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1 Appled Energy 86 (2009) Contents lsts avalable at ScenceDrect Appled Energy ournal homepage: Enhancement of combned heat and power economc dspatch usng self adaptve real-coded genetc algorthm P. Subbara a,, R. Rengara b, *, S. Salvahanan c,2 a Kalasalngam Unversty, Srvllputhur, Tamlnadu , Inda b Electrcal and Electroncs Engneerng, S.S.N. College of Engneerng, Old Mahabalpuram Road, Thrupporur (T.K), Kalavaam, Kancheepuram (Dst.) 603 0, Tamlnadu, Inda c S.S.N. College of Engneerng, Old Mahabalpuram Road, Thrupporur (T.K), Kalavaam, Kancheepuram (Dst.) 603 0, Tamlnadu, Inda artcle nfo abstract Artcle hstory: Receved 2 March 2008 Receved n revsed form 4 October 2008 Accepted 4 October 2008 Avalable onlne 22 November 2008 Keywords: Smulated bnary crossover Polynomal mutaton Co-generaton Combned heat and power economc dspatch In ths paper, a self adaptve real-coded genetc algorthm (SARGA) s mplemented to solve the combned heat and power economc dspatch (CHPED) problem. The self adaptaton s acheved by means of tournament selecton along wth smulated bnary crossover (SBX). The selecton process has a powerful exploraton capablty by creatng tournaments between two solutons. The better soluton s chosen and placed n the matng pool leadng to better convergence and reduced computatonal burden. The SARGA ntegrates penalty parameterless constrant handlng strategy and smultaneously handles equalty and nequalty constrants. The populaton dversty s ntroduced by mang use of dstrbuton ndex n SBX operator to create a better offsprng. Ths leads to a hgh dversty n populaton whch can ncrease the probablty towards the global optmum and prevent premature convergence. The SARGA s appled to solve CHPED problem wth bounded feasble operatng regon whch has large number of local mnma. The numercal results demonstrate that the proposed method can fnd a soluton towards the global optmum and compares favourably wth other recent methods n terms of soluton qualty, handlng constrants and computaton tme. Ó 2008 Elsever Ltd. All rghts reserved.. Introducton Increasng demand for power and heat resulted n the exstence of co-generaton unts whch smultaneously produce power and heat. The obectve of economc dspatch (ED) problem n a conventonal power plant s to fnd the optmal pont for the power producton such that the total demand matches the generaton wth mnmum fuel cost. However, the obectve of combned heat and power economc dspatch (CHPED) s to fnd the optmal pont of power and heat generaton wth mnmum fuel cost such that both heat and power demands are met whle the combned heat and power unts are operated n a bounded heat versus power plane. The mutual dependences of heat and power generaton ntroduce a complcaton n the ntegraton of co-generaton unts nto the power system economc dspatch. A technque was developed n [] to solve the CHPED problem usng separablty of the cost functon and constrants. Here two-level strategy s adopted; the lower * Correspondng author. Tel.: /45/46 (Moble: ); fax: /64/65. E-mal addresses: subbara_pott@yahoo.com (P. Subbara), rengara8@gmal.- com (R. Rengara), salvahanans@hotmal.com (S. Salvahanan). Present address: Then Kammavar sangam College of Technology, Koduvlarpatt, Then , Tamlnadu, Inda. Moble: Moble: , Tel./fax: /45/46, /64/ 65. level solves economc dspatch for the values of power and heat lambdas and the upper level updates the lambda s senstvty coeffcents. The procedure s repeated untl the heat and power demands are met. It was clamed n [2] that the Lagrangan Relaxaton method cannot deal wth dscontnuous and/or nonmonotonc nput/output model for generator fuel cost characterstcs. However, a two-layer Lagrangan relaxaton algorthm was developed and solved