An Intelligent Approach Based on Meta-Heuristic Algorithm for Non-Convex Economic Dispatch

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1 Journal of Operaton and Automaton n Power Engneerng Vol. 3, No. 1, Wnter & Sprng 015, Pages: An Intellgent Approach Based on Meta-Heurstc Algorthm for Non-Convex Economc Dspatch F. Namdar, R. Sedaghat* Department of Electrcal Engneerng, Faulty of Engneerng, Lorestan Unversty, Khorram abad, Iran ABSTRACT One of the sgnfcant strateges of the power systems s Economc Dspatch (ED) problem, whch s defned as the optmal generaton of power unts to produce energy at the lowest cost by fulfllng the demand wthn several lmts. The undenable mpacts of ramp rate lmts, valve loadng, prohbted operatng zone, spnnng reserve and mult-fuel opton on the economc dspatch of practcal power systems are scrutnzed n ths paper. Thus, the proposed nonlnear non-convex formulaton s solved by a new modfed verson of bo-nspred bat algorthm. Due to the complextes assocated wth the large-scale optmzaton problem of economc dspatch, adaptve modfcatons are added to the orgnal bat algorthm. The modfcaton methods are appled at two separate stages and pledge augmentaton n convergence rate of the algorthm as well as extrcatng the algorthm from local optma. Veracty of the proposed methodology are corroborated by performng smulatons on three IEEE test systems. KEYWORDS: Non-convex economc dspatch, Modfcaton mechansm, Meta-heurstc algorthm, Nonlnear constraned optmzaton. 1. INTRODUCTION Scarcty of energy resources, ncreasng generaton cost of the power systems, and ever-growng load demand for electrc energy necesstate optmal Economc Dspatch (ED) n the current electrc power systems. As power demand ncreases and consderng that the fuel cost of the power generaton s exorbtant, reducng the operaton costs of power systems has turned to a sgnfcant topc [1]. Economc dspatch s an essental and sgnfcant optmzaton problems task for the operaton of power systems. The man objectve of ED n power systems s to economcally dstrbute the total requred generaton between the generaton unts, whle satsfyng the load demand and system equalty and nequalty constrants []. Receved: 7 May 014 Revsed: 10 Feb. 015 Accepted: 7 Feb. 015 *Correspondng author: R. Sedaghat (E-mal: reza_sedaghat@yahoo.com) 015 Unversty of Mohaghegh Ardabl Improvements n the schedulng of the unt power outputs can contrbute to sgnfcant cost savngs. Furthermore, t offers nformaton n formng market-clearng prces. Prevously, dfferent algorthms to fnd rate of optmum product for each power generaton unt are proposed n the lterature. Conventonal algorthms such as gradent method, Lambda Iteraton Method (LIM), Lnear Programmng (LP), Quadratc Programmng (QP), Lagrangan multpler method, and classcal technque based on coordnaton equatons can solve the ED problems. There are complex and nonlnear characterstcs wth equalty and nequalty constrants assocated wth the practcal ED, whch may be mposed to the problem to ensure the system operator of system relablty durng dsturbances and a secure operaton. Many generatng unts are suppled wth multple fuel sources and should be scheduled by the most economc fuel to burn [3]. Takng everythng 47

