DSpace at VNU: Cuckoo search algorithm for non-convex economic dispatch

www.ietdl.org Published in IET Generation, Transmission & Distribution Received on 8th March 2012 Revised on 14th January 2013 Accepted on 10th February 2013 doi: 10.1049/iet-gtd.2012.0142 ISSN 1751-8687 Cuckoo search algorithm for non-convex economic dispatch Dieu N Vo1, Peter Schegner2, Weerakorn Ongsakul3 Department of Power Systems, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam Institute of Electrical Power Systems and High Voltage Engineering, Technische Universität Dresden, 01069 Dresden, Germany Energy Field of Study, School of Environment, Resources and Development, Asian Institute of Technology, Pathumthani 12120, Thailand E-mail: ongsakul@ait.asia Abstract: This study proposes a cuckoo search algorithm (CSA) for solving non-convex economic dispatch (ED) considering generator and system characteristics including valve-point effects, multiple fuels, prohibited zones, spinning reserve and power loss CSA is a new meta-heuristic optimisation method inspired from the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds of other species When the host birds discover an alien egg in their nest, they can either throw it away or simply abandon their nest and build a new one elsewhere The CSA idealised such breeding behaviour in combination with Lévy flights behaviour of some birds and fruit flies for applying to various constrained optimisation problems The effectiveness of the proposed method has been tested on different non-convex ED problems Test results have indicated that the proposed method can obtain less expensive solutions than many other methods reported in the literature Accordingly, the proposed CSA is a promising method for solving the practical nonconvex ED problems Nomenclature ai, bi, ci aij, bij, cij ei, fi eij, fij Bij, B0i, B00 N ni Pi Pi, max Pi, Pij, Puik Plik PD PL Si Si, max SR fuel cost coefficients of unit i fuel cost coefficients for fuel type j of unit i fuel cost coefficients of unit i reflecting valve-point effects fuel cost coefficients for fuel type j of unit i reflecting valve-point effects B-matrix coefficients for transmission power loss total number of generating units number of prohibited zones of unit i power output of unit i maximum power output of unit i minimum power output of unit i minimum power output for fuel j of unit i upper bound for prohibited zone k of unit i lower bound for prohibited zone k of generator i total system load demand total transmission loss spinning reserve from unit i maximum spinning reserve contribution of unit i total system spinning reserve requirement IET Gener Transm Distrib., 2013, Vol 7, Iss 6, pp 645–654 doi: 10.1049/iet-gtd.2012.0142 Introduction Economic dispatch (ED) is to optimally allocate the real power output among the online thermal units so that their total production cost is minimised while satisfying the unit and system operating constraints [1, 2] Conventionally, the objective function of the ED was approximated by a single quadratic function for mathematical convenience [3] Nevertheless, the input–output characteristics of thermal generating units are essentially more complicated because of the effects of valve point effects [4], multiple fuels (MFs) [5] or prohibited zones [6] Therefore the practical ED problem can be formulated as non-convex objective function subject to non-linear constraints, which is difficult to be solved by the classical mathematical programming techniques Several conventional methods have been applied for solving ED problems such as gradient search, Newton’s method, dynamic programming (DP) [3], hierarchical approach based on the numerical method (HNUM) [5], decomposition method [6] and Maclaurin series-based Lagrangian (MSL) method [7] Among these methods, only MSL method can directly deal with the non-convex ED problem with non-differentiable objective by using the Maclaurin expansion of non-convex terms in the objective function Although this method can quickly find a solution 645 & The Institution of Engineering and Technology 2013 www.ietdl.org for the problem, the obtained result is still far from optimum, especially for the large-scale systems In general, the conventional methods are not effective for non-convex ED problems Recently, many methods based on artificial intelligence have been developed for solving ED problems such as Hopfield neural network (HNN) [8], genetic algorithm (GA) [9–12], evolutionary programming (EP) [13], Taguchi method (TM) [14], biogeography-based optimisation (BBO) [15], and particle swarm optimisation (PSO) [16–26] Among of them, the HNN method based on the minimisation of its energy function can be only applied to the convex optimisation problems with differentiable