Constrained Evolutionary Programming Approaches to Power System Economic Dispatch

Transcription

1 Proceedngs of the 6th WSEAS Int. Conf. on EVOLUTIONARY COMPUTING, Lsbon, Portugal, June 16-18, 2005 (pp ) Constraned Evolutonary Programmng Approaches to Power System Economc Dspatch K. Shant Swarup Abstract - Ths paper proposes a novel methodology of Constrant evolutonary programmng for solvng dynamc economc dspatch. Dynamc Economc Dspatch s one of the man functons of power generaton operaton and control. It determnes the optmal settngs of generator unts wth predcted load demand over a certan perod of tme. The objectve s to operate an electrc power system most economcally whle the system s operatng wthn ts securty lmts. Ten unts test system wth smooth and nonsmooth fuel cost functons are consdered to llustrate the sutablty and effectveness of the proposed method. Keywords: Constraned Optmzaton, Evolutonary Programmng, Stochastc Optmzaton technques, Power system operaton and control, Dynamc Economc Dspatch. I INTRODUCTION Dynamc Economc Dspatch s a method to schedule the onlne generator outputs wth the predcted load demands over a certan perod of tme so as to operate an electrc power system most economcally [1-4]. It s a dynamc optmzaton problem takng nto account the constrants mposed on system operaton by generatng rampng rate lmts. The dynamc economc dspatch s not only the most accurate formulaton of the economc dspatch problem but also the most dffcult to solve because of ts large dmensonalty. Normally t s solved by dvdng the entre dspatch perod nto a number of small tme ntervals, and then a statc economc dspatch has been employed to solve the problem n each nterval. However all of those methods may not be able to provde an optmal soluton and usually gettng stuck at a local optmal. Recently, stochastc optmzaton technques [5-15] such as Smulated Annealng (SA), Genetc Algorthm (GA) and Evolutonary Programmng (EP) have been gven much attenton by researchers due to ther ablty to seek for the near global optmal soluton. Approprate settng of the control parameters of the SA algorthm s a dffcult task and the speed of the algorthm s slow when appled to a real power system. Genetc Algorthm, nvented by Holland [15] n the early 1970s, s a stochastc global search method that mmcs the metaphor of natural bologcal evaluaton. The GA uses only the objectve functon nformaton, not dervatve or other auxlary knowledge. Therefore GA can deal wth the non-smooth, non-contnuous and non dfferentable functons, whch actually exst n a practcal optmzaton problem. Evolutonary Programmng (EP) ntroduced by Lawrence J. Fogel n the 1960s, on the other hand, s also a global search method startng from a populaton of canddate solutons, and fnds soluton n parallel usng evaluaton process. Both GA and EP can provde a near global soluton. However the encodng and decodng schemes essental n the GA approach are not needed n dynamc economc dspatch problem. Therefore EP s faster n speed than GA n ths case. Any evolutonary computaton technque appled to a partcular problem should address the ssue of handlng nfeasble ndvduals. The presence of constrants sgnfcantly affects the performance of optmzaton algorthm, ncludng evolutonary search methods. The general way of handlng constrants s by penalzng the nfeasble ponts. Constrant handlng n evolutonary computaton s more or less problem dependent [21-23]. In ths paper, the applcaton of constraned optmzaton usng Evolutonary programmng technque for solvng dynamc economc dspatch has been proposed. The proposed method starts wth only feasble canddate solutons generated by an teratve procedure of correctng volatons. In order to show the effectveness of the proposed method, 10 unts test system wth smooth and non-smooth fuel cost functons are consdered n ths paper. The results of the proposed method are compared wth those reported n lterature [18-20]. II PROBLEM FORMULATION The dynamc economc dspatch problem can be formulated as follows. The objectve functon: Where, T Mn F = ( ) N t= 1 = 1 F t P t (1)

