Internatonal Journal on Electrcal Engneerng and Informatcs Volume 3, Number 1, 011 Bee Hve Algorthm to Optmze Mult Constraned ecewse Non-Lnear Economc ower Dspatch roblem n Thermal Unts B. admanabhan 1, J. Jasper M. E., and Sva Kumar R. S. 3 1 M.E ower System Engneerng, Sardar Raja College of Engneerng Trunelvel Dst Assocate. rofessor, onjesly College of Engneerng Nagercol 3 M.E ower System Engneerng, Anna Unversty, Combatore 1 padmanabhan_balu@yahoo.co.n, maltojasper@gmal.com, 3 svars@gmal.com Abstract: Ths paper presents applcaton of a Bee Hve Algorthm to Economc Load Dspatch whch consders practcal constrants and non lnear characterstcs. The proposed ED formulaton ncludes ramp rate lmts, valve loadng effects, multple fuels, equalty and nequalty constrants, whch usually are found smultaneously n realstc power systems. Conventonal methods such as Lambda teraton and Base pont partcpaton are not able to obtan optmal soluton for unts havng dscontnuous fuel cost functons. Bee Hve Algorthm can overcome the dffcultes and provdes an almost global optmal soluton, snce they don t get stuc up at local optmum. Keywords: Economc load Dspatch, Bee Hve Algorthm, Valve-pont loadng, Ramp rate lmts, Multple Fuels 1. Introducton ECONOMIC Load Dspatch (ELD) sees the best generaton for the generatng plants to supply the requred demand plus transmsson losses wth the mnmum producton cost. Improvement n schedulng the unts output can lead to sgnfcant cost savngs. In tradtonal ELD problems, the cost functon of each generator s approxmately represented by a smple quadratc functon and s solved usng mathematcal programmng based on several optmzaton technques such as dynamc programmng, Lnear programmng, homogenous lnear programmng and quadratc programmng methods[],[3],[4]. However none of these methods may be able to provde an optmal soluton and they usually get stuc at a local optmum. Normally the nput-output characterstc of modern generatng unts are hghly nonlnear n nature due to valve-pont effect [1],[14], [15], [16], [18], [19] ramp-rate lmts, Fuel swtchng [11], [0] etc, havng multple local mnmum ponts n the cost functon. To overcome such dffcultes many heurstc search algorthms, such as Genetc algorthm [1], [5], Dfferental Evoluton [6], Tabu search [7], [19], etc., have been proposed to solve ELD problem. These technques can be used to search the global optmum wth any type of objectve functon and constrants []. In ths paper, two ED problem for 3 and 10 thermal unts wth a non smooth fuel cost functon [8] are employed to demonstrate the performance of the proposed method wth BHA and the results were compared wth GA. The rest of ths paper s organzed as follows: Secton II descrbes the formulaton of an ED problem; whle secton III explans the standards n BHA. Secton IV then detals the procedure of handlng the BHA. Secton V gves the flow chart. Secton VI gves the Data and gves the results of the optmzaton. Secton VII outlnes our concluson and future research.. roblem Descrpton The objectve of ED s to determne the generaton levels for all on-lne unts whch mnmze the total fuel cost, whle satsfyng a set of constrants. It can be formulated as follows: Receved: November nd, 010. Accepted: March 14 th, 011 109
B. admanabhan, et al. A. Economc Dspatch (Ed) roblem Formulaton The fuel cost functons of the generatng unts are usually descrbed by a quadratc functon of power output [13]. Thus the objectve functon s to mnmze: F ( ) = a + b + c (1) Where a, b,c - the fuel cost coeffcents of the th unt N- Number of generatng unts n the system - output generaton of th unt. 1. ower balance constrant: N = 1 = D + L () Where D Total power demand L Total networ losses. Capacty lmts constrants: mn max (3) Where mn mnmum generaton lmt max maxmum generaton lmt B. Valve ont Effect Large steam turbne generators wll have a number of steam admsson valves that are opened n sequence to obtan ever ncreasng output of the unt. As the unt loadng ncreases the nput to the unt ncreases and the ncremental heat rate decreases between the openng ponts for any two valves [9], however, when a valve s frst opened, the throttlng losses ncreases rapdly and the ncremental heat rate rses suddenly. Ths s valve pont effect whch leads to non-smooth, non-convex nput-output characterstcs [1], to be solved usng the heurstc technques[1]. The valve pont effect s ncorporated n ED problem by supermposng the sne component model on the quadratc cost curve whch s gven below, F *( ) ( ) sn( [ mn F + e f ]) = (4) Where F *( ) fuel cost f th unt wth valve pont effect e, f the fuel cost coeffcents of the th unt wth valve pont effect. 110