objective function This method can be implemented on large-scale problems but it suffers many drawbacks such as local optimum solution and long computation time The others are the meta-heuristic search methods which can overcome the drawbacks of the HNN method because of their ability to find near optimal solution for non-convex optimisation problems However, for the large-scale and non-smooth problems with multiple minima, these methods may suffer low solution quality and long computational time In addition, hybrid methods have been also developed for dealing with the non-convex ED problems such as combining of chaotic differential evolution and quadratic programming (DEC-SQP) [27], simulated annealing like particle swarm optimisation (SA-PSO) [28], combination of differential evolution and BBO (DE/BBO) [29], and hybrid Hopfield neural network quadratic programming based technique (HNN-QP) [30] These hybrid methods utilise the advantages of each element method to enhance their search ability for the complex problems However, the hybrid methods contain many controllable parameters which may not be properly selected In this paper, a cuckoo search algorithm (CSA) is proposed for solving non-convex ED problems considering generator and system characteristics including valve point loading effects (VPE), MF options, prohibited operating zones, spinning reserve and power loss CSA is a new meta-heuristic optimisation method inspired from the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds of other species When the host birds discover an alien egg in their nest, they can either throw it away or simply abandon their nest and build a new one elsewhere The CSA idealised such breeding behaviour in combination with Lévy flights behaviour of some birds and fruit flies for applying to various constrained optimisation problems The proposed CSA is tested on several large-scale and non-convex systems and the obtained results are compared to those from many other methods in the literature 2.1 Problem formulation ED problem with valve point effects The ED with VPE is a non-smooth and non-convex problem with multiple minima considering ripples in the heat-rate curves of boilers The model of VPE has been proposed in [9] by adding a sinusoidal function to the quadratic fuel cost function The objective of the problem is written as N Min F = Fi Pi i=1 646 & The Institution of Engineering and Technology 2013 (1) Fig Fuel cost curve of units with valve-point effects where the fuel cost function of unit i is represented by [13] Fi Pi = + bi Pi + ci Pi2 + ei × sin fi × Pi, − Pi (2) subject to Real power balance: The total real power output of generating units satisfies total real load demand plus power loss N Pi = PD + PL (3) i=1 where the power loss PL can be approximated by Kron’s formula [3] N N PL = N Pi Bij Pj + i=1 j=1 B0i Pi + B00 (4) i=1 Generator capacity limits: The real power output of generating units should be within their upper and lower operating limits as Pi, ≤ Pi ≤ Pi, max (5) The cost curve function of units with valve point effects is depicted in Fig 2.2 ED problem with multiple fuels In practical power system operation conditions, many thermal generating units being supplied with MF sources such as coal, natural gas and oil require that their fuel cost functions may be segmented as piecewise quadratic cost functions for different fuel types Therefore, in the ED problem with MFs, the piecewise quadratic function is used to represent the MFs that are available for each generating unit [5] The fuel cost function of unit i is represented by [8] ⎧ ⎪ ai1 + bi1 Pi + ci1 Pi2 , fuel 1, Pi, ≤ Pi ≤ Pi1 ⎪ ⎪ ⎨ Fi Pi = ai2 + bi2 Pi + ci2 Pi , fuel 2, Pi1 , Pi ≤ Pi2 ⎪ ⎪ ⎪ ⎩ a + b P + c P2 , fuel j, P ij ij i ij i ij−1 , Pi ≤ Pi, max (6) IET Gener Transm Distrib., 2013, Vol 7, Iss 6, pp 645–654 doi: 10.1049/iet-gtd.2012.0142 www.ietdl.org Fig Fuel cost curve of units with MFs Fig Fuel cost curve of units with prohibited zones For generator i with j fuel options in (6), its cost curve is divided into j discrete segments between lower limit Pi, and upper limit Pimax, in which each fuel type is represented by a quadratic function with lower power output limit Pij-1 and upper power output limit Pij The cost curve function of units with MFs is depicted in Fig The objective of the ED problem with MF is to minimise the total cost (1) where the fuel cost function for each generator is given in (6) subject to the real power balance constraint (3) and generator capacity limits (5) the real power balance constraint (3), generator capacity limits (5) for units having no POZ, and POZ: For units with POZ, their feasible operating points should be located at one of the sub-regions as follows 2.3 ED problem with VPE and MFs In