2 Proceedngs of the 6th WSEAS Int. Conf. on EVOLUTIONARY COMPUTING, Lsbon, Portugal, June 16-18, 2005 (pp ) F s the total operatng cost over the whole dspatch perods, T s the number of hours n the tme horzon, N s the number of dspatchable unts, F P ) s the ndvdual generaton producton t ( t cost n terms of real power output P at any tme t. The fuel cost functon of each unt wth smooth cost functon can be expressed as 2 F ( P ) = a P + b P + c (2) The fuel cost functon of each unt wth valve pont loadng effects can be expressed as 2 F P a P + b P + c + e sn f P ( ) ( ( P )) = mn Where a, b, c, e and f are constants of fuel cost functon of the unt. Subject to: 1) Real power balance constrant N = 1 P j P Dj = 0 (3) Where PDj s the total assumed load demand durng j th nterval. 2) Real power operatng lmts P P P, = 1,..., N (4) mn max Where P mn and P max are the mnmum and the maxmum real power outputs of the th generator respectvely. 3) Generatng unt ramp rate lmts Pt P( t 1) UR P( t 1) Pt DR = 1,..., N t = 1,..., T (5) Where UR and DR are the ramp-up and rampdown lmts of the th generator respectvely. III METHODOLOGY Evolutonary Programmng s a near global stochastc optmzaton method whch places emphass on the behavoral lnkage between parents and ther offsprng, rather than seekng to emulate specfc genetc operators as observed n nature to fnd a soluton. In order to obtan hgh qualty soluton, constraned optmzaton procedure has been adopted n ths paper. Constraned Optmzaton usng Evolutonary Programmng method starts from multple feasble ponts. 1) Representaton For T ntervals n the generaton schedulng horzon, there are T dspatched for the N generators. An array of control varable vectors can be shown as: P11P12... P1 T P21P22... P2 T (6)... PN 1... PNT Where Pt s the real power output of generator at ntervalt. 2) Intalzaton In order to avod usng penaltes and unnecessarly searchng n nfeasble regon, the ntal populaton s flled wth only feasble canddate solutons generated by an teratve procedure of correctng constrant volatons. 3) Ftness Evaluaton Snce the constrants are always satsfed by every member at any pont of evoluton only the objectve functon s used to calculate the cost of each canddate soluton, whch s taken as ts ftness value. 4) Creaton of offsprng A new populaton of soluton (Offsprng) s produced from the exstng populaton by selectng two dfferent generators randomly and at a randomly selected tme and transferrng some of the power from one generator to the other n such a way that the constrants are not volated. The amount to be transferred s chosen by the followng procedure: a) Calculate how much power can be removed from frst randomly selected generator so that t wll not volate ts constrants (both real power operatng lmts and generatng unt ramp rate lmts) and take a random fracton of t. b) Calculate how much power can be added to the second randomly selected generator so that t wll not exceed ts constrants and take a random fracton of t. c) Select the smaller of the two fractons and transfer t from the frst to the second generator. Snce both are n the same tme nterval power balance constrant wll not be volated. 5) Selecton and Competton The selecton process used n ths paper s the stochastc tournament method. The 2p ndvduals compete wth each other for selecton. p potental canddates are selected for the next generaton. IV COMPUTATIONAL PROCEDURE:

