the synaptic interconnections among neurons which may leadto constraint mismatch if the weighting coefficients in its energyfunction are not carefully selected. Moreover, the HNN methodalso suffers slow convergence to optimal solution and the con-straints of the problem must be linearized when applying inHNN [16]. In general, most of the artificial intelligence techniquesusually suffer slow convergence to the near optimum solution forthe HTS problems.
The cuckoo search algorithm (CSA) developed by Yang and Deb[17] is a new meta-heuristic algorithm for solving optimizationproblems inspired from the obligate brood parasitism of somecuckoo species by laying their eggs in the nests of other host birdsof other species. To verify the effectiveness of the CS algorithm,Yang and Deb compared its performance with particle swarm opti-mization (PSO) and GA for ten standard optimization benchmarkfunctions [17]. As observed from the obtained results, the CSAmethod has been outperformed both PSO and GA methods for alltest functions in terms of success rate in finding optimal solutionand the number of required objective function evaluations. Thehighlighted advantages of the CSA method are fine balance ofrandomization and intensification and less number of controlparameters. Recently, CSA has been successfully applied for solvingnon-convex economic dispatch (ED) problems considering genera-tor and system characteristics including valve point loading effects,multiple fuel options, prohibited operating zones, spinning reserveand power loss [18]. In addition, CSA has been also used for solving
278 T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287
the ED problems in practical power system and micro grid powerdispatch problem [19]. For ED problems [18,19], CSA has beentested on many systems and obtained better solution quality thanseveral methods in the literature such as HNN, GA, EP, Taguchimethod, biogeography-based optimization, and PSO, etc. More-over, for micro grid power dispatch problem [19], CSA also obtainshigher solution quality than DE and PSO. On the other hand, forPhotovoltaic system, CSA has been used to track Maximum PowerPoint [20]. A comprehensive assessment is carried out against twoother methods, namely Perturbed and Observed (P&O) and PSO.The evaluations include (1) gradual irradiance and temperaturechanges, (2) step change in irradiance and (3) rapid change in bothirradiance and temperature. These tests are carried out for bothlarge and medium-sized PV systems. It is stated in [20] that CSAoutperforms both P&O and PSO with respect to tracking capability,transient behavior and convergence. Consequently, CSA is an effi-cient method for solving optimal problems.
In this paper, a cuckoo search algorithm (CSA) is proposed forsolving short-term fixed head HTS problem considering powerlosses in transmission systems and valve point loading effects infuel cost function of thermal units. The effectiveness of the pro-posed CSA has been tested on different hydrothermal systemsand the obtained results have been compared to those from othermethods available in the literature such as existing GA (EGA), andOGB-GA in [9], Newton’s method and HNN in [7], and PSO, DE, EPand AIS in [14].
2. Problem formulation
The objective of the HTS problem is to minimize the total fuelcost of thermal generators while satisfying hydraulic, power bal-ance, and generator operating limits constraints. The short-termfixed-head hydrothermal scheduling problem having N1 thermalunits and N2 hydro units scheduled in M time sub-intervals is for-mulated as follows.
The objective is to minimize the total cost of thermal generators[14]:
MinCT ¼XM
m¼1
XN1
i¼1
tm½asiþbsiPsi;mþcsP2si;mþjdsi�sinðesi�ðPmin
si �Psi;mÞÞj�
ð1Þ
subject to:– Power balance constraint: The total power generation from ther-
mal and hydro plants must satisfy the total load demand andpower loss in each subinterval:
XN1
i¼1
Psi;m þXN2
j¼1
Phj;m � PL;m � PD;m ¼ 0; m ¼ 1; . . . ;M ð2Þ
where the power losses in transmission lines are calculated usingKron’s formula:
PL;m ¼XN1þN2
i¼1
XN1þN2
j¼1
Pi;mBijPj;m þXN1þN2
i¼1
B0iPi;m þ B00 ð3Þ
– Water availability constraint: The total available water dischargeof each hydro plant for the whole scheduled time horizon is lim-ited by:XM
m¼1
tmqj;m ¼Wj; j ¼ 1; . . . ;N2 ð4Þ
where the rate of water flow from hydro plant j in interval m isdetermined by:
qj;m ¼ ahj þ bhjPhj;m þ cjP2hj;m ð5Þ
– Generator operating limits: Each thermal and hydro units havetheir upper and lower generation limits:
3. Cuckoo search algorithm for short-term fixed-head HTSproblem
3.1. Cuckoo search algorithm
The cuckoo search algorithm (CSA) was developed by Yang andDeb [17]. In comparison with other meta-heuristic search algo-rithms, the CSA is a new and efficient population-based heuristicevolutionary algorithm for solving optimization problems withthe advantages of simple implement and few control parameters.This algorithm is based on the obligate brood parasitic behaviorof some cuckoo species combined with the Lévy flight behaviorof some birds and fruit flies. There are mainly three principal rulesduring the search process as follows [21].