2 Journal of Operaton and Automaton n Power Engneerng, Vol. 3, No. 1, Wnter & Sprng 015 nto account, the system operator must consder ramp rate lmts, Prohbted Operatng Zones (POZs), system spnnng reserve, valve loadng effects, and multple fuel source optons to solve a realstc ED problem, whch makes hard the fndng of the optmum soluton dffcult [4]. Recently, as an alternatve for conventonal mathematcal approaches, the modern stochastc optmzaton algorthms based on operatonal research and artfcal ntellgence concepts, such as Genetc Algorthms (GA), Tabu search, smulated annealng (SA), dfferental evoluton and Partcle Swarm Optmzaton (PSO) are consdered as realstc and powerful soluton schemes for obtanng the global optmums n power system optmzaton problems and due to ther ablty to fnd an almost global optmal soluton for ED problems wth operatng constrants. Whle each of the above studes has consdered dfferent parts of the practcal ED problem, none of them have consdered all the practcal constrants smultaneously [5]. In addton, the utlzed algorthms are not robust at all and have found dfferent results for dfferent runs. In response to these defcences, a practcal formulaton was devsed for ED and then a new method s proposed as ts soluton n ths paper. Thus, a modfed verson of bat algorthm (BA) as an evolutonary metaheurstc algorthm was employed to solve the proposed realstc ED problem. BA tres to formulate and smulate the journey of bats n search of nutrtous resources or chasng preys. The algorthm s smple n concept; thus, t s easy to mplement, snce many adjustng parameters are not ncluded n the formulaton. However, the orgnal algorthm suffers low convergence rate and t s destned to get trapped n local optma due to the lack of dversty n the populaton. Thus, two modfcaton stages are emplaced n the orgnal algorthm to help ncrease the convergence rate of the algorthm and dversfy the populaton. Interspersng the populaton to the entre search space mproved the odds of fndng the global optma. The robustness and capablty of the proposed methodology s demonstrated by applyng the procedure to two varous IEEE standard test systems. In sum, the man contrbutons of ths study can be summarzed as follows: Proposng a comprehensve model for ED problem to consder practcal constrants n real systems. Modfyng the orgnal BA to enable t of seekng the search space faster and more precsely.. PRACTICAL ED MATHEMATICAL DISCRIPTION The ED problem s a nonlnear optmzaton problem, the objectve of whch s to determne the optmal combnaton of power generatons, whch mnmzes the cost functon whle satsfyng an equalty constrant and an nequalty constrant. The mathematcal representaton of the classcal ED problem and the proposed practcal ED are descrbed n ths secton..1. Classc ED ED n ts classcal formulaton ams to mnmze the summng costs of thermal generatng unts whch are generally consdered as the second order polynomal functon of the generaton [], as follows: n f( X) Cost( X ) = F( P ) = (a ) n +bp +cp = 1 =1 = (1) n whch P denotes output power of the th unt and n stands for the number of generators n the network. The polynomal coeffcents of cost for the th unt are represented by a, b, and c as well. The conventonal ED optmzaton problem s subjected to the followng constrants forcng generators to produce power wthn specfc lmts so that ther total generaton equals to total power demand n the network (D). n P =D = 1 mn P P P In the above formulaton, the lower and upper bounds of power generaton for the th unt are denoted by P mn and P, respectvely. () (3) 48

3 F. Namdar, R. Sedaghat: An Intellgent Approach Based on Meta-Heurstc Algorthm for Non-Convex.. Proposed practcal ED formulaton Effects of valve-pont loadngs However, t s more practcal to consder the effect of valve pont loadng for thermal power plants [6]. These effects, whch occur as each steam admsson valve n a turbne, create a rpplng nfluence on the unt s cost curve. Typcally, as each steam valve starts to open, the valve pont results n the rpples lke n Fg. 1. Consderng the valve-pont effects, the fuel cost functon of the th thermal generatng unt s expressed as the sum of a quadratc and a snusodal functon n the followng form: mn F( P ) =a + bp+ cp + esn( f( P P)) (4) Costs of valve loadng effect are represented by coeffcents e and f n the snusodal term. F( P ) =a j +bjp + mn Fuel j: Pj P j = 1,,.., n f mn cj P + ej sn( f j( Pj P)) ; P (6) The number of dfferent fuel types that are provded s denoted by n f. Apart from the two prevously mentoned conventonal constrants, the secure operaton of network mandates respectng the followng constrants: Ramp rate lmts P P0 UR; If generaton ncreases P0 P DR; If generaton decreases (7) Ramp-up and ramp-down rate lmts of the th unt are denoted by UR and DR, respectvely. Also, P 0 s the actve power output of the th unt n the prevous hour. It s an sgnfcant matter that, due to the consderaton of ramp rates, the output power of each unt s now bounded by a new lmt as follows: mn ( P, P0 ) mn(, 0 ) DR P P P + UR = 1,,.., n (8) Fg. 1. Input output curve under valve pont loadng Multple fuels Snce the dspatchng unts are practcally provded wth mult-fuel sources, each unt should be really represented by several pecewse quadratc functons reflectng the effects of fuel type changes and a generator must dentfy the economc fuel to burn. Thus, snce dfferent fuels possess varous costs, the fnal generaton cost of the unts wll depend on ther choce of fuel, leadng to dvded cost functon for generators as follows: a +b P +c P ; Fuel 1: P P P a +b P +c P F( P ) =... mn a j +bj P +cjp ; Fuel j: Pj P P mn mn ; Fuel : P P P (5) By addng the term related to valve-loadng effect, the above formulaton s turns to the followng form: Prohbted Operatng Zones (POZs) Each generator has ts generaton capacty lmtaton, whch cannot be exceeded. The prohbted operatng zones n the nput-output performance curve, due to steam valve operatng n shaft bearng, were consdered n ths paper n order to determne the optmum ED problem. In practce, when adjustng the generaton output of a unt, operaton n the prohbted zones must be avoded. Thus, the shape of the nput-output curve n the neghborhood of the prohbted zones s dffcult to be determned and the best economcal approach s acheved by avodng the operaton n these areas, as shown n Fg.. The POZ restrctons mght be represented for the th unt as follows: mn LB P P P,1 UB LB P, j 1 P P, j; j =,3,.., NP (9) UB P, j P P For = 1,,.., N GP N GP s the number of generators ncorporatng POZ and NP s the number of POZs of the th unt. Besdes, P,j LB and P,j UB refer to the lower and upper 49

4 Journal of Operaton and Automaton n Power Engneerng, Vol. 3, No. 1, Wnter & Sprng 015 boundares of the j th POZ of the th generator respectvely. Fg.. The prohobted operatng zones and generaton lmts for a generator Spnnng reserve Due to the ncluson of POZ n the formulaton of ED problem, spnnng reserve of the system should be wrtten n the followng form: mn{( P P), S } ( Ω Θ) S = 0 Θ (10) S S r r = n = 1 S S R (11) (1) n whch, S and S refer to the spnnng reserve of the th unt and ts mum value, respectvely. The set of operatng unts s denoted by Ω and the set of operatng unts wth POZ s represented by Θ. S R refers to the total spnnng reserve requred by the system. 3. OPTIMIZATION TECHNIQUE 3.1. Orgnal bat algorthm Bats locate the poston of food or prey by echolocaton. Echolocaton s the process of fgurng out the poston of objects from ther response to some sort of subsonc sgnals. Bats spreads a sgnal n the permeter and wats to receve the reverberated sgnal from food or prey. Ths dea has been smulated to form a meta-heurstc evolutonary optmzaton algorthm called the (BA) [7]. The rulng deas behnd the BA are lsted as bellow: 1) Reverberated sgnals from varous objects are dfferent and thus bats are capable of dstngushng between food and prey; ) The generated sgnal by a bat statoned n poston X or flyng at the velocty of V s specfed by f and A as sgnal frequency and ampltude, respectvely; 3) Ampltude of the sgnal denoted by A s reduced gradually; 4) Frequency of the sgnal