3 Proceedngs of the 6th WSEAS Int. Conf. on EVOLUTIONARY COMPUTING, Lsbon, Portugal, June 16-18, 2005 (pp ) The pseudo-code for the proposed method can be descrbed as follows: Step 1: Read the system data from the nput fle. Step 2: Intalze the populaton Step 3: Check for feasblty and apply repar operator f populaton s not feasble. Step 3: Evaluate ftness functon. Step 4: Creaton of offsprng by mutaton. Step 5: Competton and selecton. Step 6: If termnaton not reached loop to Step 3. Step 7: Get the fnal result and qut the program. The above strateges are llustrated clearly n Fg 1. V SIMULATION RESULTS The 10 unts system wth smooth and non-smooth Fg 1. Flow Chart of Constraned Optmzaton. fuel cost functon s used n ths paper to demonstrate the performance of the proposed method. Unt data can be found n Appendx. The smulatons were carred out on a Pentum III 550MHz processor. Case 1: In ths case, 10 unts test system wth smooth fuel cost functon was used (Tables A.1 and A.2). The demand of the system was dvded nto 12 ntervals. In ths smulaton, populaton sze s 20 and the algorthm wll stop when generatons are reached. The generaton schedule (best soluton) for the 10 unts test system wth smooth fuel cost functon s gven n Table 1. The smulaton result of the proposed method compared to those obtaned from (Lnear Programmng) LP[19] and (Evolutonary Programmng) EP (Drect) [19] s gven n Table 2. Hgh qualty soluton can be obtaned at the expense of computng tme by ncreasng the populaton sze. The performance of the proposed method for 10-unts 12-hour demand system s gven n Fg 2. The contrbuton of each unt towards the load s gven n Fg 3. Case II: In ths case, 10 unts test system wth non-smooth fuel cost functon was used 9Tables A.3 and A.4). The demand of the system was dvded nto 24 ntervals. In ths smulaton, a populaton sze of 20 was chosen and the algorthm wll stop after teratons. The generaton schedule (best soluton) for the 10 unts test system wth non-smooth fuel cost functon s gven n Table 3. The smulaton result of the proposed method compared to those obtaned from (Sequental Quadratc Programmng) SQP and (Evolutonary Programmng) EP [20] s gven n Table 4. The performance of the proposed method for 10-unts 24-hour demand system s gven n Fg 4. The contrbuton of each unt towards the load s gven n Fg 5. VI DISCUSSION In the past, to solve the economc dspatch problem, most algorthms requre the ncremental cost curves to be of monotoncally smooth ncreasng nature and contnuous. The conventonal methods gnore or flatten out the portons of the ncremental cost curve that are monotoncally ncreasng or not contnuous. Approaches wthout restrcton on the shape of fuel cost functons are needed to gve accurate results. The constraned optmzaton usng Evolutonary Programmng approach has proved to be a very effcent method to solve the Dynamc Economc Dspatch problem wth smooth and non-smooth fuel cost functons. EP seeks a soluton wth exponental convergence rate,.e., the speed of convergence wll be very fast at the begnnng of the smulaton and wll slow down when the generaton ncreases. When coupled wth Constraned Optmzaton usng effcent constrant handlng technques, the algorthm gves an optmal soluton at a faster rate. The advantage of the proposed method s that, fuel cost functon wth any arbtrary shape can be dealt wth. Therefore, the proposed methodology can be appled to the system wth non-smooth, nondfferentable and non-contnuous functon, whch exsts n any real power system. VII CONCLUSION Ths paper presents a new methodology whch uses effcent constrant handlng technques wthout adoptng penalty methods for obtanng