1. Each cuckoo lays one egg (a design solution) at a time anddumps 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 becarried over to the next generation.
3. The number of available host nests is fixed, and a host bird candiscover an alien egg with a probability pa e [0,1]. In this case, itcan either throw the egg away or abandon the nest so as tobuild a completely new nest in a new location.
As a further approximation, the last assumption can be approx-imated by a fraction pa of the n host nests are replaced by newnests (with new random solutions). For maximization problems,the quality or fitness of a solution can simply be proportional tothe value of the objective function. Other forms of fitness can bedefined in a similar way to the fitness function in geneticalgorithms.
3.2. Calculation of power output for slack thermal and hydro units
In this research, the output power for slack hydro units is calcu-lated based on the availability water constraint while the poweroutput of thermal units is determined using the power balanceconstraint.
Suppose that the water discharges of the first (M � 1) subinter-vals of N2 hydro units are obtained, the water discharge for hydrounit j at subinterval M is calculated using the available water con-straint (4) as follows:
qj;M ¼ Wj �XM�1
m¼1
tmqj;m
!=tM; j ¼ 1; . . . ;N2 ð8Þ
Therefore, the power output of hydro unit j at subinterval m isdetermined using (5):
To guarantee that the power balance constraint (2) is alwayssatisfied, a slack thermal unit is arbitrarily selected and thus itspower output will be dependent on the power output of theremaining N1 � 1 thermal units and N2 hydro units in the system.
T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287 279
Suppose that the power outputs of (N1 � 1) thermal unit and N2
hydro units at subinterval m are known, the power output of theslack thermal unit 1 is calculated by:
Ps1;m ¼ PD;m þ PL;m �XN1
i¼2
Psi;m �XN2
j¼1
Phj;m ð10Þ
Eq. (3) is rewritten in terms of the slack thermal unit 1 as follows:
PL;m ¼ BTT;11P2s1;m
þ 2XN1
i¼2
BTT;1iPsi;m þ 2XN2
j¼1
BTH;1jPhj;m þ BT;01
!Ps1;m
þXN1
i¼2
XN1
j¼2
Psi;mBTT;ijPsj;m þXN2
i¼1
XN2
j¼1
Phi;mBHH;ijPhj;m
þ 2XN1
i¼2
XN2
j¼1
Psi;mBTH;ijPhj;m þXN1
i¼2
BT;0iPsi;m þXN2
j¼1
BH;0jPhj;m
þ B00 ð11Þ
where
Bij ¼BTT;ij BTH;ij
BHT;ij BHH;ij
��������; B0i ¼
BT;0i
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 hydrounits,BTH,ij = BHT,ij
T
Substituting (11) into (10), a quadratic equation is obtained:
A� P2s1;m þ B� Ps1;m þ C ¼ 0 ð12Þ
where
A ¼ BTT;11 ð13Þ
B ¼ 2XN1
i¼2
BTT;1iPsi;m þ 2XN2
j¼1
BTH;1jPhj;m þ BT;01 � 1 ð14Þ
C ¼XN1
i¼2
‘XN1
j¼2
Psi;mBTT;ijPsj;m þXN2
i¼1
XN2
j¼1
Phi;mBHH;ijPhj;m
þ 2XN1
i¼2
XN2
j¼1
Psi;mBTH;ijPhj;m þXN1
i¼2
BT;0iPsi;m þXN2
j¼1
BH;0jPhj;m
þ B00 þ PD;m �XN1
i¼2
Psi;m �XN2
j¼1
Phj;m ð15Þ
The solution of the second order Eq. (12) is obtained by:
Based on the three rules in Section 3.1, the CSA method isimplemented for solving the short-term fixed-head HTS problemas follows.