Bee Hve Algorthm to Optmze Mult Constraned ecewse Non-Lnear Fuel Rate n MBtu/hr B C A Generatons n MW Fgure.1. Valve pont curve C. Ramp Rate Lmts: The Ramp-Up and Ramp-Down rate lmts of th generator are gven by As generaton ncreases, 0 <= UR (5) As generaton decreases 0 <= DR (6) and max max( mn, 0 DR ) <= <= mn(, + UR ) 0 (7) Where s the current output power and 0 s the output power n the prevous nterval of the th generator unt. UR s the up-ramp rate lmt of the th generator and DR s the downramp rate lmt of the th generator. D. Multple Fuels: Some generatng unts are capable of operatng usng dfferent types of fuels. The use of multple fuel types may result n multple cost curves that are not necessarly parallel or contnuous. The lower regon of the resultng cost curve determnes whch fuel type s most economcal to bum. Fuel Fuel 3 Cost Fuel 1 Output (MW) Fgure.. Fuel Cost Functon Suppled wth Multple Fuel Types 111
B. admanabhan, et al. Ths cost functon can be represented by a pecewse curve (see Fg.), and the segments are defned by the range n whch each fuel s used. The ED problem wth pece wse quadratc cost curves s very dffcult to solve by standard technques. ecewse quadratc cost functons have as many segments as fuel types. F ( G, a a ) = a,1,, + b + b + b,1,, M + c + c + c,1,,,,, 1 3 < < < 1 (8) Where G and G are the lower and upper bound respectvely of the th and, b c are the fuel cost coeffcent of unt-. a,,.,, 11 th fuel of unt-i, 3. Optmzaton Usng Bee Hve Algorthm Bee Hve algorthm, proposed by Karaboga n 005 for real parameter optmzaton, s a recently ntroduced optmzaton algorthm and smulates the foragng behavor of bee colony [] for unconstraned optmzaton problems [1] [5]. For solvng constraned optmzaton problems, a constrant handlng method was ncorporated wth the algorthm [7]. The ABC algorthm s developed by nspectng the behavors of the real bees on fndng food source, whch s called the nectar, and sharng the nformaton of food sources to the bees n the nest. Bologcal Inspraton In a real bee colony, there are some tass performed by specalzed ndvduals. These specalzed bees try to maxmze the nectar amount stored n the hve by performng effcent dvson of labour and self-organzaton. The mnmal model of swarm-ntellgent forage selecton n a honey bee colony, that bee hve algorthm adopts, conssts of three nds of bees: employed bees, onlooer bees, and scout bees. Half of the colony comprses employed bees and the other half ncludes the onlooer bees. Employed bees are responsble from explotng the nectar sources explored before and gvng nformaton to the other watng bees (onlooer bees) n the hve about the qualty of the food source ste whch they are explotng. Onlooer bees wat n the hve and decde a food source to explot dependng on the nformaton shared by the employed bees. Scouts randomly search the envronment n order to fnd a new food source dependng on an nternal motvaton or possble external clues or randomly. The bee decdes for one of the possbltes usng the mechansm based on the characterstcs of the food source (qualty, quantty and dstance from the hve). The descrbed process contnues constantly, whle the bees from a hve collect nectar and nvestgate new areas wth possble food sources. 4. Implementaton Of Bee Hve Algorthm Man steps of the Bee Hve algorthm smulatng these behavors are gven below, here the food represents the economc generaton and the food source represents the lmts of each generator. 1. Intalze the food source postons.. Each employed bee produces a new food source n her food source ste and explots the better source. 3. Each onlooer bee selects a source dependng on the qualty of her soluton, produces a new food source n selected food source ste and explots the better source. 4. Determne the source to be abandoned and allocate ts employed bee as scout for searchng new food sources.
