Applied Energy 132 (2014) 276–287 Contents lists available at ScienceDirect Applied Energy journal homepage: www.elsevier.com/locate/apenergy 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 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, Vo Van Ngan Str., Thu Duc Dist., Ho Chi Minh City, Viet Nam b h i g h l i g h t s g r a p h i c a l a b s t r a c t A new cuckoo search method is Hydro Plants Thermal Plants Minimize fuel cost 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 ~ ~ Cuckoo Search Algorithm Electrical Load X1 X2 … XNd-1 Net solution Randomize a number of nests, Nd XNd Calculate slack thermal and hydro units Evaluate fitness funcƟon IteraƟon=1 Generate new eggs via Lévy Flights Calculate slack thermal and hydro units Evaluate fitness funcƟon Discover alien egg Host bird’s nest Host bird’s egg Alien bird’s egg Generate new eggs And randomize Calculate slack thermal and hydro units Evaluate fitness funcƟon IteraƟon=Max IteraƟon = IteraƟon + No Yes 4.15 x 10 4.1 STOP Fitness Function ($) 4.05 3.95 3.9 3.85 3.8 3.75 10 a r t i c l e i n f o Article history: Received 19 February 2014 Received in revised form July 2014 Accepted July 2014 Keywords: Cuckoo search algorithm Short-term hydrothermal scheduling Convex fuel cost function Nonconvex fuel cost function Lévy flights 10 Number of iterations = 1000 10 a b s t r a c t 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 control 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 Introduction The short term hydro-thermal scheduling (HTS) problem is to determine the power generation among the available thermal ⇑ Corresponding author Tel.: +84 88 657 296x5730; fax: +84 88 645 796 E-mail address: vndieu@hcmut.edu.vn (D.N Vo) http://dx.doi.org/10.1016/j.apenergy.2014.07.017 0306-2619/Ó 2014 Elsevier Ltd All rights reserved 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 satisfying 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 T.T Nguyen et al / Applied Energy 132 (2014) 276–287 277 Nomenclature ahj, bhj, chj asi, bsi, csi dsi, esi Bij, B0i, B00 PD,m Phj,m Phj,max Phj,min water discharge coefficients of hydro plant j fuel cost coefficients of thermal plant i fuel cost coefficients of thermal plant i reflecting valve-point effects B-matrix coefficients for transmission power loss total system load demand at subinterval m power output of hydro plant j in subinterval m maximum power output of hydro plant i minimum power output of hydro plant i [1], k–c iteration method, dynamic programming (DP) [2], Lagrange relaxation (LR) [3], decomposition and coordination method [4], and mixed integer programming (MIP) [5], Newton’s method [6,7] In the ECM method, the coordination equations are linearized and solved for the water availability constraint separately from generating units, thus the Lagrangian multiplier associated with water availability constraint is separately from the outputs of generating units Based on the obtained Lagrangian multiplier of water constraint, the Lagrangian multiplier associated with power balance constraint is determined and the outputs of thermal and hydro units are finally calculated In the k–c method, the c values of the different hydro plants are initially chosen and thereafter the k iterations are invoked for the given power demand at each interval of the scheduling period The DP method is a popular optimization method implemented for solving the hydrothermal scheduling problem However, computational and dimensional requirements in the DP method increase drastically with large-scale system planning horizon [8] On the contrary to the DP method, the LR method is more efficient for dealing with large-scale problems However, the LR method may suffer to duality gap oscillation resulting from the dual problem formulation, leading to divergence for some problems with operation limits and non-convexity of incremental heat rate curves of generators In the decomposition and coordination method, the problem is decomposed into thermal and hydro sub-problems and they are solved by network flow programming and priority list based dynamic programming methods In order to solve the hydrothermal scheduling problem, MIP requires linearization of equations whereas the decomposition and coordination method may encounter difficulties when dealing with the operation limits and 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 economic load dispatch in hydrothermal power systems However, the Newton’s method mainly depends on the formulation and inversion of Jacobian matrix, leading to restriction of applicability on large-scale problems In general, these conventional methods can be applicable for only the HTS problems with differentiable fuel cost function and constraints Recently, several artificial intelligence techniques have been proposed for solving the hydrothermal scheduling problems such as 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 GA and EP algorithms