3.3.1. InitializationA population of Np host nests is represented by X = [X1,
X2, . . ., XNp]T, in which each Xd (d = 1, . . ., Np) represents a solutionvector of variables given by Xd = [Psi,m,d, qj,m,d], where Psi,m,d is the
power out of thermal unit i at subinterval m corresponding to nestd and qj,m,d is the water discharge for hydro unit j at subinterval mcorresponding to nest d.
In the CSA, each egg can be regarded as a solution which is ran-domly generated in the initialization. Therefore, each element innest d of the population is randomly initialized as follows:
Psi;m;d ¼ Psi;min þ rand1 � ðPsi;max � Psi;minÞ; i ¼ 2; . . . ;N1; m
qN2 ;m:d� of nest d including the thermal units from 2 to N1 for M sub-intervals and water discharges for hydro units from 1 to N2 for thefirst (M � 1) subintervals. At the subinterval M, the nest d only con-tains thermal units from 2 to N1. The power output of the thermalunits and water discharges in the Np nests are randomly chosensatisfying Psi,min 6 Psi,m,d 6 Psi,max and qj,min 6 qj,m,d 6 qj,max.
Based on the initialized population of the nests, the fitness func-tion to be minimized corresponding to each nest for the consideredproblem is calculated:
FTd ¼XM
m¼1
XN1
i¼1
FiðPsi;m;dÞ þ Ks
XM
m¼1
ðPs1;m;d � Plims1 Þ
2
þ Kq
XN2
j¼1
ðqj;M;d � qlimj Þ
2 ð19Þ
where Ks and Kq are penalty factors for the slack thermal unit 1 andavailable water at subinterval M, respectively; Ps1,m,d is the poweroutput of the slack thermal unit calculated from Section 3.2 corre-sponding to nest d in the population; qj,M is the water dischargeof all hydro plants at the subinterval M calculated from Eq. (8) cor-responding to nest d in the population.
The limits for the slack thermal unit 1 and water discharge atthe subinterval M in (19) are determined as follows:
Plims1 ¼
Ps1;max if Ps1;m;d > Ps1;max
Ps1;min if Ps1;m < Ps1;min
Ps1;m;d otherwise
8>>><>>>:
ð20Þ
qlimj ¼
qj;max if qj;M;d > qj;max
qj;min if qj;M;d < qj;min
qj;M;d otherwise
8>>><>>>:
ð21Þ
where Ps1,max and Ps1,min are the maximum and minimum poweroutputs of slack thermal unit 1, respectively; qj,max and qj,min arethe maximum and minimum water discharges of hydro plant j.
The initialized population of the host nests is set to the bestvalue of each nest Xbestd (d = 1, . . ., Nd) and the nest correspondingto the best fitness function in (19) is set to the best nest Gbestamong all nests in the population.
3.3.2. Generation of New Solution via Lévy FlightsThe new solution is calculated based on the previous best nests
via Lévy flights. In the proposed CSA method, the optimal path forthe Lévy flights is calculated by Mantegna’s algorithm [22]. Thenew solution by each nest is calculated as follows:
Calculate all thermal and hydro generations based on the initialization.
- Set Xd to Xbestd for each nest- Set the best of all Xbestd to Gbest- Set iteration counter iter = 1.
Generate new solution via Lévy flightsnewdd
newd XrandXbestX Δ××+= 3α
- Check for limit violations and repairing.- Calculate all hydro and thermal generation outputs.- Evaluate fitness function to choose new Xbestd and Gbest
Discover alien egg and randomizedis disd d dX Xbest K X= + × Δ
- Check for limit violations and repairing.- Calculate all hydro and thermal generation outputs.- Evaluate fitness function to choose new Xbestd and Gbest
Iter<Itermax?
Stop
Iter = Iter + 1
Yes
No
Fig. 1. The flowchart of CSA for solving ST-HTS.
