International Journal on Electrical Engineering and Informatics Volume 4, Number 4, December 2012

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1 Internatonal Journal on Electrcal Engneerng and Informatcs Volume 4, Number 4, December 2012 Bg-Bang and Bg-Crunch (BB-BC) and FreFly Optmzaton (FFO): Applcaton and Comparson to Optmal Power flow wth Contnuous and Dscrete Control Varables C. V. Gopala Krshna Rao 1 and G. Yesuratnam 2 1 Dept., M.V.S.R Engneerng college, Hyderabad. 2 Dept., Unversty college of Engneerng, Osmana Unversty, Hyderabad. vgkrao_ch@yahoo.com, ratnamgy2003@yahoo.co.n Abstract: Bg-Bang and Bg-crunch (BB-BC), a heurstc optmzaton method s based on the concept of unversal evoluton. FreFly optmzaton (FFO), also a recent heurstc optmzaton method, s based on the concept of flashng behavour of lghtngbugs. Both the optmzaton methods are appled to obtan the soluton of the Optmal Power Flow (OPF) wth contnuous and dscrete control varables for quadratc generator output cost functons. The contnuous control varables are generatng unt actve power outputs and generator bus voltage magntudes, whle the dscrete ones are transformertap settngs and swtchable shunt devces. A number of functonal constrants such as load bus voltage magntudes, lne flows and reactve power capabltes are ncluded as quadratc penaltes n the optmzaton functon. A comparatve smulaton results for Ward Hale 6 bus system wth seven control varables and IEEE 30 bus system wth twenty-three control varables are presented. Keywords: Bng-Bang and Bg-Crunch, FreFly, optmal power flow, dscrete, contnuous NOTATIONS F T : total operatng cost,nb: number of buses, NG: number of generator buses, NT: number of Transformers, NL: number of lnes (branches),nsh: number of swtchable shunts, NPQ: number of load buses NTR: number of transformers, : actve power njecton at bus, : reactve power at bus,np: populaton sze (number of frefles n FFO/number of Bg- Bangs BB-BC)NC: number of control varables (co-ordnates of frefles FF0/dspersons n BB-BC) 1. Introducton Rapd growth n power system sze and Electrcal power demand, problem of reducng the operatng cost has ganed mportance whle mantanng voltage securty and thermal lmts of transmsson lne branches. A large number of mathematcal programmng (algorthms) and AI (Artfcal Intellgence technque) have been appled to solve OPF[1,2]. In most general formulaton, the OPF s a nonlnear, non-convex, large scale, statc optmzaton problem wth both contnuous and dscrete control Varables. Mathematcal programmng approaches such as Calculus methods, Non-lnear programmng (NLP), Lnear programmng (LP), Quadratc programmng (QP), algorthms appled to obtan OPF soluton requre smooth and contnuous cost functon. Dynamc programmng methods (DP) are good at solvng quadratc and ramp cost functons, at the cost of ncreased dmensonalty and may get struck n local optmalty [3]. In cost optmzaton problems, t s desrable to obtan global optmum soluton [2]. Recent advances n AI technques can be appled as complementary approach to pave the way towards global/near global solutons for complex optmzaton problems such as OPF [2]. All search ntellgence technques, are populaton based and stochastc n nature. Search ntellgence Receved: February 25 th, Accepted: December 23 th,