the CHPED problem [3], and a customzed branch-and-bound algorthm was also developed and solved the CHPED problem [4,5]. Alternatves to the tradtonal mathematcal approaches: evolutonary computaton technques such as genetc algorthm (GA) [6,7], evolutonary programmng (EP) [2], mult-obectve partcle swarm optmzaton (MPSO) [8], harmony search (HS) [9], fuzzy decson mang (FDM) [0] and mproved ant colony search algorthm (ACSA) [] have been successfully appled to CHPED problem. In [6], mproved genetc algorthm wth multpler updatng (IGAMU) approach s mplemented to solve the CHPED problem usng penalty based constrant handlng method. However, certan drawbacs regardng values of penalty parameters have been reported n [6,7]. The real-coded GAs are more sutable for large dmensonal search spaces than bnary-coded GAs snce they are more consstent, precse and lead to faster convergence [2 4]. In ths paper, ablty of real-coded GAs to mae par-wse comparson n tournament selecton are exploted to devse a penalty /$ - see front matter Ó 2008 Elsever Ltd. All rghts reserved. do:0.06/.apenergy

2 96 P. Subbara et al. / Appled Energy 86 (2009) functon approach that does not requre any penalty parameter and such an approach s mplemented to solve CHPED problem. The equalty and bounded heat versus power plane constrants are effectvely handled by penalty parameterless approach. The performance of the SARGA s compared wth those of Lagrangan Relaxaton method, branch-and-bound algorthms, ACSA, GA based penalty functon (GA_PF), PSO, EP, IGAMU and HS. The smulaton results show that SARGA ntegrated wth penalty parameterless approach performs better than these methods, n terms of soluton qualty, handlng constrants and computaton tme. Secton 2 descrbes the characterstcs of a co-generaton unt and formulaton of the CHPED problem, Secton 3 deals wth self adaptve real-coded genetc algorthm, Secton 4 descrbes mplementaton of SARGA to CHPED problem, and Secton 5 presents numercal results based on a smple co-generaton system. 2. Characterstcs and formulaton of the CHPED problem Combned heat and power generaton s an establshed and mature technology, whch has hgher energy effcency and less green house gas emsson as compared wth the other forms of energy supply. The basc dfference between conventonal condensng plant and combned heat and power unts s n the type of the power obtaned and the overall effcency of each plant. In conventonal condensng plants the energy from the fuel s used to produce electrcal power only, whle n combned heat and power (CHP) systems, the energy from the fuel s used to produce both electrcal and thermal power thus ncreasng ts effcency. The conventonal condensng plant delvers power at an effcency of 35 55%. Usng effcent flue gas condensaton, the total effcency of CHP unt s found to be n the range of 80 % (lower heatng value base) [5 7]. The heat producton depends on power generaton and vce versa. Ths ntroduces complexty due to the non-separable nature of electrcal power and heat n the CHP unts. The combned heat and power economc dspatch (CHPED) problem of a system s to determne the unt heat and power producton, so that the system producton cost s mnmzed, whle the heat and power demands and other constrants are met. Fg. shows the feasble bounded regon n the heat versus power plane of a combned cycle co-generaton unt. The feasble operatng regon s enclosed by the boundary curve MNOPQR. The upper and lower bounds of power and heat unts are restrcted by ther own generaton lmts. The prmary obectve of the CHPED s to M N O determne the most economc loadng ponts of the combned heat and power generaton unts such that both the heat and power demands can be met and operated wthn the bounded regon n the heat versus power plane. The