Multi-Objective Optimization using Evolutionary Computation Techniques

Transcription

1 Internatonal Journal of Computer Applcatons ) Volume 7 No.11, August 11 MultObectve Optmzaton usng Evolutonary Computaton Technques Rambabu CH Electrcal Department, Sr Vasav Engg College, Tadepallgudem, W..DT, A.P., INDIA Dr.Y.P.Obulesh Electrcal Department, L.B.R.College of Engneerng, Mylavaram, Krshna Dst, A.P, INDIA Dr.CH.Sababu Electrcal Department, JNTUK College of Engg., Kaknada, E..Dt, A.P.,INDIA ABSTRACT In ths paper an EP and PSO based optmzaton algorthms have been proposed for solvng optmal power flow problems wth multple obectve functons. These algorthms take nto consderaton all the equalty and nequalty constrants. The mprovement n system performance s based on reducton n cost of power generaton and actve power loss. The proposed algorthms have been compared wth the other methods reported n the lterature. Smulaton studes have been carred out for the optmal solutons of the IEEE 14bus and IEEE 3bus systems. Keywords EP, PSO, Actve Power Loss 1. INTRODUCTION The man obectve of electrc power utltes s to provde hgh qualty relable supply to the consumers at the lowest possble cost whle operatng to meet the lmts and constrant mposed on the generatng unts. Ths formulates the wellknown Economc Load Dspatch ELD) problem for fndng the optmal combnaton of the output power of all onlne generatng unts that mnmzes the total fuel cost, whle satsfyng all constrants [1]. The Optmal Power Flow ) s an mportant crteron n today s power system operaton and control due to scarcty of energy resources, ncreasng power generaton cost and ever growng demand for electrc energy. As the sze of the power system ncreases, load may be varyng. The generators should share the total demand plus losses among themselves. The sharng should be based on the fuel cost of the total generaton wth respect to some securty constrants. enerally, most of the approaches apply senstvty analyss and gradentbased optmzaton algorthms by lnearzng the obectve functon and system constrants around an operatng pont. Unfortunately, the problems of are hghly nonlnear and a mult model optmzaton problems,.e. there exst more than one local optmum. Therefore, conventonal optmzaton methods that make use of dervatves and gradents are, n general, not able to locate or dentfy the global optmum [7]. ELD s solved tradtonally usng mathematcal programmng based on optmzaton technques such as lambda teraton, gradent method and so on. Economc load dspatch wth pecewse lnear cost functons s a hghly heurstc, approxmate and extremely fast form of economc dspatch. Complex constraned ELD s addressed by ntellgent methods. Among these methods, some of them are genetc algorthm A) and, evolutonary programmng EP), dynamc programmng DP), tabu search, hybrd EP, neural network NN), adaptve Hopfeld neural network AHNN), partcle swarm optmzaton PSO) etc. For calculaton smplcty, exstng methods use second order fuel cost functons whch nvolve approxmaton and constrants are handled separately, although sometmes valvepont effects are consdered [13][15]. Intellgent methods are teratve technques that can search not only local optmal solutons but also a global optmal soluton dependng on problem doman and executon tme lmt. They are generalpurpose searchng technques based on prncples nspred from the genetc and evoluton mechansms observed n natural systems and populatons of lvng bengs. These methods have the advantage of searchng the soluton space more thoroughly. The man dffculty s ther senstvty to the choce of parameters. Among ntellgent methods, PSO s smple and promsng. It requres less computaton tme and memory. It has also standard values for ts parameters. In ths, the Partcle Swarm Optmzaton PSO) s proposed as a methodology for