practical power system operation conditions, thermal generating units can be supplied with MF sources and their boilers also have valve points for controlling their power outputs Therefore for more accurate in determination of the solution for the practical ED problem the fuel cost function of units should consider both VPE and MF [12] The fuel cost function of generating unit i is represented by ⎧ Fi1 Pi , fuel 1, Pi, ≤ Pi ≤ Pi1 ⎪ ⎪ ⎨ Fi Pi = Fi2 Pi , fuel 2, Pi1 , Pi ≤ Pi2 ⎪ ⎪ ⎩ Fij Pi , fuel j, Pij−1 , Pi ≤ Pi, max − Pi )) (8) ED problem with prohibited operating zones Thermal generating units may have prohibited operating zones (POZ) because of physical limitations on components of units Consequently, the whole operating region of a generating unit with POZ will be broken into several isolated feasible sub-regions [16] The fuel cost function for each unit in the ED problem with POZ can be a quadratic function (2) or a quadratic function with VPE (8) The equality and inequality constraints for this problem include IET Gener Transm Distrib., 2013, Vol 7, Iss 6, pp 645–654 doi: 10.1049/iet-gtd.2012.0142 N Si ≥ SR (10) i=1 Si = Pi, max − Pi , Si, Si = 0; subject to the real power balance constraint (3) and generator capacity limits (5) 2.4 (9) Equation (9) indicates that if unit i has ni POZ, it will have (ni + 1) feasible disjoint operating regions which will form a non-convex set The cost curve function of units with prohibited zones is depicted in Fig Spinning reserve constraint: The spinning reserve constraint for all units is defined as (7) Fij Pi = aij + bij Pi + cij Pi2 ∀i [ V where the operating margin of each unit Si is determined by where the fuel cost function for fuel type j of unit i is determined by + eij × sin( fij × (Pij, ⎧ l ⎪ ⎨ Pi, ≤ Pi ≤ Pi1 u Pi [ Pik−1 ≤ Pi ≤ Pikl ; k = 2, , ni ; ⎪ ⎩ Pu ≤ P ≤ Pi, max i ini max ∀i [ V ; ∀i Ó V (11) (12) where Ω is the set of units with POZ Equation (10) shows that the spinning reserve contribution of all units should satisfy a required threshold and the contributed spinning reserve in the system is only from the units without prohibited zones as in (11) This is because the ability to regulate system load of units with prohibited zones are strictly limited by their prohibited zones Therefore the required spinning reserve is mainly contributed from the units without prohibited zones 3.1 CSA for ED problems CSA Cuckoo search is a new meta-heuristic algorithm inspired from the nature for solving optimisation problems developed by Yang and Deb in 2009 [31] The basic idea of this algorithm is based on the obligate brood parasitic behaviour of some cuckoo species in combination with the Lévy flight behaviour of some birds and fruit flies There are three idealised rules for the new CSA described as follows [32] 647 & The Institution of Engineering and Technology 2013 www.ietdl.org † 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 a high quality of egg (better solution) will be carried over to the next generation; † A host bird can discover an alien egg in its nest with a probability of pa ∈ [0, 1] In this case, it can simply either throw the egg away or abandon the nest and find a new location to build a completely new one Based on these rules, a general mathematical model for the CSA is summarised in [31, 32] 3.2 Calculation of power output for slack unit Fig Adjustment of unit’s prohibited zone violation To guarantee that the equality constraint (3) is always satisfied, a slack generating unit is arbitrarily selected and therefore its power output will be dependent on the power output of remaining N-1 generating units in the system The method for calculation of power output for the slack unit is as follows [33] Suppose that the power output of the N − generating units are known, the power output of the slack unit is calculated by N Ps = PD + PL − Pi (13) i=1 i=s where s is an arbitrary unit selected among the N units The power loss in (4) is rewritten by considering Ps as an unknown variable ⎛ ⎞ N ⎜ PL = Bss Ps2 + ⎝2 ⎟ Bsi Pi + B0s ⎠Ps i=1 i=s N N Pi Bij Pj + (15) where the coefficients A, B and C are given by A = Bss (16) N Bsi Pi + B0s − (17) i=1 i=s N N Pi Bij Pj + i=1 j=1 i=s j=s B0i Pi + B00 + PD − i=1 i=s Pi (18) i=1 i=s The power output of the slack generator is the positive root of (15) between the two ones obtained as below −B + √ B2 − × A × C , 2A where B2 − × A × C ≥ Ps = Pinew = Pikl Piku if Pi ≤ Pikm if Pi Pikm (21) The modification of unit’s limits in (21) for the cases with prohibited zones violation is depicted in Fig A × Ps2 + B × Ps + C = B=2 (20) This middle point divides a prohibited zone in two sub-zones, the left and right prohibited sub-zones with respect to the point Therefore the operating point Pi of unit i violating its