obectve functon of the CHPED problem s gven by mn f cos t ¼ XNp ¼ C ðp Þþ XNc ¼ subected to the equalty constrants X Np ¼ X Nc ¼ P þ XNc ¼ O ¼ P d H þ XN h T ¼ H d ¼ and nequalty constrants P mn C ðo ; H Þþ XN h C ðt Þ P P max ; ¼ ;...; N p ; ð4þ O mn ðh ÞO O max ðh Þ; ¼ ;...; N c ð5þ H mn ðo ÞH H max ðo Þ; ¼ ;...; N c ð6þ T mn wth T T max ; ¼ ;...; N h ð7þ ¼ C ðp Þ¼a þ b P þ c P 2 C ðo ; H Þ¼a þ b O þ c O 2 þ d H þ e H 2 þ f O H C ðt Þ¼a þ b T þ c T 2 ðþ ð2þ ð3þ ð8þ ð9þ ð0þ where mn f cos t s the total mnmum fuel cost; C, C and C are the unt producton costs of the conventonal power, co-generaton and heat-alone unts, respectvely; a, b and c are fuel cost coeffcents of the th conventonal unt; a, b, c, d, e and f are fuel cost coeffcents of the th co-generaton unt; a, b and c are fuel cost coeffcents of the th heat-alone unt; P and O are power generatons of conventonal power and co-generaton unts; H and T are heat generaton of co-generaton and heat-alone unts; H d and P d are heat and power demands; N p, N c and N h denote the number of conventonal power, co-generaton and heat-alone unts, respectvely; P mn and P max are the mnmum and maxmum power generaton lmts of the conventonal unts; O mn and O max are the mnmum and maxmum power generaton lmts of the co-generaton unts; H mn and H max are the mnmum and maxmum heat generaton lmts of the co-generaton unts; T mn and T max are the mnmum and maxmum heat generaton lmts of the heat-alone unts. The mutual dependences of heat and power generatons from (5) and (6) ntroduce a complcaton n the ntegraton of co-generaton unts. Therefore, the optmzaton problem of the CHPED s non-lnear and hghly constraned n nature. Power (MW) R Q P Heat (MW th ) Fg.. Feasble operatng regon of a co-generaton unt. 3. Self adaptve real-coded genetc algorthm Self adaptaton s a phenomenon whch maes the genetc algorthms flexble and solves the CHPED problem wth feasble operatng regon. SARGA [8,9] nvolves two crtcal ssues: evolutonary search drecton and populaton dversty. As the evolutonary drecton s effectve n searchng, the strong evolutonary drecton can reduce the computatonal burden and ncrease the probablty of rapdly fndng an optmal soluton. Moreover, ncrease n populaton dversty creates the genotype of the offsprng that dffers more from the parents. Accordngly, a hghly dverse populaton can ncrease the probablty of explorng the global optmum and prevent the premature convergence to a local

3 P. Subbara et al. / Appled Energy 86 (2009) optmum. In the next secton, SARGA mplementaton to CHPED problem s presented. 3.. Generaton of ntal populaton In SARGA, each chromosome s encoded as a vector of floatngpont numbers, wth the same length as the vector of decson varables. The real-coded representaton of genetc algorthm s accurate and effcent because t s closest to the real desgn space. For convenence, (x,x 2,...,x,...,x N ) s represented as a vector of chromosome to the soluton of the CHPED problem. Intalzaton of M ndvdual populaton s generated usng x ¼ x l þ r ðx u x l Þ ðþ where x u and x l are the doman of x and r s a random number n the range of 0. Repeat () N tmes and produce the vector x,x 2,...,x,...,x N. Repeat the above procedure M tmes to create the M unformly dstrbuted ndvduals as ntal feasble solutons n the search space. The ftness of an ndvdual s a measure of how close the soluton s to the global optmum Tournament selecton In ths selecton, tournaments are played between two parents randomly chosen from ntal feasble soluton and better parent s selected and placed n the matng pool. Two other parents are pced up agan and another slot n the matng pool s flled wth better parents. In ths manner, each parent can be made to partcpate n exactly two tournaments. The best parents n a populaton