denoted by f and ts rate denoted by r are changed automatcally. The algorthm s ntated by generatng a prelmnary populaton of bats whch are randomly nterspersed n the search space. The process of evoluton s followed by updatng the poston of bats based on two separate steps. The frst adjustment n the bat poston s accordng to: new old V = V + f( XG X ); = 1,, NBat new old new X = X + V ; = 1,..., NBat (13) mn mn f = f + ϕ ( f f ) ; = 1,..., N 1 Bat Where, X G ndcates the best global soluton. The upper and lower frequency lmts of the th bat are represented by f and f mn, respectvely. The populaton sze s equal to the total number of bats denoted by N Bat and φ 1 s a randomly generated number between 0 and 1. The second movement n the bat poston s smulated as follows: new old old X = X + ε A ; = 1,, N (14) mean Bat where, ε s a random number n the range of [-1,1] old and A mean s the mean value of ampltude of all the bats. Once the poston of bats s mproved by the above adjustments, a new random ndvdual X new s generated n case the rate of ts sgnal r s greater than a random value β. Ths new soluton wll be nserted to the populaton n case the followng constrant s respected: [ β < A]&[ f( X) < f( Gbest)] (15) As mentoned formerly, the value of sgnal ampltudes generated by bats has a gradual decrease formulated by: new old A = αa (16) Iter + 1 r = r 0 [1 exp( γ t)] 50

5 F. Namdar, R. Sedaghat: An Intellgent Approach Based on Meta-Heurstc Algorthm for Non-Convex Where t represents teraton number and α and γ are constant parameters. 3.. Modfed BA The orgnal BA suffers some drawbacks such as the possblty of gettng trapped n local optma and low rate of convergence to the optmal soluton. Two modfcatons are devsed and added to the algorthm n order to mprove ts convergence rate and dversty as follows: Modfcaton method 1 In the frst modfcaton step, t s attempted to dversfy the bat populaton usng Lévy flght, defned as a random walk wth regular and dspersed step lengths accordng to heavy-taled probablty dstrbuton [8]. The mathematcal representaton of the Lévy flght s formulated as: Le vy( ω) ~ τ = t ω ; (1< ω 3) (17) Ths dea s borrowed to generate a new ndvdual n each teraton as follows: new old X = X + ϕ1 Levy ( ω) (18) Ths new soluton mght replace the th bat n the populaton n case t excels the objectve functon. Modfcaton method The second modfcaton step s devsed to ntersperse randomly generate solutons n the populaton based on conventonal GA operators of crossover and mutaton. To do so, three bats X b1, X b and X b3 are chosen randomly such that b 1 b b 3 are related for th bat n the populaton and two test solutons are generated as follows: XTest,1 = Xb + ϕ 1 1 ( Xb X ) b3 (19) XTest, = ϕ XG + ϕ3 ( XG X ) (0) The above ndvduals are compared to the th bat and the one whch enhances the objectve functon replaces X. 4. SOLUTION PROCEDURE In order to apply MBA to solve the ED problem, the followng steps should be mplemented: Step 1: Defnng the nput data. Here, all data ncludng the network data, algorthm data (such as number of bats, ntal postons, constant coeffcents, etc.), objectve functon parameters, constrants parameters, and etc. are defned completely. Step : Formaton of the ftness functon. It s noted that the ftness functon ncludes the objectve functon and the penalty values related to the problem constrants. Step 3: Generaton of the ntal populaton based on the nformaton gven n the prevous secton. Step 4: Evaluaton of the objectve functons for each bat separately and fndng the best soluton. Step 5: Movement of the bat populaton to the new mproved postons. Step 6: Applcaton of the proposed modfcaton methods accordng to Eqs. (17)-(0). Step 7: Updatng the value of the best ndvdual. Step 8: Checkng the termnaton crteron. If the termnaton crteron s satsfed, then the algorthm s fnshed and the results are prnted; else, step 5 and the rest of the steps of repeated. 