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

Electrical Power and Energy Systems 65 (2015) 271–281 Contents lists available at ScienceDirect 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 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 3864 7256x5730 E-mail address: vndieu@gmail.com (D.N Vo) http://dx.doi.org/10.1016/j.ijepes.2014.10.004 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 ỵ bsi Psi;m ỵ csi P2si;m ỵ jdsi mẳ1 iẳ1   i Psi;m  sin esi  P 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; iẳ1 m ẳ 1; ; M 2ị j¼1 where the power losses in transmission lines are calculated using Krons formula [2]: PL;m ẳ NX ỵN NX ỵN iẳ1 Pi;m Bij Pj;m ỵ jẳ1 NX ỵN B0i Pi;m ỵ B00 3ị iẳ1 – 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] 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 The best nests with high quality of egg (better solution) will be carried over to the next generation The number of available host nests is fixed, and a host bird can discover an alien egg with a probability pa [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]: tỵ1ị Xi ẳ X ti ỵ a È Le0 VyðkÞ ð8Þ where a > 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 4ị mẳ1 where the rate of water flow from hydro plant j in subinterval m is determined by: qj;m ẳ ahj ỵ bhj Phj;m ỵ chj P2hj;m ð5Þ – Generator operating limits: Each thermal and hydro units have their upper and lower generation limits: Psi;min Psi;m Psi;max ; Phj;min Phj;m P hj;max ; i ¼ 1; ; N ; 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 274 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 = 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 ẳ ỵ 5ị=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 P2s1;m ỵ B Ps1;m ỵ C ẳ 11ị where A ẳ BTT;11 Bẳ2 12ị N1 N2 X X BTT;1i Psi;m ỵ BTH;1j Phj;m ỵ BT;01 iẳ2 Cẳ 13ị j¼1 N1 X N1 N2 X N2 N1 X N2 X X X P si;m BTT;ij P sj;m ỵ P hi;m BHH;ij P hj;m ỵ P si;m BTH;ij P hj;m i¼2 j¼2 i¼1 j¼1 i¼2 j¼1 N1 N2 N1 N2 X X X X ỵ BT;0i P si;m þ BH;0j P hj;m þ B00 þ P D;m À P si;m P hj;m iẳ2 jẳ1 iẳ2 jẳ1 14ị The solution of the second order Eq (11) is obtained by: Ps1;m ẳ B ặ p B2 4AC 2A ð15Þ where B2À4AC 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 using (5) Then its water discharge at subinterval M is calculated using the available water constraint (4) as follows: qj;M ¼ Wj M1 X !, t m qj;m tM 16ị mẳ1 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 ẳ bhj ặ q bhj À 4chj ðahj À qj;M Þ 2chj ; j ¼ 1; 2; ; N2 ð17Þ where bhj À  chj  ðahj À qj;M Þ P 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 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 ẳ PD;m ỵ PL;m PL;m ¼ BTT;11 P2s1;m Psi;m À Phj;m ! N1 N2 X X ỵ BTT;1i Psi;m ỵ BTH;1j Phj;m ỵ BT;01 Ps1;m i¼2 j¼1 N1 X N1 N2 X N2 N1 X N2 X X X ỵ Psi;m BTT;ij Psj;m þ Phi;m BHH;ij Phj;m þ Psi;m BTH;ij Phj;m i¼2 jẳ2 iẳ1 jẳ1 iẳ2 jẳ1 N1 N2 X X ỵ BT;0i Psi;m ỵ BH;0j P hj;m ỵ B00 iẳ2 jẳ1 10ị P si;m;d ẳ P si;min ỵ rand1 P si;max P si;min ị; i ẳ 2; ; N ; m ẳ 1; ; M 18ị 275 T.T Nguyen, D.N Vo / Electrical Power and Energy Systems 65 (2015) 271281 P hj;m;d ẳ P hj;min ỵ rand2 Phj;max P hj;min ị; j ẳ 1; ; N2 ;m ẳ 1; ;M 19ị where rand1 and rand2 are uniformly distributed random numbers in [0,1] Consider vector X d ẳ ẵP 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 to N1 for M subintervals and hydro units from to N2 for the first (M-1) subintervals At subinterval M, nest d only contains thermal units from to N1 The power output of the thermal and hydro units in the Np nests are randomly chosen satisfying Psi,min Psi,m,d Psi,max and Phj,min Phj,m,d 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 F i Psi;m;d ị ỵ K s ðPs1;m;d À Plim ðqj;M;d À qlim j Þ s1 ị ỵ K q mẳ1 iẳ1 mẳ1 jẳ1 20ị where Ks and Kq are penalty factors for the