wll wn both tmes, thereby mang two copes of them n the new populaton. Usng ths, worst parents wll lose n both tournaments and wll be elmnated from the populaton. In ths way, any parents n a populaton wll have zero, one or two copes n the new populaton. Ths leads to better convergence n terms of soluton qualty and computatonal tme Smulated bnary crossover Ths crossover operator wors wth two parent solutons and creates two offsprngs. SBX smulates the worng prncple of the sngle-pont crossover operator on bnary strngs. Durng ths operaton common nterval schemata between the parents are preserved n the offsprngs. The procedure of computng the offsprngs x ð;tþþ and x ð2;tþþ from the parents x ð;tþ and x ð2;tþ s descrbed as follows: a spread factor b s s obtaned as the rato of the absolute dfference n the offsprng values to that of the parents: b s ¼ xð2;tþþ x ð2;tþ x ð;tþþ x ð;tþ ð2þ Frst, a random number u between 0 and s created. Thereafter, from a specfed probablty dstrbuton functon, the ordnate b q s found so that the area under the probablty curve from 0 to b q s equal to the chosen random number u. The probablty dstrbuton used to create an offsprng s derved to have a smlar search power to that n a sngle-pont crossover n bnary-coded GAs and s gven as follows: ( 0:5ðg c þ Þb g c ; b s s Pðb s Þ¼ ð3þ 0:5ðg c þ Þ b g c þ2 s ; b s > In (3), a large value of the dstrbuton ndex g c gves a hgher probablty for creatng offsprngs closer to parents and a smaller value of g c creates offsprng dstant from the parents. Usng (3) b q s calculated by equatng the area under the probablty curve equal to u as follows: 8 < ð2u Þg c þ ; u 0:5 b q ¼ : g c þ ; u > 0:5 2ð u Þ ð4þ After obtanng the value b q from (4), the offsprngs are calculated as follows: h x ð;tþþ ¼ 0:5 ð þ b q Þx ð;tþ þð b q Þx ð2;tþ ð5þ h x ð2;tþþ ¼ 0:5 ð b q Þx ð;tþ þðþb q Þx ð2;tþ ð6þ 3.4. Polynomal mutaton The polynomal mutaton s smlar to the non-unform mutaton operator usng polynomal probablty dstrbuton nstead of a normal dstrbuton. Here, the probablty of creatng an offsprng closer to the parents s more than the probablty of creatng one away from t. As the generaton proceeds, ths probablty of creatng offsprng closer to the parents gets hgher and hgher, and the offsprngs created are gven as follows: y ð;tþþ ¼ x ð;tþþ þðx u x l Þ d ð7þ where the parameter d s calculated from the polynomal probablty dstrbuton Pðd Þ¼0:5ðg m þ Þð d Þ g m ( d ¼ ð2r g Þ m þ ; r < 0:5 ½2ð r ÞŠg m þ ; r 0:5 ð8þ ð9þ In (8) and (9), g m the mutaton constant s any non-negatve real number and r s a random number between 0 and ; g m produces a perturbaton of the order gm n the normalzed decson varable. For handlng bounded decson varables, the mutaton operator s modfed for two regons..e., ½x l ; x Š and ½x ; x u Š, whch s smlar to non-unform mutaton. The shape of the probablty dstrbuton s drectly controlled by an external parameter g m and the dstrbuton s not dynamcally changed wth generatons. Ths leads to slght perturbaton and prevents the ndvduals from premature convergence Stoppng rules The algorthm stops when maxmum number of generatons s reached or t termnates early dependng on the unsuccessful generaton of the algorthm. Two rules for termnatng the progress of the unsuccessful generatons of the algorthm are used: () the best soluton not changng for a prespecfed nterval of generatons () otherwse, the algorthm stops f the termnaton condton. f cos t; f cos t; = fcos t; 0:00 s satsfed ð20þ where f cos t, and f cos t, are feasble solutons at generaton and, respectvely Constrant handlng strategy In penalty parameter based method [20], an external penalty parameter whch penalzes nfeasble solutons s used. Based on the constrant volaton concernng g p (x) orz q (x) a bracet-operator penalty term s added to the obectve functon and a penalzed functon s formed: F cos t ðxþ ¼f cos t ðxþþ XP p¼ R p hg p ðxþ þ XQ q¼ r q Z q ðxþ ð2þ where R P and r q are penalty parameter; P s the total number of nequalty constrants and Q s the total number of equalty