are evolutionary based method for solving optimization problems However, the essential encoding and decoding schemes in the both methods are different In the GA method, the crossover and mutation operations required to diversify the offspring may be detrimental to actually reaching an optimal solution In this regard, the EP is more likely better when overcoming these disadvantages In the EP method, the mutation is a key PL,m Psi,m Psi,max Psi,min qj,m tm Wj 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 duration for subinterval m volume of water available for generation by hydro unit j during the scheduling period search operator which generates new solutions from the current ones [15] However, one disadvantage of the EP method in solving some of the multimodal optimization problems is its slow convergence to a near optimum The DE method 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 difficult to deal with large-scale problems with slow or no convergence to the near optimum solution The AIS method is one of the efficient methods for solving the nonconvex short-term hydrothermal scheduling The most important step of the AIS method is the application of the aging operator to eliminate the old antibodies, to maintain the diversity of the population, and to avoid the premature convergence The advantages of the AIS method are few parameters and small maximum number of iterations However, the AIS method is also difficult to deal with large-scale problems like other meta-heuristic search methods Optimal gamma based genetic algorithm (OGB-GA) [9] is an improvement of GA for efficiently solving the HTS problem In the OGB-GA method, the c values of the hydro plants are considered as the GA variables and the k iterations over the scheduling period can be called to find the thermal and hydro generations for each chromosome in the population to calculate the value of the fitness function Therefore, the number of the GA variables is drastically reduced and does not even depend on the number of intervals in the scheduling period [9] 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 in its energy function are not carefully selected Moreover, the HNN method also suffers slow convergence to optimal solution and the constraints of the problem must be linearized when applying in HNN [16] In general, most of the artificial intelligence techniques usually suffer slow convergence to the near optimum solution for the HTS problems The cuckoo search algorithm (CSA) developed by Yang and Deb [17] is a new meta-heuristic algorithm for solving optimization problems inspired from the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds of other species To verify the effectiveness of the CS algorithm, Yang and Deb compared its performance with particle swarm optimization (PSO) and GA for ten standard optimization benchmark functions [17] As observed from the obtained results, the CSA method has been outperformed both PSO and GA methods for all test functions in terms of success rate in finding optimal solution and the number of required objective function evaluations The highlighted advantages of the CSA method are fine balance of randomization and intensification and less number of control parameters Recently, CSA has been successfully applied for solving non-convex economic dispatch (ED) problems considering generator and system characteristics including valve point loading effects, multiple fuel options, prohibited operating zones, spinning reserve and 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 power dispatch problem [19] For ED problems [18,19], CSA has been tested on many systems and obtained better solution quality than several methods in the literature such as HNN, GA, EP, Taguchi method, biogeography-based optimization, and PSO, etc Moreover, for micro grid power dispatch problem [19], CSA also obtains higher solution quality than DE and PSO On the other hand, for Photovoltaic system, CSA has been used to track Maximum Power Point [20] A comprehensive assessment is carried out against two other methods, namely Perturbed and Observed (P&O) and PSO The evaluations include (1) gradual irradiance and temperature changes, (2) step change in irradiance and (3) rapid change in both irradiance and temperature These tests are carried out for both large and medium-sized PV systems It is stated in [20] that CSA outperforms both P&O and PSO with respect to tracking capability, transient behavior and convergence Consequently, CSA is an efficient method for solving optimal problems In this paper, a cuckoo search algorithm (CSA) is proposed for solving short-term fixed head HTS problem considering power losses in transmission systems and valve point loading effects in fuel cost function of thermal units The effectiveness of the proposed CSA 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 existing GA (EGA), and OGB-GA in [9], Newton’s method and HNN in [7], and PSO, DE, EP