5. SIMULATION RESULTS Effectveness of the proposed approach n solvng the proposed realstc ED problem s llustrated by applyng the method to two test systems. It s worth notng that the adjustng parameters of MBA are all determned expermentally by several runnng of the algorthm. But, the sgnfcant pont s that the algorthm s not much dependent on the values of the settng parameters and the output results s robust. The frst example ncluded ten generatng unts consderng fuels valve loadng effects of multple fuels smultaneously [13]. The optmzaton problem s solved 100 tmes to generate the results accordng to Table 1. For comparson, the obtaned results from several recently publshed ED soluton methods (wth smlar tral runs) are also represented n ths Table. It should be notced that the results of the other reported methods are drectly quoted from ther respectve references. Accordng to the results, the best, average, and worst solutons of the proposed MBA are better than the best, average, and worst results of all other methods n the frst ED test systems (IEEE 10-unt system), respectvely. In addton, the optmal operatng ponts of the unts are gven n Table. 51

6 Journal of Operaton and Automaton n Power Engneerng, Vol. 3, No. 1, Wnter & Sprng 015 The mpact of the modfcatons on the convergence rate of the BA s llustrated n Fg. 3. Table 1. Cost functon optmzaton usng the proposed method on the IEEE 10-unt test system Method Best Average Worst CGA-MU [13] IGA-MU [13] DE [14] RGA [14] PSO [14] PSO-LRS [15] NPSO[15] NPSO-LRS [15] BA Proposed MBA Table. Optmal operatng pont for the generators on the IEEE 10-unt test system Unt Optmal Generaton It s obvous that the proposed MBA was effectvely successful n approachng the global optmal soluton n less than 100 teratons. Fg. 3. Convergence speed of the proposed MBA on the IEEE 10-unt test system For the sake of better demonstratng the robustness of the proposed methodology, the procedure s appled to the 40-unt IEEE power system and the second test case was fully ntroduced n [15]. Smlar to the frst case, 100 trals are done to the IEEE-40 unt system and the optmzaton results and the unt allotted outputs are llustrated n Table 3 and Table 4, respectvely. Investgaton of the results revealed the domnance of the proposed MBA n terms of fndng the optmal soluton compared to other methods. Table 3. Cost functon optmzaton usng the proposed method on the IEEE 40-unt test system Method Best Average Worst SPSO [16] 14, , NA PSO [17] 13, , NA CEP [17] 13, , ,90.89 HGAPSO [16] 1, , NA FEP [16] 1, , ,45.59 IFEP [16] 1, , , MPSO [18] 1,5.650 NA NA ESO [19] 1, , , PSO-LRS [18] 1, , , Improved GA [0] 11, , , HPSOWM[ 1] 11, ,844.4 NA IGAMU[ ] 11, NA NA NPSO [18] 11, , , HDE[3] 11, , NA NPSO-LRS [18] 11, , , KH [4] 11, , , BA 11, , , Proposed MBA 11, , , The mprovements were obvous n all average and the worst and best solutons. The convergence behavor of the proposed MBA s depcted n Fg. 4 and t can be seen that the modfcatons enhanced convergence rate of the algorthm. Table 4. Optmal operatng pont of the generators on the IEEE 40-unt test system Unt Optmal Generaton Unt Optmal Generaton