280 T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287
where a > 0 is the updated step size; rand3 is a normally distributedrandom number in [0, 1] and the increased value DXd
new is deter-mined by:
DXnewd ¼ v � rxðbÞ
ryðbÞ� ðXbestd � GbestÞ ð23Þ
where
m ¼ randx
jrandyj1=bð24Þ
where randx and randy are two normally distributed stochastic vari-ables with standard deviation rx(b) and ry(b) given by:
rxðbÞ ¼Cð1þ bÞ � sin pb
2
� �C 1þb
2
� �� b� 2
b�12ð Þ
24
35
1=b
ð25Þ
ryðbÞ ¼ 1 ð26Þ
where b is the distribution factor (0.3 6 b 6 1.99) and C(�) is thegamma distribution function. For the newly obtained solution, itslower and upper limits should be satisfied according to the unit’slimits:
qj;m;d¼qj;max if qj;m;d > qj;max
qj;min if qj;m;d < qj;min
qj;m;d otherwise
8><>: ; j¼1; . . . ;N2; m¼1; . . . ;M�1
ð27Þ
Psi;m;d¼Psi;max if Psi;m;d > Psi;max
Psi;min if Psi;m;d < Psi;min
Psi;m;d otherwise
8><>: ; i¼2; . . . ;N1; m¼1; . . . ;M
ð28Þ
The power output of N2 hydro units and the slack thermal unit 1 arethen obtained as in Section 3.2. The fitness value is calculated using(19) and the nest corresponding to the best fitness function is set tothe best nest Gbest.
3.3.3. Alien egg discovery and randomizationThe 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 prob-lem similar to the Lévy flights. The new solution due to this actioncan be found out in the following way:
Xdisd ¼ Xbestd þ K � DXdis
d ð29Þ
where K is the updated coefficient determined based on the proba-bility of a host bird to discover an alien egg in its nest:
where rand4 and rand5 are the distributed random numbers in [0,1]and randp1(Xbestd) and randp2(Xbestd) are the random perturbationfor positions of the nests in Xbestd. For the newly obtained solution,its lower and upper limits should be also satisfied constraints (27)and (28). The value of the fitness function is calculated using (19)and the nest corresponding to the best fitness function is set tothe best nest Gbest.
3.3.4. Stopping criteriaThe algorithm is stopped when the number of iterations (Iter)
reaches the maximum number of iterations (Itermax).
The flowchart of the proposed CSA for solving the problem isgiven in Fig. 1.
4. Numerical results
The proposed CSA has been tested on five systems with qua-dratic fuel cost function of thermal units and two systems withnonconvex fuel cost function of thermal units. The proposed algo-rithm is coded in Matlab platform and run on a 2 GHz PC with 2 GBof RAM.
4.1. Selection of parameters
In the proposed CSA method, three main parameters whichhave to be predetermined are the number of nests Np, maximumnumber of iterations Nmax, and the probability of an alien egg tobe discovered pa.
Among the three parameters, the number of nests has signifi-cantly effects on the obtained solution quality. Generally, thehigher number of NP is chosen the higher probability for a betteroptimal solution is obtained. However, the computational timefor obtaining the solution for case with the large numbers is long.By experiments, the number of nests in this paper is set from 10 to100 depending on system size. Similar to NP, the maximum num-ber of iterations Nmax also has an impact on the obtained solutionquality and computation time. It is chosen based on the complexityand scale of the considered problems. For the test systems in thispaper, the maximum number of Nmax ranges from 300 for small
Table 1Results by CSA for the first system with quadratic fuel cost of thermal units withdifferent values of Pa.