slack thermal unit 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 and the water discharge at the subinterval M in (20) are determined as follows: > < 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 > < 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: rx bị ẳ 31=b pb C1 ỵ bị sin ị 26ị b1 C1ỵb ị b 2 ị ry bị ẳ 27ị where b is the distribution factor (0.3 b 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 ẳ Xbest nodiscardd ỵ 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 ẳv rx bị Xbest nodiscardd Gbestị ry ðbÞ ð29Þ ð21Þ rx ðbÞ ry ðbÞ 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 22ị ẳ Xbest nodiscardr Xbest nodiscardd ị=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 = – 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 is the distributed random number in [0,1], the step size a ¼ A= G is determined and DXdiscardnew is obtained by: d rx bị new DXdiscardd ẳ v   ðXbest discardd À GbestÞ; ry ðbÞ ð24Þ where mẳ randx jrandy j1=b 28ị 25ị new DXnodiscardd ẳ Xbest nodiscardr À Xbest nodiscardd Þ=u; ð31Þ pffiffiffi where a = A and u ẳ ỵ 5ị=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 ẳ Xbest nodiscardr ỵ a rand5 DXnodiscardd 32ị new DXnodiscardd ẳ Xbest nodiscardd Xbest nodiscardr Þ=u ð33Þ pffiffiffi and a = A = and u ẳ ỵ 5ị=2 For the newly obtained solution, its lower and upper limits should be satisfied according to the unit’s limits: > < 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 > < 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 > : 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 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 ẳ Xbest d ỵ K DX d if rand6 < pa 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 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: ẳ rand7 ẵrandp1 Xbestd ị 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 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 MCSA CSA 3.88 Fitness Function ($)  Nmax ð36Þ where K is the updated coefficient determined based on the probability of a host bird to discover an alien egg in its nest: K¼ Table The sensitivity analysis with respect to the stopping criterion for the system with quadratic fuel cost function 3.86 3.84 3.82 3.8 3.78 3.76 10 10 10 Number of iterations = 5000 Fig Convergence characteristic of MCSA and CSA for the system with quadratic fuel cost function of thermal unit Table Result comparison for the system with quadratic fuel cost function of thermal units Method Newton [21] HNN [21] CSA MCSA Cost ($) 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 = T.T Nguyen, D.N Vo / Electrical Power and Energy Systems 65 (2015) 271–281 Table Results by MCSA for the first system with non-convex 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 66118.2739 66116.7871 66115.7551 66115.6562 66115.5023 66115.6378 66115.5692 66115.5053 66115.4459 66148.3443 66137.1950 66124.9015 66124.9541 66116.5400 66127.1959 66119.4853 66116.3965 66116.8292 66180.9419 66159.2199 66150.5171 66153.4599 66122.6741 66150.7313 66147.1090 66121.9564 66128.1060 14.7858 14.0093 12.3232 12.6732 1.5408 13.8924 9.0145 1.5833 2.9602 6.5 6.7 6.7 6.9 6.8 6.7 6.8 6.8 6.6 Table Results by MCSA for the second system with non-convex 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 92825.8057 92801.8044 92755.2380 92757.2273 92770.2015 92781.4987 92741.9020 92805.6040 92863.1251 93178.3231 92925.6080 92860.9817 92841.1892 92835.5498 92855.6608 92836.6646 92943.7870 93039.8177 93760.1479 93217.5736 93055.7946 92915.7016 92961.4572 92929.1243 92938.5369 93321.5465 93477.5816 263.4581 104.6282 70.6202 48.8641 47.4677 48.5824 47.7365 113.1874 140.1228 13.6 14.1 14.0 13.8 13.7 14.2 13.3 13.4 13.7 Table The sensitivity analysis with respect to the stopping criterion