and AIS in [14] Problem formulation The objective of the HTS problem is to minimize the total fuel cost of thermal generators while satisfying hydraulic, power balance, and generator operating limits constraints The short-term fixed-head hydrothermal scheduling problem having N1 thermal units and N2 hydro units scheduled in M time sub-intervals is formulated as follows The objective is to minimize the total cost of thermal generators [14]: MinC T ẳ N1 M X X tm ẵasi ỵ bsi Psi;m ỵ cs P2si;m ỵ jdsi sinesi P si P si;m ịịj mẳ1 iẳ1 1ị 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 ỵ Phj;m PL;m PD;m ẳ 0; iẳ1 m ẳ 1; ; M 2ị jẳ1 where the power losses in transmission lines are calculated using Krons formula: PL;m ẳ NX ỵN NX þN i¼1 P i;m Bij P j;m þ j¼1 NX ỵN tm qj;m ẳ W j ; B0i Pi;m ỵ B00 3ị iẳ1 j ẳ 1; ; N2 4ị mẳ1 i ẳ 1; ; N1 ; Phj;min Phj;m Phj;max ; j ¼ 1; ; N2 ; m ¼ 1; ; M m ¼ 1; ; M ð6Þ ð7Þ Cuckoo search algorithm for short-term fixed-head HTS problem 3.1 Cuckoo search algorithm The cuckoo search algorithm (CSA) was developed by Yang and Deb [17] In comparison with other meta-heuristic search algorithms, 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 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 [21] 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 e [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 3.2 Calculation of power output for slack thermal and hydro units In this research, the output power for slack hydro units is calculated based on the availability water constraint while the power output of thermal units is determined using the power balance constraint Suppose that the water discharges of the first (M À 1) subintervals of N2 hydro units are obtained, the water discharge for hydro unit j at subinterval M is calculated using the available water constraint (4) as follows: qj;M ¼ Wj À MÀ1 X ! t m qj;m =t M ; j ¼ 1; ; N2 ð8Þ Therefore, the power output of hydro unit j at subinterval m is determined using (5): Phj;m ¼ 5ị bhj ặ q bhj 4chj ahj qj;m ị 2chj j ẳ 1; 2; ; N2 where where the rate of water flow from hydro plant j in interval m is determined by: qj;m ẳ ahj ỵ bhj Phj;m ỵ cj P2hj;m Psi;min Psi;m P si;max ; m¼1 – Water availability constraint: The total available water discharge of each hydro plant for the whole scheduled time horizon is limited by: M X – Generator operating limits: Each thermal and hydro units have their upper and lower generation limits: ðbhj ; m ¼ 1; ; M; ð9Þ À 4chj ðahj À qj;m ÞÞ P 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 N1 À thermal units and N2 hydro units in the system 279 T.T Nguyen et al / Applied Energy 132 (2014) 276–287 Suppose that the power outputs of (N1 À 1) thermal unit and N2 hydro units at subinterval m are known, the power output of the slack thermal unit is calculated by: Ps1;m ẳ PD;m ỵ PL;m N1 N2 X X Psi;m Phj;m iẳ2 10ị jẳ1 Eq (3) is rewritten in terms of the slack thermal unit as follows: PL;m ẳ BTT;11 P2s1;m ỵ Psi;m BTT;ij Psj;m ỵ N2 X N2 X Phi;m BHH;ij P hj;m iẳ1 jẳ1 iẳ2 jẳ1 iẳ2 jẳ1 ỵ B00 ð11Þ where HT;ij BTH;ij ; B BT;0i B0i ¼ B HH;ij H;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 = BHT,ijT A P 2s1;m ỵ B Ps1;m ỵ C ẳ where 13ị N1 N2 X X BTT;1i Psi;m ỵ BTH;1j P hj;m ỵ BT;01 iẳ2 14ị jẳ1 N1 X N1 N2 X N2 X X C¼ ‘ P si;m BTT;ij P sj;m ỵ P hi;m BHH;ij Phj;m iẳ2 jẳ2 ỵ2 FT d ẳ ỵ B00 ỵ PD;m iẳ2 N2 X iẳ2 jẳ1 Psi;m 15ị The solution of the second order Eq (12) is obtained by: Ps1;m ẳ B ặ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 À 4AC 2A M X ðPs1;m;d À P lim s1 ị mẳ1 19ị where Ks and Kq are penalty factors for the slack thermal unit and available water at subinterval M, respectively; Ps1,m,d is the power output of the slack thermal unit calculated from Section 3.2 corresponding to nest d in the population; qj,M is the water discharge of all hydro plants at the subinterval M calculated from Eq (8) corresponding to nest d in the population The limits for the slack thermal unit and water discharge at the subinterval M in (19) are determined as follows: jẳ1 Phj;m F i Psi;m;d ị ỵ K s N2 X ỵ K q qj;M;d qlim j Þ N1 X N2 N1 N2 X X X Psi;m BTH;ij Phj;m ỵ BT;0i Psi;m ỵ BH;0j Phj;m N1 X N1 M X X m¼1 i¼1 Plim s1 i¼1 jẳ1 iẳ2 jẳ1 18ị jẳ1 12ị A ẳ BTT;11 m 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 ; q1;m;d ; q2;m;d ; ; qN2 ;m:d of nest d including the thermal units from to N1 for M subintervals and water discharges for hydro units from to N2 for the first (M À 1) subintervals At the subinterval M, the nest d only contains thermal units from to N1 The power output of the thermal units and water discharges in the Np nests are randomly chosen satisfying Psi,min Psi,m,d Psi,max and qj,min qj,m,d qj,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: Substituting (11) into (10), a quadratic equation is obtained: B¼2 j ¼ 1; ; N2 ; ¼ 1; ; M À N1 X N2 N1 N2 X X X Psi;m BTH;ij Phj;m ỵ BT;0i Psi;m ỵ BH;0j Phj;m BTT;ij Bij ẳ B m 17ị qj;m;d ẳ qj;min ỵ rand2 qj;max qj;min ị; jẳ1 