Pa Min. cost ($) Avg. cost ($) Max. cost ($) Std. dev. ($) CPU (s)
T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287 281
systems to 2500 for the large-scale systems. The value of the prob-ability for an alien egg to be discovered pa can be chosen in therange [0, 1]. However, different values of pa may lead to differentoptimal solutions for a problem. For the complicated or large-scaleproblems, the selection of the value for the probability has an obvi-ous effect on the optimal solution. In contrast, the effect of theprobability is inconsiderable for the simple problems, that is differ-ent values of the probability can also lead the same optimal solu-tion. Therefore, the best value of pa has to be tuned in its range[0, 1]. Besides, the value of distribution factor b needs be deter-mined has a significant impact on solution quality of CSA and itis suggested in the range [0.3,1.99] as in the Mantegna’s algorithm[22]. As experience in [18], the value of distribution factor in thesuggested range does not have much effect on the final solutionof economic dispatch problem and it has been fixed at 1.5 for alltest systems. Therefore, it also fixed at 1.5 for all test systems inthis paper.
4.2. Systems with quadratic fuel cost function of thermal units
In this test case, the proposed CSA is tested on five systems. Thedata for the first four system is from [1] consisting of one thermaland one hydro plant for the first system, one thermal and twohydro plants for the second system, two thermal and two hydroplants for the third system, and one thermal and one hydro plantsfor the fourth one. The fifth system from [7] consists of two ther-mal plants and two hydro plants with four scheduling subintervals.All data for the five systems are given in Appendix A. The numberof nests is set to 20 for system 1, 50 for systems 2 and 3, 10 for sys-tem 4, and 100 for system 5. The maximum number of iterationsfor the CSA is set to 1000, 1500, 2500, 300 and 1000 for the five sys-tems, respectively. The value of the probability pa will be analyzedin the range from 0.1 to 0.9 with a step of 0.1. For each case, theproposed CSA is run 10 independent trials and the obtained resultsincluding minimal total cost, average total cost, maximal total cost,standard deviation, and average computational time are given inTables 1–5. For the first four systems, the proposed CSA methodcan obtain optimal solution for the values of pa from 0.2 to 0.9.
Table 2Results by CSA for the second system with quadratic fuel cost of thermal units withdifferent values of Pa.
Pa Min. cost ($) Avg. cost ($) Max. cost ($) Std. dev. ($) CPU (s)
For the fifth system, the best value of pa for CSA to obtain the opti-mal solution is 0.9.
The values of Pa to obtain the best solution chosen are respec-tively 0.4, 0.6, 0.9, 0.1 and 0.9 for the five systems correspondingto the bold values in Tables 1–5. The minimal cost, average cost,maximum cost and average computational time obtained by theproposed CSA are then compared to those from EGA [9] and
Table 7Result comparison for the fifth test system with quadratic fuel cost function ofthermal units.
Method Newton’s method [7] HNN [7] CSA
Cost ($) 377,374.67 377,554.94 375,960.5756
100
101
102
103
9.6
9.65
9.7
9.75
9.8
9.85
9.9
9.95
10x 10
4
Number of iterations = 1000
Fitn
ess
Func
tion
($)
Fig. 2. Convergence characteristic of test system 1 with quadratic fuel costfunction.
102 103 1045.3
5.32
5.34
5.36
5.38
5.4
5.42x 10
4
Number of iterations = 2500
Fitn
ess
Func
tion
($)
Fig. 4. Convergence characteristic of test system 3 with quadratic fuel cost functionof thermal units.
282 T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287
OGB-GA [9] given in Table 6 for the first four systems and fromHNN [7] and Newton’s method [7] given in Table 7 for the fifth sys-tem. The result comparison has shown that the proposed CSA canobtain approximate or better total than EGA [9] and OGB-GA [9] forthe first four test systems. Obviously, the proposed CSA obtainsmuch better total cost than both Newton’s method and HNN. Fur-thermore, the proposed CSA is faster than EGA [9] and OGB-GA [9].Therefore, the proposed CSA is effective for solving different hydro-thermal systems with quadratic fuel cost function of thermal units.There is no computer reported in [9]. The optimal solution for eachsystem by the proposed CSA is given in Appendix B. Figs. 2–6 show
101
102
103
104
0
1
2
3
4
5
6
7
8x 10
10
Number of iterations = 1500
Fitn
ess
Func
tion
($)
Fig. 3. Convergence characteristic of test system 2 with quadratic fuel costfunction.
the convergence characteristic of CSA for systems 1–5,respectively.