for the first system with non-convex fuel cost of thermal units Nmax Min cost Avg cost Max cost Std dev CPU 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000 66116.2441 66115.6030 66115.4459 66115.4379 66115.4379 66115.4379 66115.4379 66115.4379 66115.4379 66115.4379 66130.5065 66123.4250 66116.8292 66121.315 66118.7912 66119.8527 66118.3408 66118.3637 66118.3103 66117.2630 66154.8832 66147.2104 66128.1060 66146.6594 66153.3468 66145.6194 66145.6559 66145.6107 66145.3794 66153.3462 13.1013 12.5686 2.9602 12.1452 10.2527 10.7322 8.9965 9.0359 8.9579 8.2578 2.0 4.5 6.6 8.8 10.5 13.8 16.8 17.9 20.3 23.3 Table The sensitivity analysis with respect to the stopping criterion for the second system with non-convex fuel cost of thermal units Nmax Min cost Avg cost Max cost Std dev CPU 3000 4000 5000 6000 7000 8000 9000 10,000 92813.5998 92808.9373 92741.9020 92763.7014 92749.8239 92738.4444 92754.0452 92752.6756 92991.4466 92969.6709 92836.6646 92900.4971 92812.5064 92797.726 92802.9407 92798.1902 93358.3227 93499.2851 92938.5369 93231.8716 92914.4207 92885.7751 92924.8274 92899.0808 139.9290 152.7867 47.7365 124.4226 43.6531 31.2707 41.9571 37.8289 8.8 10.5 13.3 16.2 20.1 23.6 27.6 28.9 Step 2: Initialize nests for power output of all hydro and thermal units using (18) and (19) Step 3: Evaluate fitness function to choose Xbest_discardd, Xbest_nodiscardd, Xbest_nodiscardr and Gbest based on 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’ Step 5: – Check for limit violations and repairing using (34) and (35) 277 – 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’ Step 7: – 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’ – Calculate the fitness function in (20) Step 8: Put new eggs generated in step and 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 Step 11: If Iter Iter max , Iter = Iter + and return to Step 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 implemented for solving these systems for result comparison The both algorithms are coded in Matlab platform and run on a GHz PC with 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 each [21] 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 3000, respectively The effect of the probability pa on the optimal solution by the MCSA method for this system is analyzed and the obtained results are given in Table Based on the analysis, the best value of pa of MCSA for this system is 0.7 Moreover, a further sensitive 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 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 computational time is faster Therefore, the best parameters of MCSA for this system are Np = 12, Nmax = 3000, and pa = 0.7 For implementation of the CSA method, the probability pa, number of nests, and maximum number 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 deviation, and average computational time are given in Table The optimal solution for the system by the proposed MCSA and CSA are given in Appendix B Fig shows the convergence characteristic of MCSA and CSA for the system As observed from Table 3, the MCSA method can obtain better 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 278 T.T Nguyen, D.N Vo / Electrical Power and Energy Systems 65 (2015) 271–281 Table Results obtained from MCSA and CSA of two systems with non-convex fuel cost of thermal units System Method Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s) CSA MCSA CSA MCSA 66115.5935 66115.4459 92767.0886 92741.9020 66126.9193 66116.8292 92870.8049 92836.6646 66147.8236 66128.1060 93132.5493 92938.5369 13.2565 2.9602 99.0503 47.7365 23.1 6.6 41.2 13.3 x 10 CSA MCSA 6.65 Fitness Function ($) 6.645 6.64 6.635 6.63 6.625 Table 11 Results by MCSA for the third system with non-convex fuel cost of thermal