4 98 P. Subbara et al. / Appled Energy 86 (2009) constrants. In (2), the bracet-operator hdenotes the absolute value of the operand, f the operand s negatve; f the operand s non-negatve, t returns value of zero. Snce dfferent constrants may tae dfferent orders of magntude, t s essental to normalze all constrants before usng the above equaton. A constrant g p ðxþ P b p can be normalzed by usng the followng equaton: g 0 p ðxþ g pðxþ b p P 0 ð22þ Equalty constrants Zq(x) can also be normalzed smlarly. Normalzng the constrants n ths manner, has the advantage: all normalzed constrant volatons tae more or less the same order of magntude and hence, they all can be smply added as the overall constrant volaton and thus only one penalty parameter R wll be needed to mae the overall constrant volaton of the same order as the obectve functon: " # F cos t ðxþ ¼f cos t ðxþþr XP hg p ðxþ þ XQ Z q ðxþ ð23þ p¼ The optmal soluton of F cos t (x) depends on penalty parameter R. Users usually have to try dfferent values of R to fnd whch value would steer the search towards the feasble regon. The most dffcult aspect of the penalty functon approach s to fnd approprate penalty parameters needed to gude the search towards the optmal soluton. Ths requres extensve expermentaton to fnd any reasonable soluton. In ths paper, GA s populaton based approach and ts ablty to mae par-wse comparson n tournament selecton are exploted to devse a penalty functon approach that does not requre any penalty parameter and such an approach s gven below: F cos t ðxþ ¼f cos t ðxþ; ¼ f cos t;max þ XP f x s feasble p¼ hg p ðxþ þ XQ q¼ q¼ Z q ðxþ; otherwse ð24þ In (24), f cos t,max s the obectve functon value of the worst feasble soluton n the populaton. The fundamental dfference between ths approach and that usng the penalty parameter s that the obectve functon value s not computed for any nfeasble soluton. Snce all feasble soluton has zero constrant volaton and all nfeasble solutons are evaluated accordng to ther constrant volaton only, both obectve functon value and constrant volaton are not combned n any soluton n the populaton. Thus, there s no need to have any penalty parameter for ths approach. The SARGA wth ths constrant handlng approach has been mplemented on CHPED problem. From the numercal results, the penalty parameterless approach has repeatedly found soluton closer to the global optmal soluton. 4. Implementaton of SARGA to CHPED problem The real-coded genetc algorthm combnes the SBX along wth the polynomal mutaton. The tournament selecton s nserted between ntalzaton of populaton and SBX crossover. Then, the systematc reasonng ablty s ncorporated n the crossover operatons to select the better genes for crossover, and consequently enhance the real-coded genetc algorthm. The steps of the SARGA approach are depcted n Fg. 2 and are descrbed as follows: Step Parameter settng. Input: populaton sze M, crossover rate p c, SBX crossover constant g c, mutaton rate p m, mutaton constant g m and number of generatons. START Intalzaton Tournament selecton Smulated bnary crossover operaton Polynomal mutaton operaton Offsprng populaton generaton Sortng of ftness values n ncreasng order Selecton of better chromosomes as parents for next generaton Stoppng crteron? Dsplayng the optmal chromosome END Yes Fg. 2. Flowchart for mplementaton of SARGA. Step 2 Intal populaton s generated. The functon values of the populaton are then calculated usng f cos t. Step 3 Tournament selecton operaton s performed. Step 4 Crossover operatons usng smulated bnary crossover. The probablty of crossover s determned by crossover rate p c. Step 5 Mutaton operatons usng polynomal mutaton. The probablty of mutaton s determned by mutaton rate p m. Step 6 Offsprng populaton s generated. Step 7 Sort the ftness values n ncreasng order among the generated populaton. Step 8 Select the better M, chromosomes as parents of the next generatons. Step 9 Chec for the stoppng crtera. Step 0 Dsplay the optmal chromosome and the optmal ftness value. No