4.3. Systems with valve point effects on fuel cost function of thermalunits
For this case, cost curve with valve point loading effects forthermal units is considered. The proposed CSA is tested on two sys-tems from [14] where the first system comprises two hydro plantsand two thermal plants and the second one consists of two hydroplants and four thermal plants. The data of the test systems is givenin Appendix A.
The number of nests is set to 50 and 100 for the first and secondsystem, respectively. The maximum number of iterations for theCSA is set to 1500 for the both systems. For the probability pa,the proposed CSA is run for its different values from 0.1 to 0.9 witha step size of 0.1 to find the best one. For each case of each system,the proposed CSA is performed 10 independent runs and theobtained results including minimal total cost, average total cost,
101
102
103
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
11
Number of iterations = 300
Fitn
ess
Func
tion
($)
Fig. 5. Convergence characteristic of test system 4 with quadratic fuel cost functionof thermal units.
101
102
103
3.75
3.8
3.85
3.9
3.95
4
4.05
4.1
4.15x 10
5
Number of iterations = 1000
Fitn
ess
Func
tion
($)
Fig. 6. Convergence characteristic of test system 5 with quadratic fuel cost functionof thermal units.
Table 8Results by CSA for the first system with nonconvex fuel cost of thermal units withdifferent values of pa.
pa Min. cost ($) Avg. cost ($) Max. cost ($) Std. dev. ($) CPU (s)
Fig. 7. Convergence characteristic of test system 1 with non-convex fuel cost ofthermal units.
102
103
104
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10
10.1x 10
4
Number of iterations = 1500
Fitn
ess
Func
tion
($)
Fig. 8. Convergence characteristic of test system 2 with non-convex fuel cost ofthermal units.
T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287 283
maximal total cost, standard deviation, and average computationaltime are respectively given in Tables 8 and 9. Based on the analysis,the value of pa for CSA to obtain the best solution is 0.8 for the bothsystems.
The obtained results are compared to those from other methodsavailable in the literature including AIS, EP, PSO, and DE in [14]
Table 10Result comparison for two test systems with non-convex fuel cost of thermal units.
Method AIS [14] EP [1
System 1 Cost ($) 66,117 66,19CPU time (s) 53.43 75.48
System 2 Cost ($) 93,950 94,25CPU time (s) 59.14 67.82
given in Table 10. The result comparison in the table has indicatedthat the proposed CSA can obtain better solution quality than theothers in terms of total cost and computational time. Therefore,the proposed is very effective for solving the HTS problem withnonconvex fuel cost function of thermal units. Note all methodsin [14] have been implemented on a Pentium-IV 3.0 GHz PC. Theoptimal solutions for the systems are given in Appendix B. Figs. 7and 8 show the convergence characteristic of CSA for both systems,respectively.
4] PSO [14] DE [14] CSA
8 66,166 66,121 66,115.4471.62 60.76 6.7
0 94,126 94,094 92,725.134380.37 83.54 18.6
284 T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287
5. Conclusions
In this paper, the CSA method has been successfully applied forsolving short-term hydrothermal scheduling problem with smoothand nonsmooth fuel cost curves of thermal units. The CSA methodis a new meta-heuristic algorithm inspired from the obligate broodparasitism of some cuckoo species by laying their eggs in the nestsof other host birds of other species for solving optimization prob-lems. The highlighted advantages of the CSA method are few con-trol parameters and effective for optimization problems withcomplicated constraints. The proposed CSA has been tested on sev-eral hydrothermal systems with different fuel cost functions ofthermal units. The result comparisons with other methods in theliterature have indicated that the proposed CSA is more efficientthan many other methods. Therefore, the proposed CSA can be a
Table A2Hydro system data of test systems 1, 2 and 3 with quadratic fuel cost function of therma
System Hydro plant ahj (MCF/h)
1 1 61.532 1 0.2
2 0.43 1 1.98
2 0.936
Table A3Thermal generator and hydro system data of the test system 4 with quadratic fuel cost fu
Thermal plant Hydro plant
asi ($/h) bsi ($/MW h) csi ($/MW2 h) ahj (acre-ft/h)
575 9.2 0.00184 330
Table A4Thermal generator data of the test system 5 with quadratic fuel cost function of thermal
Thermal plant asi ($/h) bsi ($/MW h)
1 380 6.752 600 5.28
Table A5Hydro system data of the test system 5 with quadratic fuel cost function of thermal units
T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287 287
References
[1] Rashid AHA, Nor KM. An efficient method for optimal scheduling of fixed headhydro and thermal plants. IEEE Trans Power Syst 1991;6:632–6.