units with different values of pa pa Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 208762.7817 204341.9458 203707.1445 203065.9561 203726.0858 203909.0909 204162.857 206244.1729 203876.9789 213515.558 206511.495 205084.626 204599.077 205179.783 205603.094 206661.253 208659.567 213134.302 218874.476 208536.579 205884.938 206267.853 206515.485 208225.794 209396.63 213045.941 227258.392 3026.5798 1081.6181 584.0704 738.2208 659.2901 1091.5761 1350.5882 1960.5150 5582.4592 30.9 30.5 30.6 30.5 30.2 30.6 31.0 30.8 30.5 6.62 total cost than the others Therefore, MCSA is very effective for solving the short-term fixed-head HTS with quadratic fuel cost function of thermal units 6.615 6.61 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Number of iterations = 5000 Systems with valve point effects on fuel cost function of thermal units Fig Convergence characteristic of MCSA and CSA for the first system with nonconvex fuel cost of thermal units x 10 1.8 CSA MCSA 1.7 Fitness Function ($) 1.6 1.5 1.4 1.3 1.2 1.1 1000 2000 3000 4000 5000 6000 7000 Number of iterations = 7000 Fig Convergence characteristic of MCSA and CSA for the second system with non-convex fuel cost of thermal units better solution quality than the conventional CSA The best total cost for the system obtained by the proposed MCSA and CSA are compared to those from HNN and Newton’s method in [21] as given in Table Obviously, the proposed MCSA can obtain better The proposed MCSA and CSA methods are tested on two systems from [20] where the first system comprises two hydro plants 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 Appendix 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 3000 and 5000 for the two systems, respectively The effect of pa on the optimal solution of MCSA for the two systems is analyzed in Tables and Based on the analyses, the best values of pa for 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 stopping criterion is given in Tables and Similar to the case 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 For the CSA method, the probability pa and the number of nests 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 independent trials for the two systems and the obtained results including minimal total cost, average total cost, maximal total cost, standard deviation, and average computational time are given in Table The optimal solutions obtained from the MCSA and CSA methods for the systems are given in Appendix B The convergence characteristics of MCSA and CSA for the both systems are shown in Figs and 3, respectively Table 10 Result comparison for two test systems with non-convex fuel cost of thermal units Method System System Cost ($) CPU time (s) Cost ($) CPU time (s) AIS [20] EP [20] PSO [20] DE [20] CSA MCSA 66,117 53.43 93,950 59.14 66,198 75.48 94,250 67.82 66,166 71.62 94,126 80.37 66,121 60.76 94,094 83.54 66,115.6 23.1 92,767 41.2 66,115.4 6.6 92,741 13.3 279 T.T Nguyen, D.N Vo / Electrical Power and Energy Systems 65 (2015) 271–281 Table 12 The obtained result from MCSA and CSA for the third system with non-convex fuel cost of thermal units Method Min ($) Avg ($) Max ($) Std dev ($) CPU (s) CSA MCSA 203401.1430 203065.9561 204685.7176 204599.0770 207491.1176 206267.8530 852.3547 738.2208 66.4 30.5 x 10 CSA MCSA 2.9 Fitness Function ($) 2.8 2.7 2.6 2.5 2.4 independent trials and the obtained results including minimal total cost, average total cost, maximal total cost, standard deviation, and average computational time are given in Table 12 There 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 in Fig Conclusions 2.3 2.2 2.1 2000 4000 6000 8000 10000 12000 Number of iterations = 12000 Fig Convergence characteristic of MCSA and CSA for the third system with nonconvex