7 F. Namdar, R. Sedaghat: An Intellgent Approach Based on Meta-Heurstc Algorthm for Non-Convex Table 6. Optmal operatng pont of the generators on the IEEE 10-unt wth 4 perods and consderng power losses Unt Load P1(MW) P(MW) P3(MW) P4(MW) P5(MW) P6(MW) P7(MW)P8(MW)P9(MW) P10(MW) P loss (MW) Fnally, Table 6 shows the optmal generaton values of the power unts assocated wth the best result found by MBA. It s worth notng that these results belonged to the cost functon optmzaton and the emsson results were not shown to avod repetton. 6. CONCLUSION Fg. 4. Convergence speed of the proposed MBA on the IEEE 40-unt test system Table 5. Optmal operatng pont of the generators on the IEEE 10-unt test system optmzng both cost and emsson functons Unt Optmal Generaton Best Average Worst EP [7] NA EP-SQP [7] NA IPSO [8] NA AIS [9] Cost TVAC-IPSO Functon [30] EBSO [31] BA Proposed MBA Emsson BA Functon Proposed MBA Accordng to the followng results, the proposed MBA could acheve better results from both cost and emsson targets. Ths paper addressed the mpacts and constrans mposed to the ED formulaton of practcal power systems n order to optmally solve t. In a practcal system the exstence of lmtatons such as multple fuel opton, valve loadng effects, power generaton lmts, spnnng reserve, ramp rate lmts, and POZs made solvng the ED problem complcated. The system operator faced a complcated nonlnear nonconvex optmzaton problem mandatng the use of power full optmzaton technques. As a result, the MBA was proposed to add dversty to the algorthm and ensure fast convergence to the global optmal soluton. Contrbutons of the proposed methodology were corroborated through the numercal analyss of two varous test systems. In order to see the effect of power unts on the emsson functon, ths target was also consdered n the paper. Smulaton results 53

8 Journal of Operaton and Automaton n Power Engneerng, Vol. 3, No. 1, Wnter & Sprng 015 showed hgh capablty of the algorthm for solvng ths target as well. REFERENCES [1] K. Afshar and A. Shokr Gazafroud, Applcaton of stochastc programmng to determne operatng reserve wth consderng wnd and load uncertantes, Journal of Operaton and Automaton n Power Engneerng, vol. 1, no., pp , 013. [] M.A. Ortega-Vazquez and D.S. Krschen, Estmatng the spnnng reserve requrements n systems wth sgnfcant wnd power generaton penetraton, IEEE Transactons on Power Systems, vol. 4, no.1, pp , 009. [3] H. Khorramdel, B. Khorramdel, M. Tayeb Khorram and H. Rastegar, A mult-objectve economc load dspatch consderng accessblty of wnd power wth here-and-now approach, Journal of Operaton and Automaton n Power Engneerng, vol., no. 1, pp , 014. [4] A. Hatef and R. Kazemzadeh, Intellgent tuned harmony search for solvng economc dspatch problem wth valve-pont effects and prohbted operatng zones, Journal of Operaton and Automaton n Power Engneerng, vol. 1, no., pp: 84-95, 013. [5] J. B. Park, K.S. Lee, J.R. Shn and K.Y. Lee, A partcle swarm optmzaton for economc dspatch wth non-smooth cost functons, IEEE Transactons on Power Systems, vol. 0, no. 1, pp. 34-4, 005. [6] T. Nknam, R. Azzpanah-Abarghooee, R. Sedaghat and A. Kavous-Fard, An Enhanced hybrd partcle swarm optmzaton and smulated annealng for practcal economc dspatch, Energy Educaton Scence and Technology Part A: Energy Scence and Research, vol. 30, no. 1, pp , 01. [7] G.. Komarasamy, A. Wah, An Optmzed k-means clusterng technque usng bat algorthm, European Journal of Scentfc Research, vol. 84, no., pp , 01. [8] C. Brown, L.S. Lebovtch and R. Glendon, Lévy flghts n Dobe Juhoans foragng patterns, Human Ecology, vol. 35, no. 1, pp , 007. [9] A. Kavous-Fard, T. Nknam, H. Taherpoor, A. Abbas, Mult-objectve probablstc reconfgureton consderng uncertanty and mult-level load model, IET Scence, Measurment and Technology, vol. 10, no. 1, pp. 1-8, 014. [10] A. Kavous-Fard and T. Nknam, Optmal stochastc capactor placement problem from the relablty and cost vews usng frefly algorthm, IET Scence, Measurment and Technology, vol. 8, no. 5, pp , 014. [11] A. Kavous-Fard and H. Samet, Mult-objectve performance management of the capactor allocaton Problem n dstrbuted system based on modfed HBMO evolutonary algorthm, Electrc Power and Component systems, vol. 41, no. 13, pp , 013. [1] A. Kavous-Fard, T. Nknam, Optmal