[2] Wood AJ, Wollenberg BF. Power generation, operation and control. 2nded. New York: John Wiley & Sons; 1996.
[3] Salam Nor KM, Hamdan AR. Hydrothermal scheduling based Lagrangianrelaxation approach to hydrothermal coordination. IEEE Trans Power Syst1998;13:226–35.
[4] Li C, Svoboda AJ, Chung-Li Tseng, Johnson RB. Hydro unit commitment inhydro-thermal optimization. IEEE Trans Power Syst 1997;12:764–9.
[5] Nilsson O, Sjelvgren D. Mixed-integer programming applied to short-termplanning of a hydro-thermal system. IEEE Trans Power Syst 1996;11:281–6.
[6] Zaghlool MF, Trutt FC. Efficient methods for optimal scheduling of fixed headhydrothermal power systems. IEEE Trans Power Syst 1988;3:24–30.
[7] Basu M. Hopfield neural networks for optimal scheduling of fixed headhydrothermal power systems. Electr Power Syst Res 2003;64:11–5.
[8] Yang PC, Yang HT, Huang CL. Scheduling short-term hydrothermal generationusing evolutionary programming techniques. IEEE Proc Gener Trans Distrib1996;143:371–6.
[9] Sasikala J, Ramaswamy M. Optimal gamma based fixed head hydrothermalscheduling using genetic algorithm. Expert Syst Appl 2010;37:3352–7.
[10] Chan PH, Chang HC. Genetic aided scheduling of hydraulically coupled plantsin hydro-thermal coordination. IEEE Trans Power Syst 1996;11:975–81.
[11] Orero SO, Irving MR. A genetic algorithm modeling framework and solutiontechnique for short term optimal hydrothermal scheduling. IEEE Trans PWRS1998;13:501–18. May.
[12] Gil E, Bustos J, Rudnick H. Short-term hydrothermal generation schedulingmodel using a genetic algorithm. IEEE Trans PWRS 2003;18:1256–64. Nov.
[13] Lakshminarasimman L, Subramanian S. Short-term scheduling ofhydrothermal power system with cascaded reservoirs by using modifieddifferential evolution. IEEE Proc Gener Trans Distrib 2006;153:693–700.
[14] Basu M. Artificial immune system for fixed head hydrothermal power system.Energy 2011;36:606–12.
[15] Dieu VN, Ongsakul W. Enhanced merit order and augmented LagrangeHopfield network for hydrothermal scheduling. Int J Electr Power EnergySyst 2008;30:93–101.
[16] Dieu VN, Ongsakul W. Improved merit order and augmented LagrangeHopfield network for short term hydrothermal scheduling. Energy ConversManage 2009;50:3015–23.
[17] Yang X-S, Deb S. Cuckoo search via Lévy flights. In: Proc. World congress onnature & biologically inspired computing (NaBIC 2009), India; 2009. p. 210–214.
[18] Dieu NV, Peter Schegner, Weerakorn Ongsakul. Cuckoo search algorithm fornon-convex economic dispatch. IET Gener Transm Distrib 2013;7:645–54.
[19] Basu M, Chowdhury A. Cuckoo search algorithm for economic dispatch. Energy2013;60:99–108.
[20] Ahmed Jubaer, Salam Zainal. A maximum power point tracking (MPPT) for PVsystem using Cuckoo search with partial shading capability. Appl Energy2014;119:118–30.
[21] Xiangtao Li, Minghao Yin. A hybrid cuckoo search via Lévy flights for thepermutation flow shop scheduling problem. Int J Prod Res 2013;51:4732–54.
[22] Mantegna RN. Fast, accurate algorithm for numerical simulation of Levy stablestochastic processes. Phys Rev E 1994;49:4677–83 [Fast, accurate algorithmfor numerical simulation of Levy stable stochastic processes].