Bee Hve Algorthm to Optmze Mult Constraned ecewse Non-Lnear 5. Memorze the best food source found so far. 6. Repeat steps -5 untl the stoppng crteron s met. The process of the Bee hve algorthm s presented as follows: Step 1. Intalzaton: Spray percentage of the populatons nto the soluton space randomly, and then calculate ther ftness values, whch are called the nectar amounts, where represents the rato of employed bees to the total populaton. Once these populatons are postoned nto the soluton space, they are called the employed bees. Step. Move the Onlooers: Calculate the probablty of selectng a food source, select a food source to move to by roulette wheel selecton for every onlooer bees and then determne the nectar amounts of them. Step 3. Move the Scouts: If the ftness values of the employed bees do not be mproved by a contnuous predetermned number of teratons, whch s called, those food sources are abandoned, and these employed bees become the scouts. Step 4. Update the Best Food Source Found So Far: Memorze the best ftness value and the poston, whch are found by the bees. Step 5. Termnaton Checng: Chec f the amount of the teratons satsfes the termnaton condton. If the termnaton condton s satsfed, termnate the program and output the results; otherwse go bac to the Step. F ( θ = ) S F ( θ = 1 ) (9) where denotes the poston of the employed bee, represents the number of employed bees, and s the probablty of selectng the employed bee. xj ( t + 1) = θj + φ( θj () t θj ( t)) (10) where denotes the poston of the onlooer bee, denotes the teraton number, s the randomly chosen employed bee, represents the dmenson of the soluton and ( ) produces a seres of random varable n the range [ 1, 1]. θj = θj mn + r.( θj max θj mn ) (11) where s a random number and [0, 1]. 113
B. admanabhan, et al. 5. Flowchart The dagrammatc representaton of the Bee Hve algorthm s gven below: Intalze a populaton of n Scout Bees Evaluate the ftness of the populaton Select m stes for Neghborhood Search Determne the sze of Neghborhood (atch Sze) Select the fttest Bees from each patch Recrut bees for Selected Stes (More Bees for the Best e ste) Assgn the (n m) Remanng Bees to Random Search New opulaton of Scout Bees Fgure 5.1. Bee Hve Algorthm 6. Data And Results A. TEST CASE I 3-generator System: The unt characterstcs data are gven [1]. The load demand s 850MW. The constrants whch have been ncluded are Valve pont loadng, Ramp Rate Lmts, rohbted Operatng zones and transmsson losses. The output power for each generator among the three and Fuel cost for the same has been represented below and also the total power along wth the fuel cost has been calculated. Comparson has been made for the calculated result n Bee Hve Algorthm wth Genetc Algorthm. 114
Bee Hve Algorthm to Optmze Mult Constraned ecewse Non-Lnear Table 1. Convergence Results for 3 Generatng Unts Wth Valve ont Effect & Losses Quanttes Optmal Values BHA GA 1(MW) 145.56 146.56 (MW) 93.4 93.1 3(MW) 4.36 41.87 F1($/hr) 1365.4 1379.4 F($/hr) 87.5 584.3 F3($/hr) 4357.1 4377.7 loss(mw) 11.139 13.139 Total Gen(MW) 861.34 861.55 Total Fuel Cost ($/hr) 8010 8341.4 B. TEST CASE II 10-generator systems: The unt characterstcs data are gven [1]. The load demand s 000MW. System data of unts consderng b loss coeffcents are gven n [7]. The constrants whch have been ncluded n ths ten