5 P. Subbara et al. / Appled Energy 86 (2009) M N M N O Power (MW) Power (MW) 44 R Q P O 40 P Heat (MW th ) 04.8 Heat (MWth) 80 Fg. 4. Feasble operatng regons of co-generaton unt Numercal results based on a smple co-generaton system Ths secton consders a sngle area co-generaton system to llustrate the effectveness of the proposed SARGA n terms of qualty of soluton and computaton tme. The example [3] conssts of one conventonal power unt, two co-generaton unts and one heat-alone unt. The power generaton lmts of the power unt are 0 and 50 MW and heat generaton lmts of heat-alone unts are 0 and MW th. The feasble operatng regons of the two co-generaton unts are gven n Fg. 3 and 4. The system power demand P d and the heat demand H d are 200 MW and 5 MW th, respectvely. The fuel cost characterstcs of conventonal, co-generaton and heat-alone unts are gven n (26) (29). The obectve functon of the CHPED problem s mn f cos t ¼ C ðp Þþ X2 where Fg. 3. Feasble operatng regons of co-generaton unt. ¼ C ðo ; H ÞþC ðt Þ C ðp Þ¼50P C ðo ; H Þ¼2650 þ 4:5O þ 0:0345O 2 þ 4:2H ð25þ ð26þ subected to the equalty constrants: Z : P þ O þ O 2 ¼ P d Z 2 : H þ H 2 þ T ¼ H d and the nequalty constrants: g : : H O 05: g 2 : 0: H þ O 247:0 0 g 3 : 0: H O þ 98:8 0 g 4 : : H 2 O 2 46: g 5 : 0:56279H 2 þ O 2 30: g 6 : 0: H 2 O 2 þ 45: g 7 : 0:0 P 0 g 8 : P 50:0 0 g 9 : 0:0 T 0 and g 0 : T 2695:2 0 The system conssts of sx decson varables (P, O, H, O 2, H 2, T ) power balance constrant, heat balance constrant and ten nequalty constrants. 5.. Smulaton results and dscusson þ 0:03H 2 þ 0:03O H C 2 ðo 2 ; H 2 Þ¼250 þ 36O 2 þ 0:0435O 2 2 þ 0:6H 2 þ 0:027H 2 2 þ 0:0O 2H 2 C ðt Þ¼23:4T ð27þ ð28þ ð29þ Ths secton presents the smulaton results of the chosen CHPED problem wth focus on the comparson of SARGA wth Lagrangan Relaxaton [3], branch-and-bound algorthm [4,5], ACSA [], GA_PF [7], PSO [8], EP[2], IGAMU [6] and HS [9]. All these methods are coded n MATLAB and executed usng a Pentum Table Comparson of varous methods for CHPED problem. Power/heat Lagrange relaxaton technques Branch-and-bound algorthm ACSA GA_PF PSO EP IGAMU HS SARGA P (MW) O (MW) O 2 (MW) H (MW th ) H 2 (MW th ) T (MW th ) P d (MW) H d (MW th ) Total cost (US$) Executon tme n seconds

6 920 P. Subbara et al. / Appled Energy 86 (2009) Table 3 Influence of populaton sze n SARGA for CHPED problem. Polpulaton sze Best soluton Worst soluton Mean soluton Standard devaton No. of hts to global mnmum CPU tme Table 4 Performance of SARGA based on dfferent constrant handlng strategy. Fg. 5. Convergence characterstcs of SARGA n CHPED problem. IV based PC as the test platform. To verfy the performance of the SARGA, the program s run a hundred tmes on the example. The resultng fuel costs and average CPU tmes are used to compare the performance of the SARGA wth those of other methods. Durng ths evolutonary process, the followng parameter settng s used n SARGA: populaton sze M = 00 crossover rate p c = 0.9 mutaton rate p m = 0.0 crossover constant g c = 5 and mutaton g m =. The parameter settngs for ACSA, GA_PF, PSO, EP, IGAMU and HS are taen from [,7,8,2,6,9], respectvely. The best optmal soluton obtaned for ths example by [3,2] s $ The results obtaned for