economc load dspatch [].. BY EVOLUTIONARY COMPUTATION TECHNIQUES.1 Evolutonary Programmng EP) Evolutonary Programmng EP) s an optmzaton technque based on the natural generaton. It nvolves random number generaton at the ntalzaton process. The generated random numbers represent the parameters responsble for the optmzaton of the ftness value. In addton, EP also nvolves statstcs, ftness calculaton, mutaton and the new generaton wll be bred by mode of selecton. EP s a global optmzaton technque that starts wth the populaton of randomly generated canddate soluton and evolves a better soluton over a number of generatons or teratons. It s more sutable to effectvely handle noncontnuous and nondfferentable functon. The man stage of ths technque ncludes ntalzaton, mutaton, competton and selecton [13]. EP Algorthm. An Intal populaton of N p parent vectors s consdered as the tral soluton. From these parents off sprngs are created by mutaton, hence N p off sprngs are obtaned. By combnng the parents and off sprngs, N p solutons are obtaned v. Through competton and selecton, frst N p optmal solutons are selected 19

2 Internatonal Journal of Computer Applcatons ) Volume 7 No.11, August 11 v. The selected solutons are consdered as parents for the next teraton v. After the requred number of teratons, the best optmal soluton s obtaned.. Partcle Swarm Optmzaton PSO shares many smlartes wth evolutonary computaton technques such as enetc Algorthms A). The system s ntalzed wth a populaton of random solutons and searches for optma by updatng generatons. However, unlke A, PSO has no evoluton operators such as crossover and mutaton [3]. Velocty of each agent can be modfed by the followng equaton: k 1 k k k v wv c rand pbest s ) c rand * gbest s ) 1 1 *.1) w w w wmn ) / ter ))* ter.) The current poston searchng pont n the soluton space) can be modfed by the followng equaton k 1 k k 1 s s v.3) PSO Algorthm Step 1: eneraton of ntal condton of each agent. Intal searchng ponts s ) and veloctes v ) of each agent are usually generated randomly wthn the allowable range. The current searchng pont s set to pbest for each agent. The best evaluated value of pbest s set to gbest, and the agent number wth the best value s stored. Step : Evaluaton of searchng pont of each agent. The obectve functon value s calculated for each agent. If the value s better than the current pbest of the agent, the pbest value s replaced by the current value. If the best value of pbest s better than the current gbest, gbest s replaced by the best value and the agent number wth the best value s stored. Step 3: Modfcaton of each searchng pont. The current searchng pont of each agent s changed usng eqns..1),.), and.3). Step 4: Checkng the ext condton. The current teraton number reaches the predetermned mum teraton number, then exts. Otherwse, the process proceeds to step. 3. MATHEMATICAL FORMULATION OF PROBLEM The problem s to optmze the steady state performan a power system n terms of an obectve functon whle satsfyng several equalty and nequalty constrants. Mathematcally, the problem can be formulated as gven [4][5] Mn F x, u) 3.1) Subect to g x, u) 3.) h x, u) 3.3) Where x s a vector of dependent varables consstng of slack bus power P, load bus voltages 1 V, generator reactve power L outputs Q, and the transmsson lne loadngs S, Hence, x can l be expressed as gven T x P, V... V, Q... Q, S... S ] 3.4) [ 1 L1 LNL 1 N l lnl Where NL,N and nl are number of load buses, number of generators and number of transmsson lne respectvely. u s the vector of ndependent varables consstng of generator voltages V, generator real power outputs P except at the slack bus P, transformer tap settngs T, and shunt VAR 1 compensatons Q. Hence u can be expressed as gven C T u V... V, P... P, T... T, Q... Q ] 3.5) [ 1 N