dstrbuton feeder reconfguraton for relablty mprovement consderng uncertanty, IEEE Transactons on Power Delvery, vol. 9, no. 3, pp , 014. [13] C. L. Chang, Improved genetc algorthm for power economc dspatch of unts wth valve-pont effects and multple fuels, IEEE Transactons on Power Systems, vol. 0, no. 4, pp , 005. [14] P. S. Manoharan, P.S. Kannan, S. Baskar, and M.W. Iruthayarajan, Penalty parameter-less constrant handlng scheme based evolutonary algorthm solutons to economc dspatch, IET Proceedngs on Generaton Transmsson and Dstrbuton, vol., no. 4, pp , 008. [15] A. I. Selvakumar and K. Thanushkod, A new partcle swarm optmzaton soluton to nonconvex economc dspatch problems, IEEE Transactons on Power Systems., vol., no. 1, pp. 4-51, 007. [16] N. Snha, R. Chakrabat and P.K. Chattopadhyay, Evolutonary programmng technques for economc load dspatch, IEEE Transactons on Evolutonary Computaton, vol. 7, no. 1, pp , 003. [17] T.A.A. Vctore, A.E. Jeyakumar, Hybrd PSO- SQP for economc dspatch wth valve-pont effect, Electrc Power Systems Research, vol. 71, no. 1, pp , 004. [18] J.-B. Park, K.-S. Lee, J.-R. Shn and K. Y. Lee, A partcle swarm optmzaton for economc dspatch wth non-smooth cost functons, IEEE Transactons on Power Systems, vol. 0, no. 1, pp. 34-4, 008. [19] A. Perera-Neto, C. Unshuay and O. R. Saavedra, Effcent evolutonary strategy optmzaton procedure to solve the non-convex economc dspatch problem wth generator constrants, IEE Proceedngs on Generaton, Transmsson and Dstrbuton, vol. 15, no. 5, pp , 005. [0] S.H. Lng and F.H.F. Leung, An mproved genetc algorthm wth average-bound crossover and wavelet mutaton operatons, Soft Computng, vol. 11, no. 1, pp. 7-31, 007. [1] S.H. Lng, H.H.C. Iu, K.Y. Chan, H.K. Lam, B.C. 54

9 F. Namdar, R. Sedaghat: An Intellgent Approach Based on Meta-Heurstc Algorthm for Non-Convex W. Yeung and F.H. Leung, Hybrd partcle swarm optmzaton wth wavelet mutaton and ts ndustral applcatons, IEEE Transactons on Systems, Man, and Cybernetcs, Part B: Cybernetcs, vol. 38, no. 3, pp , 008. [] C.-L. Chang, Genetc-based algorthm for power economc load dspatch, IET Proceedng on Generaton, Transmsson and Dstrbuton, vol. 1, no., pp , 007. [3] S.-K. Wang, J.-P. Chou, and C.-W. Lu, Nonsmooth/non-convex economc dspatch by a novel hybrd dfferental evoluton algorthm, IET Proceedngs on Generaton, Transmsson and Dstrbuton, vol. 1, no. 5, pp , 007. [4] A.H. Gandom and A. H. Alav, Krll herd: A new bo-nspred optmzaton algorthm, Communcatons n Nonlnear Scence and Numercal Smulaton, vol. 17, no. 1, pp , 01. [5] C.K. Pangrah, P.K. Chattopadhyay, R.N. Chakrabart, and M. Basu, Smulated annealng technque for dynamc economc dspatch, Electrc Power Components and Systems, vol. 34, no. 5, pp , 006. [6] T.A.A. Vctore and A.E. Jeyakumar, A modfed hybrd EP-SQP approach for dynamc dspatch wth valve-pont effect, Electrc Power & Energy Systems, vol. 7, no. 8, pp , 005. [7] P. Attavryanupap, H. Kta, E. Tanaka, and J. Hasegawa, A hybrd EP and SQP for dynamc economc dspatch wth nonsmooth fuel cost functon, IEEE Transactons on Power Systems, vol. 17, no., pp , 007. [8] X. Yuan, A. Su, Y. Yuan, H. Ne, and L. Wang, An mproved PSO for dynamc load dspatch of generators wth valve-pont effects, Energy, vol. 34, no. 1, pp , 009. [9] S. Hemamaln and S.P. Smon, Dynamc economc dspatch usng artfcal mmune system for unts wth valve-pont effect, Electrc Power Energy Systems, vol. 33, no. 4, pp , 011. [30] B. Mohammad-vatloo, A. Rabee and M. Ehsan, Tme-varyng acceleraton coeffcents IPSO for solvng dynamc economc dspatch wth nonsmooth cost functon, Energy Converson and Management, vol. 56, pp , 01. [31] T. Nknam and F. Golestaneh, Enhanced bee swarm optmzaton algorthm for dynamc economc dspatch, IEEE Systems Journal, vol. 3, pp. 19-8,