prohibited zone k will be adjusted by B0i Pi + B00 i=1 i=s By substituting (14) into (13), a quadratic equation is obtained as follows N Pikm = Pikl + Piku /2 N i=1 j=1 i=s j=s N When a unit operates in one of its POZ, a repairing strategy is used to force the unit either to move towards the lower bound or upper bound of that zone For making a decision based on the operating point of a unit located in one of its prohibited zones, the middle point of each prohibited zone is firstly determined as follows (14) + C= 3.3 Handling of prohibited operating zones violation 648 & The Institution of Engineering and Technology 2013 (19) 3.4 Implementation of CSA The proposed CSA is a population-based method similar to other meta-heuristic methods The structure of the CSA includes two main operations including a direct search based on Lévy flights and a random search based on the probability for a host bird to discover an alien egg in its nest With the combined two operations, the proposed CSA becomes a more powerful search method than other meta-heuristic search methods for complex and large-scale optimisation problems Therefore, the proposed CSA is very effective in solving non-convex and large-scale ED problems In the proposed CSA method, each nest represents a solution and a population of nests is used for finding the best solution for the problem similar to many other meta-heuristic search methods The main steps for the proposed CSA are described as follows: Initialisation: A population of Np host nests is represented by X = [X1, X2,…, XNp]T, where each nest Xd = [Pd1,…, Pds-1, Pds + 1,…, PdN] (d = 1,…, Np) represents power output of units except the slack unit is initialised by Xdi = Pi, + rand∗1 Pi, max − Pi, (22) IET Gener Transm Distrib., 2013, Vol 7, Iss 6, pp 645–654 doi: 10.1049/iet-gtd.2012.0142 www.ietdl.org where rand1 is a uniformly distributed random number in [0, 1] for each population of the host nests This initial solution is further checked for POZ violation If the violation is found, the repairing strategy in Section 3.3 is used to move the operating point to a feasible region Based on the initial population of nests, the fitness function to be minimised corresponding to each nest for the considered problem is calculated N FTd = Fi Xdi + Ks × Pds − Pslim i=1 N + Kr × max 0, SR − (23) Sdi i=1 where Ks and Kr are penalty factors for the slack unit and spinning reserve constraint, respectively; Pds is power output of the slack unit calculated from Section 3.2 corresponding to nest d in the population; and Sdi is the spinning reserve of unit i corresponding to nest d in the population calculated from (11) and (12) The limits for the slack unit in (23) are determined based on its calculated power output as follows Pslim ⎧ ⎨ Ps, max = Ps, ⎩ Pds if Pds Ps, max if Pds , Ps, otherwise (24) where Ps, max and Ps, are the maximum and minimum power outputs of the slack unit, respectively The initial population of the host nests is set to best value of each nest Xbestd (d = 1,…, Nd) and the nest corresponding to the best fitness function in (23) is set to the best nest Gbest among all nests in the population Generation of new solution via Lévy flights: The new solution is calculated based on the previous best nests via Lévy flights In the proposed method, the optimal path for the Lévy flights is calculated by Mantegna’s algorithm [34] The new solution by each nest is calculated as follows Xdnew = X bestd + a × rand2 × DXdnew (25) where α > is the updated step size; rand2 is a normally distributed stochastic number; and the increased value ΔXd new is determined by DXdnew = n × sx (b) × X bestd − Gbest sy (b) n= randx randy (26) (27) 1/b where randx and randy are two normally distributed stochastic variables with standard deviation σx(β) and σy(β) given by sx (b) = G(1 + b) × sin (pb)/2 G (1 + b)/2 × b × 2((b−1)/2) 1/b sy (b) = (28) (29) where β is the distribution factor (0.3 ≤ β ≤ 1.99) and Γ(.) is the gamma distribution function IET Gener Transm Distrib., 2013, Vol 7, Iss 6, pp 645–654 doi: 10.1049/iet-gtd.2012.0142 For the newly obtained solution, its lower and upper limits should be satisfied according to the unit’s limits Xdinew ⎧ ⎨ Pi, max = Pi, ⎩ Xdi if Xdinew Pi, max if Xdinew , Pi, ; otherwise i=s (30) In addition, the newly adjusted solution should be further checked for POZ violation and the repairing strategy in Section 3.3 is used to move the solution out of the prohibited zones if any violation is found The fitness function (23) will be re-evaluated for the new solution to determine the newly best value of each nest Xbestd and the best nest of all nests Gbest by comparing the stored fitness values in Section 3.4.1 and the newly calculated ones Alien egg discovery and randomisation: 