N 1 NT C1 CNC Where NT and NC are the number of the regulatng transformers and shunt compensators, respectvely. F s the obectve functon to be mnmzed. g s the equalty constrants that represents typcal load flow equatons and h s the system operatng constrants Obectves The obectves consdered for mnmzaton are as follows. Obectve Functon 1: Fuel cost of generatng unts f 1 ) Obectve Functon : Actve power loss f ) Obectve Functon 3: Weghted mult obectve functon f ) 3 where f 1 = mn N ap b P C ) 3.6) 1 NL f = mn V V V V cos )) ) 3.7) 1 f 3 w1 * f1 w * f 3.8) Here w 1 w 1 Constrants The problem has two categores of constrants: Equalty Constrants: These are the sets of nonlnear power flow equatons that govern the power system,.e. n P P V V Y cos ) 3.9) D 1 n Q Q V V Y sn ) 3.1) D 1 where P and Q are the real and reactve power outputs nected at bus respectvely, the load demand at the same bus s represented by P and D matrx are represented by Q D, and elements of the bus admttance Y and. Inequalty Constrants: These are the set of constrants that represent the system operatonal and securty lmts lke the bounds on the followng: 1) enerators real and reactve power outputs mn P P P, 1,, N 3.11) mn Q Q Q, 1,, N 3.1) ) Voltage magntudes at each bus n the network mn V V V, 1,, NL 3.13) 3) transformer tap settngs mn T T T, 1,, NT 3.14) 4) Reactve power nectons due to capactor banks mn QC QC QC, 1,, CS 3.15) 5) Transmsson lnes loadng S S, 1,, nl 3.16) 6) Voltage stablty ndex

3 Obectve functon1 Obectve functon Obectve functon Obectve functon Obectve functon3 Cost) Obectve functon3 Cost) Obectve functon3 Loss) Obectve functon3 Loss) Internatonal Journal of Computer Applcatons ) Volume 7 No.11, August 11 L L, 1,, NL 3.17) Handlng of Constrants: There are dfferent ways to handle constrants n evolutonary computaton optmzaton algorthms. In ths thess, the constrants are ncorporated nto ftness functon by means of penalty functon method, whch s a penalty factor multpled wth the square of the volated value of varable s added to the obectve functon and any nfeasble soluton obtaned s reected. To handle the nequalty constrants of state varables ncludng load bus voltage magntudes and output varables wth real power generaton output at slack bus, reactve power generaton output, and lne loadng, the extended obectve functon can be defned as: N N NL NL 3.18) OF F P ) K h P ) K h Q ) K h V ) K h S ) 1 where K, K, p q v p 1 q 1 v 1 s 1 K, K are penalty constants for the real power s generaton at slack bus, the reactve power generaton of all generator buses or PV buses and slack bus, the voltage magntude of all load buses or PQ buses, and lne or transformer loadng, respectvely. h ), h Q ), h V ), h S ) are the P 1 penalty functon of the real power generaton at slack bus, the reactve power generaton of all PV buses and slack bus, the voltage magntudes of all PQ buses, and lne or transformer loadng, respectvely. NL s the number of PQ buses. The penalty functon can be defned as: h x) x x, f x x ) x, f x xmn mn x) = =, f x mn x x 3.19) where hx) s the penalty functon of varable x, x and mn x are the upper lmt and lower lmt of varable x, respectvely. In ths secton descrbe the dataset and how t s used to detect ntrusons. I frst examne what type of data was present n the dataset, what ntruson types were represented and what features were extracted. 