fuel cost of thermal units As observed from Table 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 systems 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 compared to those from other methods including AIS, EP, PSO, and DE in [20] as given in Table 10 The result comparison in the table has indicated that the proposed MCSA can obtain better solution quality in terms of total cost and computational time than the other methods Note all methods in [20] have been implemented 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 pa on the final solution by the proposed MCSA method is given in Table 11 As observed from the table, the proposed MCSA method can reach the best optimal solution for the system at pa = 0.4 In the CSA method, the probability pa, number of nests, and maximum number of iterations are set to 0.75, 50, and 12,000, respectively Both the MCSA and CSA methods are run 20 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 modifications of the MCSA method based on the conventional CSA method are classification of nest in two groups based on their fitness 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 comparisons with the conventional CSA and other methods in the literature 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 See Tables A1–A8 The transmission loss coefficients for the third system with non-convex fuel cost of thermal units are as follows: 0:39 0:1 0:12 0:12 0:15 0:16 B ¼ 10À4  0:39 0:1 0:12 0:12 0:15 0:16 0:1 0:12 0:12 0:15 0:16 0:39 0:1 0:12 0:12 0:15 0:16 0:4 0:14 0:1 0:15 0:2 0:1 0:4 0:14 0:1 0:15 0:2 0:14 0:35 0:11 0:2 0:18 0:12 0:14 0:35 0:11 0:2 0:18 0:1 0:11 0:36 0:17 0:15 0:12 0:1 0:11 0:36 0:17 0:15 0:15 0:2 0:17 0:49 0:14 0:15 0:15 0:2 0:17 0:49 0:14 0:2 0:18 0:15 0:14 0:45 0:16 0:2 0:18 0:15 0:14 0:45 0:1 0:12 0:12 0:15 0:16 0:39 0:1 0:12 0:12 0:15 0:16 0:4 0:14 0:1 0:15 0:2 0:1 0:4 0:14 0:1 0:15 0:2 0:14 0:35 0:11 0:2 0:18 0:12 0:14 0:35 0:11 0:2 0:18 0:1 0:11 0:36 0:17 0:15 0:12 0:1 0:11 0:36 0:17 0:15 0:15 0:2 0:17 0:49 0:14 0:15 0:15 0:2 0:17 0:49 0:14 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 See Tables B1–B4 Table A1 Data of thermal units in the system with quadratic fuel cost function of thermal units Thermal plant asi ($/h) bsi ($/MW h) csi ($/MW2 h) Psi,min (MW) Psi,max (MW) 380 600 6.75 5.28 0.00225 0.0055 47.5 100 450 1000 280 T.T Nguyen, D.N Vo / Electrical Power and Energy Systems 65 (2015) 271–281 Table A2 Data of hydro units in the system with quadratic fuel cost function of thermal units Hydro plant ahj (acre-ft/h) bhj (acre-ft/MW h) chj (acre-ft/MW2 h) Wj (acre-ft) Phj,min (MW) Phj,max (MW) 260 250 8.5 9.8 0.00986 0.0114 125,000 286,000 0 250 500 Table A3 Data of thermal units in the first system with non-convex fuel cost of thermal units Thermal plant asi ($/h) bsi ($/MW h) csi ($/MW2 h) dsi ($/h) esi (1/MW) Psi,min (MW) Psi,max (MW) 25 30 3.2 3.4 0.0025 0.0008 12 14 0.0550 0.0450 50 50 300 700 Table A4 Data of hydro units in the first system with non-convex fuel cost of thermal units Hydro plant ahj (MCF/h) bhj (MCF/MW h) chj (MCF /MW2 h) Wj (MCF) Phj,min (MW) Phj,max (MW) 1.980 0.936 0.306 0.612 0.000216 0.000360 2500 2100 0 400 300 Table A5 Data of thermal units in the second system with non-convex fuel cost of thermal units Thermal plant asi ($/h) bsi ($/MW h) csi ($/MW2 h) dsi ($/h) esi (rad/MW) Psi,min (MW) Psi,max (MW) 10 10 20 20 3.25 2.00 1.75 1.00 0.0083 0.0037 0.0175 0.0625 12 18 16 14 0.0450 0.0370 0.0380 0.0400 20 30 40 50 125 175 250 300 Table A6 Data of hydro units in the second system with non-convex fuel cost of thermal units Hydro plant ahj (acre-ft/ h) bhj (acre-ft/ MW h) chj (acre-ft/ MW2 h) Wj (acreft) Phj,min (MW) 260 250 8.5 9.8 0.00986 0.01140 125,000 286,000 0 Phj,max (MW) 250 