2 C. V. Gopala Krshna Rao, et al. technques developed by scentfc communty from the nspraton of natural socal behavour of dfferent organsms/natural processes, offer multple feasble solutons per teraton/generaton. Genetc algorthms and ts varants, Swarm ntellgence, Bactera foragng, ant-colony search technques are appled to obtan qualty solutons [4] to optmzaton problems. Bg-Bang and Bg-Crunch (BB-BC) developed by Erol and Eksn [5] from the concept of unversal evoluton, s also a populaton based search technque. BB-BC method has been proved to outperform genetc algorthm for benchmark test functons [5]. FreFly Optmzaton (FFO) also a heurstc algorthm that smulates the flashng behavour of frefles(lghtng bugs) developed by Dr. Xn-she yang[6] has been appled to solve a number of complex optmzaton problems[7]. The conflctng objectves of Economc Load Dspatch (ELD) and thermal emsson Pareto [8, 9] usng tradtonal B mn (real power loss coeffcents) s solved by FFO. Ths paper ams at solvng complex OPF problem wth contnuous and dscrete control varables usng BB-BC and FFO. Contnuous varables are generator real power outputs and generator termnal voltages. Dscrete varables are transformer tap settngs and swtchable shunts at power system buses. Each method s run 10 tmes wth dfferent ntal control varables for 200 generatons. The best results of each run are presented. Tme taken by both optmzaton methods are compared along wth relablty n arrvng at qualty solutons. Results of these two optmzaton methods are also compared wth Genetc approaches avalable n lterature [10, 11] of OPF for IEEE-30 bus system. 2. Optmal power flow problem formulaton: OPF problem can be stated as follows, (1)., 0 (2), 0 (3) (4) where, T (5) x s a state vector of the system wth bus bar angles and load bus voltages V L. Control varables to optmze equaton 1 are real power generaton of generator loadng unts( P g ), termnal voltages of generators(v g ), tap-settng of transformers(t tap ) and swtchable shunts(q sh ),,, (6) Equaton (1) s consdered as sum of quadratc cost functons of thermal generatng real power loadng unts wth usual a,b,c cost coeffcents of equaton (7) = Ng = 1 a + b P + c P g 2 g $/h (7) subject to equalty constrants of equaton (2) () actve power balance n the network - 0 (=1, 2, 3, NB) () reactve power balance n the network - 0 (=NG+1, NB) of equaton 4 s feasble control vectors of nequalty constrants,they are () actve power generaton of generator buses (=1, 2, NG) () lmts on voltage magntudes of generator buses (=1, 2,..NG) () lmts on swtchable shunts (v) lmts on tap settng of transformers (=1,..NSH) (=1,..NT) 576

3 Bg-Bang and Bg-Crunch (BB-BC) and FreFly Optmzaton (FFO) Equaton (3) has functonal operatng constrants whch are as follows () lmts on reactve power generaton of generator buses (=1, 2, NG) () lmts on voltage magntudes of load buses (=NG+1,.NB) () thermal lmts of transmsson lnes (=1,..NL) The lmts on the control varables of real power generatons, voltage magntudes of generators, transformer tap settngs and swtchable shunt devces are mplctly handled whle generatng the parameters randomly. Power flow soluton to equaton 2, results n state vectors x (bus bar angles, load bus voltages) of the power system network. The functonal operatng constrants are handled by a quadratc penalty functon approach [12]. Due to ncluson of penalty terms, equaton (7) transforms to a pseudo objectve functon (FF) mn (8) here,,, are penalty terms for the slack bus generator MW lmt volaton, Load bus voltage lmt volatons, generator reactve power lmt volatons and volatons for thermal lmts of lnes respectvely. 3. Bg-Bang and Bg-Crunch (BB-BC) Bg-Bang and Bg-crunch (BB-BC) optmzaton, s developed from the concept of unversal evoluton. Bg-Bang Phase relates to energy dsperson n random state before evoluton of unverse. The dspersed energy s drawn nto an order for the formaton of unverse. The stage of drawng the energy to an ordered state s Bg-crunch phase. Ths concept can be mathematcally smulated by obtanng object functon values by creatng random control varables (Bg -Bang) phase. The Centre of Mass (CM) of Bg-Bang phase s drawn nto an ordered state by a Bg- crunch phase. Crunch phase control varables emerge as best control varables from Bg-Bang phase. Sequental repetton of Bg-Bang around CM eventually leads to the global control varables of the functon to be optmzed. In the Bg Bang phase control matrx (U) of dmenson (NP*NC) s generated wthn lower and upper lmts of control varables. Each row of control varable s substtuted n functon to be optmzed to obtan NP number of functon values. Then centre of mass u CM of frst phase dspersons can be computed usng equaton 9. NP NP u CM = (1/ f ) u /( 1/ f ) (9) = 1 = 1 Computaton of u CM s crunch phase of the optmzaton. In equaton 9, u s th row of U. f s the functon value correspondng to u. Ths completes k th generaton of optmzaton method. For (k+1) th generaton, each row of control vector s updated around equaton 10. u CM usng u = u CM + lmt ( u * randn )/ k (10) 577