4. COMPUTATIONAL PROCEDURE Step 1: Input the system data for load flow analyss Step : Run the power flow Step 3: At the generaton en =; set the smulaton parameters of EP/PSO parameters and randomly ntalze k ndvduals wthn respectve lmts and save them n the archve. Step 4: For each ndvdual n the archve, run power flow to determne load bus voltages, angles, load bus voltage stablty ndces, generator reactve power outputs and calculate lne power flows. Step 5: Evaluate the penalty functons Step 6: Evaluate the obectve functon values and the correspondng ftness values for each ndvdual. Step 7: Fnd the generaton local best xlocal and global best xglobal and store them. Step 8: Increase the generaton counter en = en+1. Step 9: Apply the EP/PSO operators to generate new k ndvduals Step 1: For each new ndvdual n the archve, run power flow to determne load bus voltages, angles, load bus voltage stablty ndces, generator reactve power outputs and calculate lne power flows. Step 11: Evaluate the penalty functons Step 1: Evaluate the obectve functon values and the correspondng ftness values for each new ndvdual. Step 13: Apply the selecton operator of EP/PSO and update the ndvduals. Step 14: Update the generaton local best xlocal and global best xglobal and store them. Step 15: If one of stoppng crteron have not been met, repeat steps 415. Else go to step 16 Step 16: Prnt the results There are two stoppng crteron for the optmzaton algorthm. The algorthm can be stopped f the mum number of generatons s reached en = en ) or there s no soluton mprovement over a specfed number of generatons. The frst crteron s used n ths paper. 5. SIMULATION RESULTS The proposed EP and PSO algorthms for solvng optmal power flow problems are tested on standard IEEE 14 & IEEE 3bus test systems. The EP and PSO parameters used for the smulaton are summarzed n Table 1 Table 1 Optmal parameter settngs for EP and PSO Parameter EP PSO Populaton sze Number of teratons Cogntve constant, c1 Socal constant, c Inerta weght, W IEEE 14bus system results Fgures 1 shows the convergence characterstcs of the three obectve functons. It can be observed that the EP converge to lower values than PSO durng ntal evolutons and the PSO converge to a mnmum value than EP after teratons. obectve obectve obectve functonfuncton1 functon 3Cost) EP PSO No. of teratons No. of teratons No. of teratons No.of teratons No. of teratons No.of teratons c e of obectve functon 3Loss) Fg 1 multple obectve functons by EP and PSO No. of teratons No. of teratons 1

4 Internatonal Journal of Computer Applcatons ) Volume 7 No.11, August 11 Control varables P 1 P P 3 P 4 P 5 Q 1 Q Q 3 Q 4 Q 5 V 1 V V 3 V 4 V 5 T 1 T T 3 Table Optmal settngs of control varables Obectve functon1 Obectve functon Obectve functon 3 EP PSO EP PSO EP PSO Q sh1 Q sh Q sh3 Q sh4 Q sh5 Cost $/hr) Loss p.u.mw CPU T me The above table presents the optmal settngs of the controlvarables wth the three obectve functons. From the Table, t was found that all the state varables satsfy ther lower and upper lmts. It can be observed that the PSO algorthm s able to reduce the cost of generaton less than that of the cost of generaton obtaned by the EP method. It s also evdent from the results that partcle swarm optmzaton technque outperforms n achevng mnmum of the specfed obectve under dfferent network contngences when compared wth evolutonary programmng method. obectve functon1 obectve functon obectve functon3 Lndces Lne loadngs Load voltages Fg Lndces, Lne loadngs and Load voltages of 14 bus by EP and PSO for multple obectve functons Fgures shows the percentage lne loadngs, load bus voltages and voltage stablty ndces after the optmzaton by EP and PSO methods wth the three obectve functons. 5. IEEE 3bus system results The proposed PSO algorthm was appled to fnd the optmal schedulng of the power system for the base case loadng condton to mnmze specfed obectve functons. enerator actvepower outputs, generator termnal voltages, transformer tap settngs and shunt reactve power compensatng elements were taken as control varables. The control varables are represented as floatng pont numbers n the populaton. The upper and lower voltage lmts of load buses were taken as 1.6 and.95 respectvely.