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 because of this action is calculated as follows Xddis = X bestd + K × DXddis (31) where K is the updated coefficient determined based on the probability of a host bird to discover an alien egg in its nest K= if rand3 , pa otherwise (32) and the increased value ΔXd dis is determined by DXddis = rand4 × randp1 X bestd − randp2 X bestd (33) where rand3 and rand4 are the distributed random numbers in [0, 1] and randp1(Xbestd) and randp2(Xbestd) are the random perturbation for positions of nests in Xbestd Similar to the solution obtained via Lévy flights, this new solution is also redefined as in (30) if the upper or lower limit is violated and Section 3.3 if any prohibited zones are violated The newly best value for each nest Xbestd and the best value of all nests Gbest are also determined based on comparing the calculated fitness function in (23) from this new solution and the stored one in Section 3.4.2 Stopping criteria: The proposed algorithm is terminated when the predefined maximum number of iterations is reached The flowchart of the proposed CSA method for solving non-convex ED problem is given in Fig Numerical results The proposed CSA is coded in Matlab platform and run 100 independent trials for each test case on a 2.1 GHz PC with GB of RAM 4.1 Selection of parameters In the proposed CSA method, four main parameters that have to be predetermined are the number of nests Np, maximum number of iterations Nmax, distribution factor β, 649 & The Institution of Engineering and Technology 2013 www.ietdl.org Table Results for 40-unit system with valve-point loading effects Fig Flowchart of the proposed CSA method for solving non-convex ED problem and the probability of an alien egg to be discovered in host nests pa Among these parameters, the number of nests can be easily fixed Since CSA is a powerful search method, it only needs a small number of nests for dealing with different systems By experiment, the number of nests is fixed at 10 for all test systems On the other hand, the maximum number of iterations for the CSA can be also easily fixed depending on the complexity and scale of the considered problems The maximum number of iterations for the CSA ranges from 300 for small systems up to 10 000 for large-scale systems The value of distribution factor β can be fixed in the range [0.3, 1.99] as in the Mantegna’s algorithm However, different values of β have not much impact on the final solution Therefore the value of β is fixed at 1.5 as in [31] for all test systems in this paper The value of the probability for an alien egg to be discovered can be chosen in the range [0, 1] However, different values of pa may lead to different optimal solutions for large-scale systems To select the optimal probability, its value is varied from 0.1 to 0.9 with the step size of 0.1 for large-scale problems with complicated objective function and fixed at a certain value in the range [0.1, 0.5] for small-scale problems pa Min total cost, $/h Avg total cost, $/h Max total cost, $/h 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 121 419.1726 121 412.7163 121 412.6466 121 415.0031 121 412.5355 121 420.8949 121 421.1218 121 420.8949 121 435.6459 121 567.6383 121 494.7789 121 502.9722 121 513.5459 121 520.4106 121 579.6568 121 645.0421 121 862.9429 122 105.6920 121 961.0546 121 657.9782 121 811.3775 122 095.5398 121 810.2538 122 076.7172 122 264.3211 122 898.0568 123 330.6021 111.2286 47.8170 79.9659 111.4726 81.5705 131.5436 166.3095 306.0020 429.1208 2.98 3.00 3.01 3.02 3.03 3.02 3.01 3.01 3.03 case is neglected The maximum number of iterations for the CSA is fixed at 8000 The results obtained by the proposed CSA including the minimum total cost, average total cost, maximum total cost, standard deviation, and average computational time with different values of pa from 0.1 to 0.9 with the increase step of 0.1 are given in Table For this system, the best of minimum total costs is $/h 121 412.5355 obtained at the probability of 0.5 whereas the best of average total costs, maximum total costs and standard deviations are respectively 121 494.7789, $/h 121 657.9782 and $/h 47.8170 obtained at the probability of 0.2 As observed from Table for this case, a good value of pa lies in the range [0.2, 0.5] The smaller and larger values of pa will not lead to optimal solution The average computational time for the proposed method to obtain the optimal solution of this system is around s The optimal solution for the system is given in the Appendix The best total cost and average computational time obtained by the proposed CSA for this system is compared to those from many other methods such as MSL [7], improved fast evolutionary programming (IFEP) [13], TM [14], modified PSO (MPSO) [17], combining of chaotic differential evolution and quadratic programming (DEC-SQP) [27], new PSO with local random search (NPSO-LRS) [22], self-organising hierarchical PSO (SOH_PSO) [20], PSO with recombination and dynamic linkage discovery (PSO-RDL) [23], quantum-inspired PSO (QPSO) [21], SA-PSO [28], BBO [15], improved coordinated aggregation-based PSO (ICA-PSO) [24, 25], DE/BBO [29], new adaptive PSO (NAPSO) [26] and CCPSO [18, 19] as shown in Table The minimum total cost obtained by the CSA is less than that from the other methods Moreover, the CSA method can obtain better solution in a faster computing manner than many other methods except MSL and BBO methods The computational times for the MSL, IFEP, NPSO-LRS, BBO and DE-BBO, ICA-PSO, and NAPSO methods are from a Pentium IV 1.5-GHz with 512-MB RAM PC, Pentium II 350-MHz with 128-MB RAM PC, Pentium IV 1.5-GHz with 128-MB RAM processor, Pentium IV 2.3-GHz PC with 512-MB RAM, Pentium IV 1.4-GHz PC and Pentium IV 3-GHz PC with 2-GB RAM, respectively There is no computational time or computer processor reported for the other methods 4.3 4.2 Std Avg dev., $/h CPU, s Systems with multiple fuels Systems with VPE The test system consists of 40 units with VPE supplying to a load demand of 10 500 MW [13] System power loss in this 650 & The Institution of Engineering and Technology 2013 The test systems here [8] comprise 10, 30, 60 and 100 units The basic 10-unit system supplies to a load demand of 2700 MW neglecting power loss For obtaining the large-scale IET Gener Transm Distrib., 2013, Vol 7, Iss 6, pp 645–654 doi: 10.1049/iet-gtd.2012.0142 www.ietdl.org Table Comparison of best total cost and average CPU time for 40-unit system with valve-point loading effects Method Total cost, $/h CPU time, s MSL [7] IFEP [13] TM [14] MPSO [17] DEC-SQP [27] NPSO-LRS [22] SOH_PSO [20] PSO-RDL [23] QPSO [21] SA-PSO [28] BBO [15] ICA-PSO [24, 25] DE/BBO [29] NAPSO [26] CCPSO [18, 19] CSA 122 406.1000 122 624.3500 122 477.7800 122 252.2650 121 741.9793 121 664.4308 121 501.1400 121 468.8200 121 448.2100 121 430.0000 121 426.9530 121 422.1000 121 420.8900 121 412.5700 121 412.5362 121 412.5355 0.047 1167.35 94.28 — — 20.74 — — — 23.89 1.1749 139.92 12 12.7 19.3 3.03 Table Results for systems with MFs No of units 10 30 60 100 total cost, $/h 623.8092 1871.4275 3742.8559 6238.1144 avg total cost, $/h 623.8092 1871.4603 3743.2089 6240.4449 max total cost, $/h 623.8097 1872.9701 3753.3834 6250.5385 std Deviation, $/h 0.0001 0.2158 1.1403 3.3561 CPU time, s 0.679 2.517 5.765 10.268 systems with 30, 60 and 100 units, the basic 10-unit system is duplicated with the load demand proportionally adjusted to the system size The maximum numbers of iterations for the CSA for these systems are set to 300, 1000, 2000 and 3000, respectively The value of pa is fixed at 0.25 for all systems The results obtained by the CSA are given in Table Table shows a comparison from the average total costs and computational times obtained by the CSA for the systems to those from conventional GA (CGA) and IGA_AMUM in [10] Test result has indicated that the proposed CSA can obtain less total costs and faster computational times than both CGA and IGA-AMUM methods for all systems Note the computational times for both CGA and IGA AMUM methods were from a PIII-700 PC 4.4 Systems with prohibited operating zones The test systems from [11] include 15, 30, 60 and 90 units The basic 15-unit system with four units having prohibited Table Results for systems with POZ No of units 10 30 60 90 cost, 32 544.9704 65 084.9949 130 170.3949 195 258.7847 $/h avg cost, 32 545.0068 65 085.1878 130 171.5986 195 264.3818 $/h max cost, 32 546.6734 65 089.8697 130 174.0722 195 271.7057 $/h Std dev., 0.2386 0.6779 0.7531 2.4857 $/h CPU 0.589 1.169 2.028 3.036 time, s zones supplies to a load demand of 2650 MW neglecting power loss with a required spinning reserve of 200 MW The large-scale systems including 30, 60 and 90 units are formed by duplicating the basic 15-unit system with the corresponding load demand proportionally adjusted to the system size The maximum number of iterations for the CSA is set to 400, 1000, 1500 and 1900 for the systems, respectively The probability pa is fixed at 0.25 for all systems The results obtained by the CSA for the systems are given in Table A comparison of the average total costs and computational times from the CSA and other methods for these systems is shown in Table The proposed CSA can obtain better solution quality than CGA and improved GA with multiplier updating method (IGAMUM) in [11] in terms of total cost and computational time for the large-scale systems The computational times for CGA and IGAMUM were from a PIII-700 PC 4.5 Systems with valve point effects and multiple fuels