ths example usng SARGA method are gven n Table, and the results are compared wth those of Lagrangan relaxaton technque, branch-and-bound algorthm, ACSA, GA_PF, PSO, EP, IGAMU and HS. Among these algorthms, PSO, GA_PF and ACSA converge to a much hgher cost due to ther premature convergence. Even though PSO requres less computatonal tme, t fals to approach near global optmal soluton. Lagrangan relaxaton technque, branch-and-bound algorthms, IGAMU and HS algorthms obtan better soluton at the expense of computatonal tme. From Table, t can be seen that the proposed method, SARGA obtans mnmum cost wth less computatonal tme. The convergence nature of the dfferent algorthms s shown n Fg. 5 from whch t s evdent that SARGA and PSO have the fastest convergence. However, PSO fals to approach towards global optmal soluton whch has been shown also n Table. In order to get more nsghts nto the worng of SARGA, the nfluence of the evolutonary parameters on the performance of SARGA s brought out as shown n Tables 2 and 3. The parameters: crossover constant g c and mutaton constant g m n SARGA affect the soluton qualty and convergence. Influence of these parameters n CHPED problem s shown n Table 2. For g c = 5 and g m = the soluton exhbts better results n terms of standard devaton, mean soluton and worst soluton. Moreover, out of 00 trals the soluton hts towards global optmal soluton 00 tmes. Populaton sze 00, g c =5, g m = Wth penalty parameter Wthout penalty parameter Penalty R Best soluton Infeasble Worst soluton Infeasble Mean soluton Infeasble Standard devaton -NA No. of hts to global mnmum CPU tme The best solutons obtaned by SARGA are summarzed n Table 3 for dfferent szes of populaton. Durng ths performance analyss, p c = 0.9, g c =5,p m = 0.0, g m = are fxed. When populaton sze s small, the algorthm does not guarantee a 00% ht towards global optmal soluton. Moreover, the standard devaton, worst and mean solutons obtaned for smaller populaton sze, exhbt larger devaton as shown n Table 3. For populaton sze below 50 the algorthm may not approach towards global optmal soluton. Durng ths smulaton process, SARGA s mplemented and tested for constrants handlng strategy. The constrant handlng abltes are evaluated next and the results are presented n Table 4. Wth R = 0 t s not able to fnd a sngle feasble soluton n 00 trals. Ths happens because smaller values of R do not force the feasblty of the soluton. Wth large penalty parameters, the pressure for the soluton to become feasble s more and t hts 97 tmes towards global optmal soluton resultng n smaller value of standard devaton. In penalty parameterless approach, there s no need to have any penalty parameter: the obectve functon value s not computed for any nfeasble soluton. All feasble solutons and all nfeasble solutons are handled accordng to (24). SARGA searches from a populaton of ponts and hence, dscovers the nearest global pont and drectly use the ftness functon nformaton n the search procedure. The search s based on the stochastc operatons. In SBX crossover, the two offsprngs are symmetrc about the parent solutons thereby avodng a bas towards any partcular parent soluton n a sngle crossover operaton. Bnary GA requres large populaton sze for larger strngs. The large populaton sze ncreases the computatonal complexty. The Table 2 Influence of g m and g c n SARGA for CHPED problem. Populaton sze 00 p c = 0.9, p m = 0.0, g c =5 p c = 0.9, p m = 0.0, g m = g m =5 g m =0 g m =5 g m =20 g c =20 g c =5 g c =0 g c =5 Best soluton Worst soluton Mean soluton Standard devaton No. of hts to global mnmum CPU tme