5 Obectve functon1 Obectve functon Obectve functon Obectve functon Obectve functon3 Cost) Obectve functon3 Cost) Obectve functon Loss) Obectve functon3 Loss) Internatonal Journal of Computer Applcatons ) Volume 7 No.11, August 11 EP PSO obectve functon1 No.of teratons No. of teratons obectve functon No. of teratons No. of teratons c e of obectve functon 3Cost) No. of teratons No. of teratons obectve functon 3Loss) Fg 3 multple obectve functons by EP and PSO Fgure 3 shows the convergence characterstcs of the three obectve functons. It can be observed that the EP converge to lower values than PSO durng ntal evolutons and the PSO converge to a mnmum value than EP after teratons. Table 3 Optmal settngs of control varables Contro l varabl es P 1 P P 3 P 4 P 5 P 6 Q 1 Q Q 3 Q 4 No. of teratons No. of teratons Obectve functon1 Obectve functon Obectve functon3 EP PSO EP PSO EP PSO Q 5 Q 6 V 1 V V 3 V 4 V 5 V 6 T 1 T T 3 T 4 Q sh1 Q sh Q sh3 Q sh4 Q sh5 Q sh6 Q sh7 Q sh8 Q sh Cost $/hr) Loss p.u.mw) CPU Tme Table 3 presents the optmal settngs of the controlvarables wth the three obectve functons. From the Table 3, t was found that all the state varables satsfy ther lower and upper lmts. From the Table 5.3, t can be observed that the PSO algorthm s able to reduce the cost of generaton less than that of the cost of generaton obtaned by the EP method. It s also evdent from the results that partcle swarm optmzaton technque outperforms n achevng mnmum of the specfed obectve under dfferent network contngences when compared wth evolutonary programmng method. obectve functon1 obectve functon obectve functon3 Lndces Lne loadngs Load voltages Fg 4 Lndces, Lne loadngs and Load voltages of 14 bus by EP and PSO for multple obectve functons 3

6 Internatonal Journal of Computer Applcatons ) Volume 7 No.11, August 11 Fgure 4 shows the percentage lne loadngs, load bus voltages and voltage stablty ndces after the optmzaton by EP and PSO methods wth the three obectve functons. The comparson of fuel cost of the proposed methods wth those of the methods reported n the lterature for IEEE 3bus system s gven n Table 4. It can be seen from the Table 5.4 that the PSO algorthm gves less cost of generaton compared wth the cost of generaton obtaned wth other methods. Table 4 Comparson of fuel costs Method Fuel Cost $/hr) EP [13] 8.97 TS [13] 8.5 TS/SA [13] ITS [13] IEP [13] SADE_ALM [15] 8.44 PSO [14] 8.41 MDE [16] enetc Algorthm [17] 83.5 radent method [18] 8.43 PSO proposed) Here an IEEE3 bus system s consdered. Evolutonary programmng EP) method has been appled on the IEEE3 bus system. Here we have consdered three obectve functons. Obectve functon1 s the cost obectve functon. Obectve functon s the loss obectve functon. Obectve functon 3 s the mult obectve functon.e. both cost and losses are taken as obectves. The ftness functon s taken as the recprocal of the obectve functon. 6. CONCLUSION An EP and PSO based optmzaton algorthms have been proposed for solvng optmal power flow problems wth dfferent obectve functons. These algorthms take nto consderaton all the equalty and nequalty constrants. The mprovement n system performance s based on reducton n cost of power generaton and actve power loss. The proposed algorthms have been compared wth the other methods reported n the lterature. Smulaton studes have been carred out for the optmal solutons of the IEEE 14bus and IEEE 3bus systems. It was observed that the results obtaned by the proposed algorthms can be mplemented n real lfe power systems for operaton and analyss. Based on the overall observatons from the results obtaned on varous IEEE test systems, t can be concluded that the proposed methods for optmal solutons are sutable for mplementng n modern power system operaton. 