The test systems in [12] consist of 10, 20, 40, 80 and 160 units The basic 10-unit system supplies to a load demand of 2700 MW neglecting power loss The large-scale systems are created by duplicating the basic 10-unit system with the load demand proportionally adjusted to the system size The maximum number of iterations for the CSA for these systems is set to 500, 1000, 2000, 4000 and 6000, respectively The value of pa is fixed at 0.1 for all systems Table shows the results obtained by the CSA for these systems Table Comparison of average total costs and CPU times for Table Comparison of average total costs and CPU times for systems with MFs systems with POZ Method No of units Total cost, $ CPU time, s Method No of units Total cost, $ CPU time, s CGA [10] 10 30 60 100 10 30 60 100 10 30 60 100 623.8106 1873.691 3748.761 6251.469 624.2896 1872.047 3744.722 6242.787 623.8092 1871.4603 3743.2089 6240.4449 29.15 263.64 517.88 873.70 29.41 80.47 157.19 275.67 0.690 2.517 5.765 10.268 CGA [11] 15 30 60 90 15 30 60 90 15 30 60 90 32 804.736 65 784.740 131 992.310 198 831.690 32 544.990 65 089.954 130 180.030 195 274.060 32 545.007 65 085.188 130 171.599 195 264.382 142.18 275.73 563.81 940.93 42.62 79.80 162.58 255.45 0.589 1.169 2.028 3.036 IGA_AMUM [10] CSA IET Gener Transm Distrib., 2013, Vol 7, Iss 6, pp 645–654 doi: 10.1049/iet-gtd.2012.0142 IGAMUM [11] CSA 651 & The Institution of Engineering and Technology 2013 www.ietdl.org Table Results for systems with VPE and MF No of units cost, $/h avg Cost, $/h max cost, $/h Std dev., $/h CPU s 10 20 40 80 160 623.8684 623.9495 626.3666 0.2438 1.587 1247.8395 1247.9980 1249.7969 0.1934 3.378 2495.9664 2496.2777 2497.7957 0.2775 7.197 4992.6853 4993.7307 5003.4294 1.0931 18.257 9990.6548 9996.6390 10 014.0183 4.9268 75.429 Table Comparison of average total costs and CPU times for systems with VPE and MF Method No of units Total cost, $ CPU time, s 10 20 40 80 160 10 20 40 80 160 10 20 40 80 160 627.6087 1249.3893 2500.9220 5008.1426 10 143.7263 625.8092 1249.1179 2499.8243 5003.8832 10 042.4742 623.9495 1247.9980 2496.2777 4993.7307 9996.6390 26.64 80.48 157.39 309.41 621.30 7.32 21.64 43.71 85.67 174.62 1.587 3.378 7.197 18.257 45.429 CGA_MU [12] IGA_MU [12] CSA In Table 8, the average total costs and computational times obtained by the CSA are compared to those from CGA_MU and IGA_MU in [12] In all cases, the CSA achieves better solution quality than both CGA_MU and IGA_MU methods, especially for the large-scale systems Note the CGA_MU and IGA_MU methods were implemented on a PIII-700 PC 4.6 Systems with valve point effects and prohibited operating zones The test systems for this problem consist of 40 and 140 units The data for the 40-unit system with VPE are given in [13] where units 10–14 have POZ as given in [26] (POZ 2) The load demand for this system is 10 500 MW The complete data of the 140-unit system can be found in [18], in which 12 units have the fuel cost function with VPE and four other units have POZ The total load demand of this system is 49 342 MW Power loss is neglected in both systems The maximum number of iterations for the CSA for both systems is set to 10 000 The results obtained by the CSA for these systems with different values of pa are given in Table The optimal solution for the 40-unit system is given in Appendix A comparison of best total cost and average computational time for the 40 and 140-unit systems is shown in Table 10 Obviously, the CSA can obtain less total cost than PSO, fuzzy adaptive PSO (FAPSO), and NAPSO in [26] for the 40-unit system and conventional PSO with the constraint treatment strategy (CTPSO), PSO with chaotic sequences (CSPSO), PSO with crossover operation (COPSO) and PSO with both chaotic sequences and crossover operation (CCPSO) in [18] for the 140-unit system Moreover, the computational time from the CSA is also faster than the other methods for the two systems except NAPSO for the 40-unit system Note that the PSO methods in [18] were implemented on a Pentium IV 2.0-GHz PC In all test cases, the proposed CSA method dominates many other methods in the literature in finding better optimal solutions with faster computational times for the non-convex ED problems In the CSA method, two main features contributing to its search process are the Lévy flights and the behaviour of a host to discover an alien egg in its nest, in which the Lévy flights are used to guide the search direction while the behaviour of alien egg discovery is used to search the optimal solution The two features are combined together to constitute a powerful search ability for the CSA method Between the two features, the behaviour of a host bird to discover an alien egg in its nest is the most effective one since it plays a key role to find