7 P. Subbara et al. / Appled Energy 86 (2009) real-coded GA uses real parameters drectly wthout any strng codng thereby reducng the computatonal complexty. 6. Concluson Ths paper presents an approach for solvng the combned heat and power economc dspatch problem (CHPED) consderng the feasble operatng regon. SARGA has effectvely provded the global optmal soluton satsfyng both equalty and nequalty constrants. For the chosen example, SARGA has superorty to the other algorthms vz. ACSA, GA_PF, PSO, EP, IGAMU and HS n terms of soluton accuracy, handlng constrants and computaton tme. Moreover, the results of SARGA method are very close to those of the conventonal numercal methods. Combned wth ther relatvely low computatonal requrements as well as ther sutablty for parallel mplementaton, the algorthm provdes global optmal soluton to a real world CHPED problem. Hence, SARGA has the merts of global exploraton, fast convergence, robustness and statstcal soundness. References [] Rooers FJ, Van Amerongen RAM. Statc economc dspatch for co-generaton systems. IEEE Trans Power Syst 994;9(3): [2] Wong KP, Alge C. Evolutonary programmng approach for combned heat and power dspatch. Electr Power Syst Res 2002;6: [3] Guo T, Henwood MI, Van Ooen M. An algorthm for combned heat and power economc dspatch. IEEE Trans Power Syst 996;(4): [4] Maonen S, Lahdelma R. Non-convex power plant modelng n energy optmzaton. Eur J Oper Res 2006;7(3):3 26. [5] Rong A, Lahdelma R. An effcent envelope-based branch-and-bound algorthm for non-convex combned heat and power producton plannng. Eur J Oper Res 2007;83():42 3. [6] Su CT, Chang CL. An ncorporated algorthm for combned heat and power economc dspatch. Electr Power Syst Res 2004;69(2 3): [7] Song YH, Xuan QY. Combned heat and power economc dspatch usng genetc algorthm based penalty functon method. Electr Power Comp Syst 998;26(4): [8] Wang LF, Sngh C. Stochastc combned heat and power dspatch based on mult-obectve partcle swarm optmzaton. Int J Electr Power Energy Syst 2008;30: [9] Vaseb A, Fesanghary M, Bathaee SMT. Combned heat and power economc dspatch by harmony search algorthm. Int J Electr Power Energy Syst 2007;29:73 9. [0] Chang CS, Fu W. Stochastc multobectve generaton dspatch of combned heat and power systems. IEE Proc Gener Transm Dstrb 998;45(5): [] Song YH, Chou CS, Stonham TJ. Combned heat and power dspatch by mproved ant colony search algorthm. Electr Power Syst Res 999;52:5 2. [2] Basar S, Subbara P, Chdambaram P. Applcaton of genetc algorthms to unt commtment problem. J Inst Eng (Inda) 200;8:95 9. [3] Basar S, Subbara P, Rao MVC, Tamlselv S. Genetc algorthms soluton to generator mantenance schedulng wth modfed genetc operators. IEE Proc Gener Transm Dstrb 2003;50(): [4] Basar S, Subbara P, Rao MVC. Performance of real coded genetc algorthms wth new genetc operators. J Inst Eng (Inda) 2004;85:24 8. [5] Wahlund B, Yan J, Westermar M. Comparatve assessment of bofuel based combned heat and power generaton plants n Sweden. In: Frst world conference on bomass for energy and ndustry, Sevlla, Span; p [6] Wahlund B, Yan J, Westermar M. A total energy system of fuel upgradng by dryng bomass feedstoc for co-generaton: a case study of Selleftea boenergy combne. Bomass Boenergy 2002;23:27 8. [7] Wahlund B, Yan J, Westermar M. Increasng bomass utlzaton n energy systems: a comparatve study of CO 2 reducton and cost for dfferent boenergy processng optons. Bomass Boenergy 2004;26: [8] Deb K, Goyal M. A combned genetc adaptve search (GeneAS) for engneerng desgn. Comput Sc Infor 996;26(4): [9] Deb K, Beyer HG. Self-adaptaton n real parameter genetc algorthms wth smulated bnary crossover. Proc Int Conf Genet Evol Comput 999:72 9. [20] Deb K. An effcent constrant handlng method for genetc algorthms. Comput Meth Appl Mech Eng 2000;86:3 38. [2] Nagendra Rao PS. Combned heat and power economc dspatch: a drect soluton. Electr Power Comp Syst 2006;34(9):