unt system was Valve pont loadng, Ramp-Rate lmts, rohbted Operatng Zones along wth Multple Fuel Swtchng. Table. Convergence Results for 10 Generatng Unts Wth Valve ont Effect Quanttes Optmal Values BHA GA 1(MW) 195.193 5.64 (MW) 86.109 33.786 3(MW) 340 330 4(MW) 300 300 5(MW) 43 4 6(MW) 160 160 7(MW) 130 130 8(MW) 10 118 9(MW) 80 80 10(MW) 3.36 45.9484 Total ower Output(MW) 057.53 056.875 loss(mw) 57.53 56.87 Total Generaton Cost($/h) 13916.573 15975.5063 7. Concluson In ths paper, a comprehensves ED model ncludng ramp rate lmts, valve loadng effects, Multple Fuels and transmsson losses together s presented. In ths method, the Bee Hve algorthm method s found best suted for the fuel cost functons of non-smooth, multple fuel curves when compared wth GA. The proposed BHA can provde a more dverse search of soluton space and so better optmum solutons wth low computaton burden can be found. The research wor s under way n order to ncorporate more securty ssues of power system n the ED model wth other constrants. 115
B. admanabhan, et al. 8. References [1] admanabhan, Sva Kumar R. S., J. Jasper : Optmzaton of pecewse non-lnear mult constraned Economc power dspatch problem usng an Improved Genetc Algorthm, JEE Trans. Indus elect pow syst., 010, 10, (3), pp. 106-11 [] Lee, F. N., and Brepohl, A. M.: Reserve constraned economc dspatch wth prohbted operatng zones, IEEE Trans. ower Syst., 1993, 8, (1), pp. 46 54 [3] Chen,. H., and Chang, H. C.: Large-scale economc dspatch by genetc algorthm, IEEE Trans. ower Syst., 1995, 10, (4), pp. 1919 196 [4] Km, J. O., Shna, D. J., ara, J. N., and Sngh, C.: Atavstc genetc algorthm for economc dspatch wth valve pont effect, Electr. ower Syst. Res., 00, 6, (3), pp. 01 07 [5] Walters, D. C., and Sheble, G. B.: Genetc algorthm soluton of economc dspatch wth valve pont loadng, IEEE Trans. ower Syst., 1993, 8, (3), pp. 135 133 [6] J. Wood and B. F. Wollenberg, ower generaton, operaton and control, New Yor: Wley, 1996. [7] IEEE Commttee Report, "resent ractces n the Economc Operaton of ower Systems," IEEE Transactons on ower Apparatus and Systems, Vol. AS-90, July/August 1971, pp. 1768-1775. [8] C. Walters, G. B. Sheble, "Genetc Algorthm Soluton Of Economc Dspatch Wth Valve ont Loadng," IEEE Trans. ower Systems, Vol. 8, No. 3, pp. 135-133, August 1993. [9] K. Wong, Y. Wong, "Genetc and genetc/smulated-annealng approaches to economc dspatch," IEE roceedngs Gener, Trans and Dstr, Vol. 141, No. 5, pp. 507-513, Sep 1994. [10] K. Wong, B. Lau, A. Fry, "Modellng Generator Input-Output Characterstcs wth Valve- ont Loadng Usng Neural Networs," IEE nd Internatonal Conference on Advances n ower System Control Operaton and Management, pp. 843-848, 7-10 Dec 1993. [11] H. Yang,. Yang, C Huang, "Evolutonary rogrammng Based Economc Dspatch for Unts wth Non-Smooth Fuel Cost Functons," IEEE Trans. ower Systems, Vol. 11, No. 1, pp. 11-118, February 1996. [1] J. ar; K. Lee; J. Shn; K. Lee, "A partcle swarm optmzaton for economc dspatch wth nonsmooth cost functons," IEEE Trans. On ower Systems, Vol. 0, No. 1, pp. 344, Feb. 005. [13] E. Ln, G. L. Vvan, "Herarchcal Economc Dspatch for ecewse Quadratc Cost Functons," IEEE Trans. ower Apparatus and Systems, Vol. AS-103, No. 6, pp. I 1 70-1175, June 1984. [14] El-Gallad, M. El-Hawary, A. Sallam, A. Kalas, "Swarm Intellgence for Hybrd Cost Dspatch