iẳ2 jẳ2 ỵ2 i ẳ 2; ; N1 ; ! iẳ2 ỵ Psi;m;d ẳ P si;min ỵ rand1 P si;max Psi;min ị; ẳ 1; ; M N1 N2 X X BTT;1i Psi;m ỵ BTH;1j Phj;m ỵ BT;01 Ps1;m N1 X N1 X power out of thermal unit i at subinterval m corresponding to nest d and qj,m,d is the water discharge for hydro unit j at subinterval m corresponding to nest d In the CSA, 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: ð16Þ where B2 À 4AC P 3.3 Implementation of cuckoo search algorithm Based on the three rules in Section 3.1, the CSA method is implemented for solving the short-term fixed-head HTS problem as follows 3.3.1 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, qj,m,d], where Psi,m,d is the qlim j Ps1;max > > > P s1;m;d > > : qj;max > > > < qj;min ¼ > qj;M;d > > : if Ps1;m;d > Ps1;max if Ps1;m < Ps1;min otherwise ð20Þ if qj;M;d > qj;max if qj;M;d < qj;min otherwise ð21Þ 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 initialized population of the host nests is set to the best value of each nest Xbestd (d = 1, , Nd) and the nest corresponding to the best fitness function in (19) is set to the best nest Gbest among all nests in the population 3.3.2 Generation of New Solution via Lévy Flights The new solution is calculated based on the previous best nests via Lévy flights In the proposed CSA method, the optimal path for the Lévy flights is calculated by Mantegna’s algorithm [22] The new solution by each nest is calculated as follows: X new ¼ Xbestd ỵ a rand3 DX new d d ð22Þ 280 T.T Nguyen et al / Applied Energy 132 (2014) 276–287 where a > is the updated step size; rand3 is a normally distributed random number in [0, 1] and the increased value DXnew is deterd mined by: DX new ẳv d rx bị Xbest d Gbestị ry bị 23ị where mẳ randx 24ị jrandy j1=b where randx and randy are two normally distributed stochastic variables with standard deviation rx(b) and ry(b) given by: À Á31=b C1 ỵ bị sin p2b rx bị ẳ b1 C 1ỵb b 2 ị 25ị ry bị ẳ 26ị The flowchart of the proposed CSA for solving the problem is given in Fig Numerical results The proposed CSA has been tested on five systems with quadratic fuel cost function of thermal units and two systems with nonconvex fuel cost function of thermal units The proposed algorithm is coded in Matlab platform and run on a GHz PC with GB of RAM 4.1 Selection of parameters where b is the distribution factor (0.3 b 1.99) and C(Á) is the gamma distribution function For the newly obtained solution, its lower and upper limits should be satisfied according to the unit’s limits: > < qj;max if qj;m;d > qj;max qj;m;d ¼ qj;min if qj;m;d < qj;min ; j ¼ 1; ;N2 ; m ¼ 1; ; M À > : qj;m;d otherwise ð27Þ > < Psi;max if Psi;m;d > Psi;max P si;m;d ¼ Psi;min if Psi;m;d < Psi;min ; i ¼ 2; ;N1 ; m ¼ 1; ;M > : Psi;m;d otherwise Initialize population of host nests Psi ,m ,d = Psi ,min + rand1 * ( Psi ,max − Psi ,min ) ð28Þ The power output of N2 hydro units and the slack thermal unit are then obtained as in Section 3.2 The fitness value is calculated using (19) and the nest corresponding to the best fitness function is set to the best nest Gbest 3.3.3 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 out in the following way: X dis d ẳ Xbestd ỵ K DX dis d ð29Þ where K is the updated coefficient determined based on the probability of a host bird to discover an alien egg in its nest: & K¼ if rand4 < pa otherwise 30ị ẳ rand5 ½randp1 ðXbestd Þ À randp2 ðXbestd Þ q j ,m ,d = q j ,min + rand * (q j ,max − q j ,min ) 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 = Generate new solution via Lévy flights X dnew = Xbest d + α × rand × ΔX dnew - 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 randomize and the increased value DXdis d is determined by: DX dis d In the proposed CSA method, three main parameters which have to be predetermined are the number of nests Np, maximum number of iterations Nmax, and the probability of an alien egg to be discovered pa Among the three parameters, the number of nests has significantly effects on the obtained solution quality Generally, the higher number of NP is chosen the higher probability for a better optimal solution is obtained However, the computational time for obtaining the solution for case with the large numbers is long By experiments, the number of nests in this paper is set from 10 to 100 depending on system size Similar to NP, the maximum number of iterations Nmax also has an impact on the obtained solution quality and computation time It is chosen based on the complexity and scale of the considered problems For the test systems in this paper, the maximum number of Nmax ranges from 300 for small X ddis = Xbestd + K ì X ddis 31ị where rand4 and rand5 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 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 to the best nest Gbest - Check for limit violations and repairing - Calculate all hydro and thermal generation outputs - Evaluate fitness function to choose new Xbestd and Gbest No Iter