4 Proceedngs of the 6th WSEAS Int. Conf. on EVOLUTIONARY COMPUTING, Lsbon, Portugal, June 16-18, 2005 (pp ) the global soluton of the Dynamc Economc Dspatch problem. Smulaton results demonstrate that the proposed method gve a cheaper total producton cost than those obtaned by other classcal technques and Evolutonary Paradgms. Ten unts system wth smooth and non-smooth fuel cost functons were used n ths paper. Increase n the complexty of the objectve functon won t affect the complexty of the proposed method. VIII REFERENCES [1] A.J. Wood, B. F. Wollenberg, Power Generaton Operaton and Control, 2 nd ed., New York: John Wley & Sons, Inc., [2] F. LI, R. Morgan and D. Wllams, Hybrd Genetc Approaches to rampng rate constraned Dynamc Economc Dspatch, Electrc Power Systems Research, vol. 43, no. 2, pp , Nov [3] Q. Xa, Y. H. Song, BM. Zhang, CQ. Kang and N. Xang, Dynamc queung approach to power system short term economc and securty dspatch, IEEE Trans. Power Systems, vol. 13, no. 2, pp , May [4] X. S. Han, H. B. Goo and D. S Krschen, Dynamc Economc Dspatch: Feasble and Optmal Solutons, IEEE Trans. Power Systems, vol. 16, no. 1, pp , Feb [5] D. B. Fogel, Evolutonary Computaton, Toward a New Phlosophy of Machne Intellgence, 2nd ed., New York: IEEE Press, [6] Z. Mchalewcz, Genetc Algorthm + Data Structures = Evoluton Programs, Sprnger-Verlag, Berln Hedelberg, [7] D.E. Goldberg, Genetc Algorthm n Search, Optmzaton and Machne Learnng, Addson Wesley, Readng, [8] T. Back, Evolutonary Algorthm n Theory and Practce, New York: Oxford Unv. Press, [9] V. Mranda, D. Srnvasan and L.M. Proenca, "Evolutonary computaton n power systems," Electrcal Power & Energy Systems, vol. 20, No. 2, pp , [10] L.J. Fogel, A.J. Owens and M.J. Walsh, Artfcal Intellgence through Smulated Evoluton, New York: John Wley & Sons, Inc., [11] K.P. Wong and C.C. Fung, "Smulated annealng based economc dspatch algorthm," IEE Proc. C, vol. 140, no. 6, pp , Nov [12] H.T. Yang, P.C. Yang and C.L. Huang, "Evolutonary Programmng based economc dspatch for unts wth non-smooth fuel cost functons," IEEE Trans. Power Systems, vol. 11, no. 1, pp , Feb [13] D. Walters, C. Sheble and B. Gerald, "Genetc algorthm soluton of economc dspatch wth valve pont loadng," IEEE Trans. Power Systems, vol. 8, no. 3, pp , Aug [14] K.P. Wong and J. Yuryevch, "Evolutonary- Programmng-Based Algorthm for Envronmentally-Constraned Economc Dspatch," IEEE Trans. Power Systems, vol. 13, no. 2, pp , May [15] J.H. Holland, Adaptaton n Natural and Artfcal Systems, The Unversty of Mchgan Press, Ann Arbor, USA, [16] H. Duo, H. Sasak, T. Nagata and H. Fujta, "A soluton for unt commtment usng Lagrangan relaxaton combned wth evolutonary programmng," Electrc Power Systems Research, vol. 51, pp , [17] F. L, R. Morgan and D. Wllams, "Hybrd genetc approaches to rampng rate constraned dynamc economc dspatch," Electrc Power Systems Research, vol. 43, pp , [18] P. Attavryanupap, H. Kta, E. Tanaka and J. Hasegawa, "A hybrd method of EP and SQP for Dynamc Economc Dspatch," n 2001 Natonal Conventon Records (6), IEE Japan, pp [19] P. Attavryanupap, H. Kta, E. Tanaka and J. Hasegawa, "Dynamc economc dspatch usng evolutonary programmng combne wth sequental quadratc programmng," n The Intellgent Systems Applcaton to Power Systems Conference 2001, pp [20] P. Attavryanupap, H. Kta, E. Tanaka and J. Hasegawa, A Hybrd EP and SQP for Dynamc Economc Dspatch wth Non- Smooth Fuel Cost Functon, IEEE Transactons on Power Systems, vol. 17, no. 2, pp , May [21] Zbgnew Mchalewcz. A Survey of Constrant Handlng Technques n Evolutonary Computaton Methods. In J. R. McDonnell, R. G. Reynolds and D. B. Fogel, edtors, Proceedngs of the 4th Annual Conference on Evolutonary Programmng, pages The MIT Press, Cambrdge, Massachusetts, [22] Zbgnew Mchalewcz and Cezary Z. Jankow. Handlng Constrants n Genetc Algorthms. In R. K. Belew and L. B. Booker, edtors, Proceedngs of the Fourth Internatonal Conference on Genetc Algorthms, pages , San Mateo, Calforna, Morgan Kaufmann Pub. [23] Zbgnew Mchalewcz and Marc Schoenauer. Evolutonary Algorthms for Constraned Parameter Optmzaton Problems. Evolutonary Computaton, 4(1):1-32, 1996.