an Table Results for 40-unit and 140-unit systems with VPE and POZ No of units 40 140 pa Min total cost, $/h Avg total cost, $/h Max total cost, $/h Std dev., $/h Avg CPU, s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 121 492.3248 121 491.0666 121 492.2345 121 491.7629 121 487.7727 121 511.8571 121 491.0662 121 511.8583 121 543.1578 655 746.14 655 874.47 656 007.74 655 976.15 655 914.54 655 985.00 655 932.81 655 940.76 655 813.74 121 611.4894 121 559.5748 121 552.1587 121 565.8166 121 611.3170 121 657.7282 121 717.7892 121 890.3793 122 256.7077 655 904.66 656 054.85 656 198.57 656 242.44 656 244.99 656 270.76 656 296.94 656 407.02 656 852.75 122 419.0885 121 896.0831 121 890.6644 122 022.0934 122 162.9295 122 142.2630 223 94.6958 122 835.1465 124 250.1555 661 572.41 656 285.99 656 549.16 656 736.36 656 874.80 658 397.32 660 875.30 660 096.81 662 039.84 134.5791 63.1918 48.1342 74.0059 135.1073 150.6573 186.1335 323.6328 516.1229 592.70 70.00 108.21 144.54 157.35 343.29 647.04 589.10 1303.25 14.92 14.74 14.88 14.69 14.71 14.78 14.57 14.51 14.60 38.90 38.42 38.80 38.48 38.14 38.74 38.40 38.35 38.76 652 & The Institution of Engineering and Technology 2013 IET Gener Transm Distrib., 2013, Vol 7, Iss 6, pp 645–654 doi: 10.1049/iet-gtd.2012.0142 www.ietdl.org Table 10 Comparison of best total costs and average CPU times for 40-unit and 140-unit systems with VPE and POZ No of units 40 140 Method Total cost, $/h CPU time, s PSO [26] FAPSO [26] NAPSO [26] CSA CTPSO [18] CSPSO [18] COPSO [18] CCPSO [18] CSA 124 875.8523 122 261.3706 121 491.0662 121 487.7727 657 962.73 657 962.73 657 962.73 657 962.73 655 746.14 35.87 19.6 12.7 14.71 100 99 150 150 38.90 optimal solution and can be independently used or combined with other methods Conclusion In this paper, the CSA method has been efficiently implemented for solving non-convex ED with practical nonlinear characteristics of generators The proposed CSA is a powerful search method with few controllable parameters The obtained results from the several test systems have indicated that the proposed CSA method has a much better performance than the other optimisation methods reported 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based technique’, Electr Power Compon Syst., 2009, 37, (3), pp 253–264 31 Yang, X.-S., Deb, S.: ‘Cuckoo search via Lévy flights’ Proc World Congress on Nature and Biologically Inspired Computing (NaBIC 2009), India, 2009, pp 210–214 32 Yang, X.-S., Deb, S.: ‘Engineering optimisation by cuckoo search’, Int J Math Model Numer Optim., 2010, 1, (4), pp 330–343 33 Dieu, V.N., Schegner, P., Ongsakul, W.: ‘A newly improved particle swarm optimization for economic dispatch with valve point loading effects’ Proc IEEE Power and Energy Society General Meeting, USA, July 2011 34 Mantegna, R.N.: ‘Fast, accurate algorithm for numerical simulation of Levy stable stochastic processes’, Phys Rev E, 1994, 49, (5), pp 4677–4683 Appendix The optimal solutions for the 40-unit system from different problems are given in Table 11 653 & The Institution of Engineering and Technology 2013 www.ietdl.org Table 11 Optimal solutions of 40-unit system for different problems Unit 10 11 12 13 14 15 16 17 18 19 20 With VPE With VPE and POZ Unit With VPE With VPE and POZ 110.7998 110.7998 97.3999 179.7331 87.7999 140.0000 259.5997 284.5997 284.5997 130.0000 94.0000 94.0000 214.7598 394.2794 394.2794 394.2794 489.2794 489.2794 511.2794 511.2794 110.7998 110.7999 97.3998 179.7331 87.7998 140.0000 259.5996 284.5995 284.5997 130.0000 168.7982 168.0414 125.0000 400.0000 394.2790 394.2792 489.2794 489.2793 511.2793 511.2793 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 523.2794 523.2794 523.2794 523.2794 523.2794 523.2794 10.0000 10.0000 10.0000 87.7999 190.0000 190.0000 190.0000 164.7998 194.3978 200.0000 110.0000 110.0000 110.0000 511.2794 523.2794 523.2794 523.2794 523.2793 523.2794 523.2794 10.0000 10.0000 10.0000 87.7998 190.0000 190.0000 190.0000 164.7997 164.7998 164.7998 110.0000 110.0000 109.9988 511.2793 654 & The Institution of Engineering and Technology 2013 IET Gener Transm Distrib., 2013, Vol 7, Iss 6, pp 645–654 doi: 10.1049/iet-gtd.2012.0142 ... optimization for different economic load dispatch problems’, IEEE Trans Power Syst., 2010, 25, (2), pp 1064–1077 16 Gaing, Z.-L.: ‘Particle swarm optimization to solving the economic dispatch considering... programming for economic dispatch optimization with valve-point effect’, IEEE Trans Power Syst., 2006, 21, (2), pp 989–996 28 Kuo, C.-C.: ‘A novel coding scheme for practical economic dispatch by... CSA for ED problems CSA Cuckoo search is a new meta-heuristic algorithm inspired from the nature for solving optimisation problems developed by Yang and Deb in 2009 [31] The basic idea of this algorithm