7. REFERENCES [1] Kennedy J, Eberhart R. Partcle swarm optmzaton Proceedngs of IEEE Internatonal Conference on Neural Networks ICNN 95) Perth, Australa: IEEE Press; Vol. IV. p [] Sh Y, Eberhart R. A modfed partcle swarm optmzer Proceedngs of IEEE Internatonal Conference on Evolutonary Computaton ICEC 98). Anchorage: IEEE Press; p [3] Sh Y, Eberhart R. Parameter selecton n partcle swarm optmzaton Proceedngs of the 1998 Annual Conference on Evolutonary Programmng. San Dego: MIT Press; [4] Fukuyama Y. et al. A partcle swarm optmzaton for reactve power and voltage control consderng voltage securty assessment. IEEE Trans Power Systems ; 154): [5] Naka S, en T, Yura T, Fukuyama Y. A hybrd partcle swarm optmzaton for dstrbuton state estmaton IEEE Trans Power Systems 3; 181):6 68. [6] IEEE Specal Publcaton 9 TH 358PWR, Voltage Stablty of Power Systems: Concepts, Analytcal Tools and Industry Experence", The IEEE Workng roup on Voltage Stablty, 199. [7] A J Wood and B F Wollenberg, Power generaton operaton and control, John Wley and Sons Inc.Sngapore, [8] Bran Stott, Ongun Alsac, and Alcr J. Montcell, Securty analyss and optmzaton, Proc. of IEEE, Vol.75, No.1, December 1987, pp [9] Parker, C.J., I. F Morrson, and D. Sutanto, Applcaton of an optmzaton method for determnng the reactve margn from voltage collapse n reactve power plannng, IEEE Trans. on Power Systems, Vol.11, No. 3, August 1996, pp [1] Venkov, V.A., V.A Stroev, V.I. Idelchck, and V.I. Tarasov, Estmaton of electrc power system steady state stablty n load flow calculatons", IEEE Trans. on PAS, Vol.PAS94, No.3, May/June 1975, pp [11] P.A. Lof,. Andersson, D. Hll, Voltage dependent reactve power lmts for voltage stablty studes, IEEE Trans.on power systems, Vol. 1, No. 1, Feb. 1995, pp. 8. [1] P Kessel and H lavtsch, Estmatng the voltage stablty of a power system, IEEE Trans. on PD, Vol.1, No.3, pp , [13] W. Ongsakul and T. Tantmaporn, Optmal power flow by mproved evolutonary programmng, Electrc Power Components and Systems, 6, 34:pp.7995,. [14] Abdo MA. Optmal power flow usng partcle swarm optmzaton, Electrc Power Energy Syst ; 47): pp [15] Peerapol Jrapong and Weerakorn Ongsakul Optmal placement of mult type FACTS devces for total transfer capablty enhancement usng hybrd evolutonary algorthm, Electrc power componenets and systems, 1 September 7, 35:pp [16] Ramasubramanan Jayashree and Mohammed Abdullah Khan A unfed optmzaton approach for the enhancement of avalable transfer capablty and congeston management usng unfed power flow controller, Serban 4

7 Internatonal Journal of Computer Applcatons ) Volume 7 No.11, August 11 ournal of electrcal engneerng, Vol.5, No., November 8, pp [17] D.Devara and B.Yegnanarayana, enetc Algorthm Based Optmal Power Flow for Securty Enhancement, IEE Proceedngs on eneraton, Transmsson and Dstrbuton 5, 156), pp [18] X.P.Zhang, S..Petousss and K.R.odfrey Nonlnear nteror pont optmal power flow method based on a current msmatch formulaton, IEE Proc.ener. Transm. Dstrb. Vol.15, No.6, January 5, pp AUTHORS PROFILE Mr. Ch.Rambabu receved the Bachelor of Engneerng degree n Electrcal & Electroncs Engneerng from Madras Unversty, n and Master s degree from JNTU Anantapur n 5. He s a research student of JNTU Kaknada. Currently, he s an Assocate Professor at Sr Vasav Engneerng College. Hs nterests are n power system control and FACTS. Dr. Y.P.Obulesh receved hs B.E degree n Electrcal Engneerng from Andhra Unversty, Vsakhapatnam n M.Tech., degree from IIT, Kharagpur, n He receved hs Ph.D. degree from Jawaharlal Nehru Technologcal Unversty, Hyderabad, n 6. Currently he s workng as a Professor and head of Dept. of EEE at LBRCEC, Mylavaram. He has publshed several Natonal and Internatonal Journals and Conferences. Hs area of nterest s the smulaton and desgn of power electroncs systems, DSP controllers, fuzzy logc and neural network applcaton to power electroncs and drves. Dr. Ch. Sa Babu receved the B.E from Andhra Unversty Electrcal & Electroncs Engneerng), M.Tech n Electrcal Machnes and Industral Drves from REC, Warangal and Ph.D. n Relablty Studes of HVDC Converters from JNTU, Hyderabad. Currently he s workng as a Professor n Dept. of EEE n JNTUCEK, Kaknada. He has publshed several Natonal and Internatonal Journals and Conferences. Hs area of nterest s Power Electroncs and Drves, Power System Relablty, HVDC Converter Relablty, Optmzaton of Electrcal Systems and Real Tme Energy Management. 5