5 Proceedngs of the 6th WSEAS Int. Conf. on EVOLUTIONARY COMPUTING, Lsbon, Portugal, June 16-18, 2005 (pp ) Anand N receved B.Tech. Degree n Electrcal Engneerng n 2003 from IIT Madras. Hs research nterests are Evolutonary Computaton, Optmzaton technques and ther applcaton to Power Systems,. K.S. Swarup s wth the Department of Electrcal Engneerng, Indan Insttute of Technology, Madras, Inda. Pror to jonng the department as vstng faculty, he held postons at the Mtsubsh Electrc Corporaton, Osaka, Japan, and Ktam Insttute of Technology, Hokkado, Japan, as vstng research scentst and vstng professor, respectvely durng 1992 to Hs areas of research are AI, knowledge-based systems, computatonal ntellgence, soft computng and Optmzaton Technques for electrc power systems x 106 Contrbuton of each unt towards total load Fg 2. Performance of CEP for 10-unts 12-hour demand test system Generaton Hours T/U P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 Fg 3. Load dstrbuton among unts for 12-hour demand 1.07 x Generaton Contrbuton of each unt towards total load Hours x 10 4 Unts/Hours P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 Fg 4. Performance of CEP for 10-unts 24-hours Fg 5. Load dstrbuton among unts for 24-hours TABLE 1. GENERATION SCHEDULE FOR 10-GENERATING UNIT 12-HOUR SYSTEM GENERATING UNITS Hours Load (MW)

6 Proceedngs of the 6th WSEAS Int. Conf. on EVOLUTIONARY COMPUTING, Lsbon, Portugal, June 16-18, 2005 (pp ) TABLE 2. COMPARISON OF TOTALPRODUCTION COST OBTAINED BY CONSTRAINED EVOLUTIONARY PROGRAM Methods Lnear Programmng [19] Evol. Prog (Drect) [19] Constraned EP Total Cost ($) TABLE 3. GENERATION SCHEDULE FOR 10-UNIT 24-HOUR SYSTEM GENERATING UNITS Tme LOAD (hrs) (MW) TABLE 4. COMPARISON OF TOTAL PRODUCTION COST AND SIMULATION TIME OBTAINED BY CONSTRAINED EP Methods Seq. Quad. Prog [20] Evol. Prog [20] Constraned EP Total Cost ($) CPU Tme

7 Proceedngs of the 6th WSEAS Int. Conf. on EVOLUTIONARY COMPUTING, Lsbon, Portugal, June 16-18, 2005 (pp ) VII APPENDIX TABLE A.1 LOAD PATTERN FOR 12-HOUR SYSTEM [19] Hours Load(MW) TABLE A.2 DATA FOR 10-UNIT SYSTEM (SMOOTH FUEL COST FUNCTION) [19] UNITS P mn P max a b a UR DR (MW) (MW) ($/MW 2 h) ($/MWh) ($/h) (MW/h) (MW/h) TABLE A.3 LOAD PATTERN FOR 24-HOUR SYSTEM [20] Hour Load(MW) Hour Load(MW) TABLE A.4 DATA FOR 10-UNIT SYSTEM (NON-SMOOTH FUEL COST FUNCTION) [20] UNITS P mn P max a b c e F UR DR (MW) (MW) ($/MW 2 h) ($/MWh) ($/h) ($/h) (rad/mw) (MW/h) (MW/h) Notatons Used n Tables A.2 and A.4: UR= Up Ramp Rate; DR= Down Ramp Rate.