500 Table A7 Data of thermal units in the third system with non-convex fuel cost of thermal units Thermal plant asi ($/h) bsi ($/MW h) csi ($/MW2 h) dsi ($/h) esi (rad/MW) Psi,min (MW) Psi,max (MW) 10 10 20 20 10 10 20 20 3.25 2.00 1.75 1.00 3.25 2.00 1.75 1.00 0.0083 0.0037 0.0175 0.0625 0.0083 0.0037 0.0175 0.0625 12 18 16 14 12 18 16 14 0.0450 0.0370 0.0380 0.0400 0.0450 0.0370 0.0380 0.0400 20 30 40 50 20 30 40 50 125 175 250 300 125 175 250 300 Table A8 Data of hydro units in the third system with non-convex fuel cost of thermal units Hydro ahj bhj chj Wj Phj,min Phj,max plant (acre-ft/h) (acre-ft/MW h) (acre-ft/MW2 h) (acre-ft) (MW) (MW) 260 250 260 250 8.5 9.8 8.5 9.8 0.00986 0.01140 0.00986 0.01140 125,000 286,000 125,000 286,000 0 0 250 500 250 500 Table B1 Optimal solutions obtained by MCSA and CSA for the system with quadratic fuel cost function of thermal units Sub-interval Duration (h) PD (MW) 12 1200 12 1500 12 1400 12 1700 MCSA Ps1 (MW) Ps2 (MW) Ph1 (MW) Ph2 (MW) 432.5053 326.2515 164.0322 308.6007 449.9445 449.0193 240.6800 409.3062 448.8348 402.4822 221.7214 369.5016 449.9593 570.3089 494.0726 494.0726 CSA Ps1 (MW) Ps2 (MW) Ph1 (MW) Ph2 (MW) 437.9985 324.1876 164.4848 304.7345 450.0000 445.9295 232.6844 420.5458 450.0000 385.2570 228.6392 378.8693 450.0000 584.6512 250.0000 478.5961 Table B2 Optimal solutions obtained by MCSA and CSA for the first system with non-convex fuel cost of thermal units Sub-interval Duration (h) PD (MW) 900 1200 1100 MCSA Ph1 (MW) Ph2 (MW) Ps1 (MW) Ps2 (MW) 238.2119 82.9216 220.0598 399.0655 323.9756 183.8083 221.3594 538.6915 274.3932 125.4489 221.3566 538.6903 CSA Ph1 (MW) Ph2 (MW) Ps1 (MW) Ps2 (MW) 237.7067 83.254 220.2122 399.0658 323.8045 183.966 221.3596 538.6922 275.0592 124.9636 221.3596 538.5385 T.T Nguyen, D.N Vo / Electrical Power and Energy Systems 65 (2015) 271–281 Table B3 Optimal solutions obtained by MCSA and CSA for the second system with non-convex fuel cost of thermal units Sub-interval Duration (h) PD (MW) 12 900 12 1100 12 1000 12 1300 MCSA Ph1 (MW) Ph2 (MW) Ps1 (MW) Ps2 (MW) Ps3 (MW) Ps4 (MW) 174.954 318.8056 89.3994 175 110.3697 50 243.9809 413.3416 124.8057 175 121.3553 50 207.3797 348.6341 123.15 174.9944 118.6775 50 249.9228 499.9853 124.9727 174.9979 221.0397 69.1368 CSA Ph1 (MW) Ph2 (MW) Ps1 (MW) Ps2 (MW) Ps3 (MW) Ps4 (MW) 188.5668 310.4464 90.0708 174.9986 104.4073 50 249.8855 406.8059 124.918 174.9991 121.8029 50.0011 187.6565 363.5403 124.859 174.9979 121.7986 50.0912 249.9993 499.9825 124.9984 175 221.9405 68.1483 Table B4 Optimal solutions obtained by MCSA and CSA for the third system with non-convex fuel cost of thermal units Sub-interval Duration (h) PD (MW) 12 1800 12 2200 12 2000 12 2600 MCSA Ps1 (MW) Ps2 (MW) Ps3 (MW) Ps4 (MW) Ps5 (MW) Ps6 (MW) Ps7 (MW) Ps8 (MW) Ph1 (MW) Ph2 (MW) Ph3 (MW) Ph4 (MW) 118.7578 174.9629 122.5802 50.0501 123.1402 174.9995 122.3478 51.5102 200.5213 290.2943 138.9413 300.4622 123.9536 174.9995 131.1588 50.4618 124.9743 174.9938 138.6556 50.0509 247.9007 431.8541 249.9482 406.3396 122.2651 174.9994 121.0466 50.0038 124.6315 174.7414 120.8343 50.001 178.9172 358.7702 236.0955 374.6472 124.9988 174.8741 245.8808 76.6911 124.9951 174.9812 248.3071 84.5742 248.8821 496.9859 247.5457 498.7531 CSA Ps1 (MW) Ps2 (MW) Ps3 (MW) Ps4 (MW) Ps5 (MW) Ps6 (MW) Ps7 (MW) Ps8 (MW) Ph1 (MW) Ph2 (MW) Ph3 (MW) Ph4 (MW) 118.8687 174.9948 121.525 50.6977 110.0656 174.9994 122.8098 50.0006 151.6292 304.6483 173.3393 315.47 124.2754 174.9968 121.6558 51.0102 124.9991 174.9998 164.243 50.0006 248.0488 466.3494 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for solving optimization problems with... 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... 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