roblem," Canadan Conf. on Electrcal and Computer Engneerng, Vol., pp. 753-757, 13-16 May 001. [15] W. Ln, F. Cheng, M. Tsay, "Nonconvex Economc Dspatch by Integrated Artfcal Intellgence," IEEE Trans. on ower Systems, Vol. 16, No., pp. 307-311, May 001. [16] J. ar, S. Yang, K. Mun, H. Lee, J. Jung, "An applcaton of evolutonary computatons to economc load dspatch wth pecewse quadratc cost functons," The 1998 IEEE Internatonal Conference on Evolutonary Computaton, Vol. 8, No. 3, pp. 89-94, 4-9 May 1998. [17] K. Y. Lee, A. Sode Yone, J. Ho ar, "Adaptve Hopfeld Neural Networs for Economc Load Dspatch," IEEE Trans. on ower Systems, Vol. 13, No., pp. 519-56, May 1998. [18] N. Lee, A. M. Brepohl, "Reserve Constraned Economc Dspatch Wth rohbted Operatng Zones," IEEE Trans. ower Systems, Vol. 8, No. 1, pp. 46-54, February 1993. [19] J. Y. Fan, J. D. McDonald, "A ractcal Approach to Real Tme Economc Dspatch Consderng Unt's rohbted Operatng Zones," IEEE Trans. ower Systems, Vol. 9, No. 4, pp. 1737-1743, November 1994. 116
Bee Hve Algorthm to Optmze Mult Constraned ecewse Non-Lnear [0] S. 0. Orero, M. R. Irvng, "Economc Dspatch Of Generators Wth rohbted Operatng Zones: A Genetc Algorthm Approach," IEE roceedngs - Gener. Transm. Dstrb., Vol. 143, No. 6, pp. 59-534, November 1996. [1]. Chen, H. Chang, "Large-Scale Economcc Dspatch by Genetc Algorthm," IEEEE Trans. on ower Systems, Vol. 10, No. 4, pp. 1919-196, November 1995. [] T. Jayabarath. G. Sadasvam, V. Ramachandran, "Evolutonary rogrammng Based Economc Dspatch of Generators wth rohbted Operatng Zones," Electrc ower Systems Research, Vol. 5, No. 3, pp.61-66, December 1999. [3] anchol R. K, Swarup K. S., artcle swarm optmzaton for securty constraned economc dspatch. In: resented at Internatonal Conference on Intellgen Sensng and Informaton rocessng. (IEEE Cat. No. 04EX783), Chenna, Inda; 004. [4] EL-Sharawy M, Neebur D., Artfcal neural networs wth applcaton to power systems, IEEE ower Eng Soc, A Tutoral Course 1996. [5] Yalcnoz T, Short M. J., Neural networs approach for solvng economc dspatch problem wth transmsson capacty constrants, IEEE Trans ower Syst 1998;13:307 13. [6] Youssef H. K., El-Naggar K. M., Genetc based algorthm for securty constraned power system economc dspatch, Elect ower Syst Res 000;53:47 51. [7] Chao-Lung Chang, Improved Genetc Algorthm for ower Economc Dspatch of Unts Wth Valve-ont Effects and Multple Fuels, IEEE Trans. ower Systems, Vol. 0, No. 4, November 005 B. admanabhan was born on June 6, 1989. He s dong hs M.E. (ower Systems Engneerng) degree n Sardar Raja College of Engneerng, Trunelvel Dst, TamlNadu, Inda. Hs area of nterest ncludes power system optmzaton, and Soft Computng Technques. J. Jasper was born on February 8, 1981. He s currently worng toward the h.d. degree wth the Faculty of Electrcal Engneerng, Anna Unversty of Technology, Combatore, Inda. He s presently an Assocate rofessor n the Departmentt of Electrcall Engneerng, onjesly College of Engneerng, and TamlNadu, Inda. Hs major research nterest ncludes power system operaton, Dstrbuted Generaton, ntellgent control and Electrcal Machnes. Sva Kumar R. S. was born on May 1, 1988. He s dong hs M.E. (ower Systems Engneerng) degree Combatore, TamlNadu, Inda. n Anna Unversty of Technology, Hs area of nterest ncludes power system optmzaton and Artfcal Intellgence. 117