4 C. V. Gopala Krshna Rao, et al. lmt Where u s scale of upper U upper and lower U lower lmts of the control varables, K s generaton number, randn s normally dstrbuted random number between -1 and +1. Repetton of Bg Bang followed by crunch results n optmum value of the functon. 4. Frefly Optmzaton (FFO) Frefles, randomly dstrbuted n space, emt lght due to photogenc organs on ther surface for varous socal behavour such as prey attracton, warnng sgnals to a predator. The poston of each frefly can be located usng co-ordnate ponts. The brghter frefly emts more lght to attract other frefles. The other frefles, whch are lesser n brghtness, get attracted towards brghter one, by updatng ther postons. Thus, frefles keep movng n space tll all of them reach same poston (towards brghter one). Ths socal behavour of frefles s mathematcally smulated by ntroducng an attracton factor that depends on the poston of Frefly. The brghtness of frefly s proportonal to the maxmum of functon to be optmzed. The coordnates of each frefly are analogous to control varables of the optmzaton functon to be optmzed. The attracton towards brghter one s smulated as monotoncally decreasng functon. β = exp (-γ ) (11) In the above equaton s the dstance between any two frefles, s the ntal attractveness and γ s an absorpton co-effcent whch controls the lght ntensty between two frefles. The movement of frefly j, wth row vector as co-ordnates can be moved to a more brghter frefly, wth row vector co-ordnates by usng the followng update equaton for frefly j, = exp γ. alpha*rand (1, NC)-0.5) (12) Where alpha s step sze, rand s unform random number between 0 and 1.In equaton 12, frst term s current poston (co-ordnates) of frefly j, second term s the attractveness factor and last term allows random movement of frefly. FFO s maxmzaton algorthm. In ths paper, n each generaton of FFO, functon values are sorted n descendng order. Mnmum of functon value s consdered as the brghtest frefly, all other frefles are moved to the brghter one as per equaton 12. The mplementaton of optmzaton methods to OPF s presented n what follows. 5. Steps to mplement BB-BC and FFO to OPF In general evolutonary approach appled to OPF conssts of smlar steps, the specfcty of approach dffers only n updatng the control varables from current generaton to the new generaton durng optmal search. The followng steps are common to both optmzatons of ths paper appled to solve OPF. The specfcty of each optmzaton s ndcated after the followng steps. 1. Read OPF data (cost coeffcents of objectve functon, Lne, bus data and locaton of control varables) n power system network. 2. Generate ntal control varable matrx U of sze (NP*NC) wthn the lower and upper lmt of control varables.e th row of U can be generated as u = U lower *(U upper U lower )*rand (1, NC). Where, rand s unform random number [0, 1], U lower and U upper are lower and upper lmts of control varables respectvely. Typcally, U lower and U upper are row vectors of dmenson (1*NC). 3. Set generaton count k=1. 578

5 Bg-Bang and Bg-Crunch (BB-BC) and FreFly Optmzaton (FFO) 4. Intalze FF count to 1. Row select of U to Fetch the row correspondng to Row select from U, modfy lne and bus data of power system network. Solve for power balance equaton of OPF by usng Newton Raphson (NR)/Fast decoupled load Flow (FDLF). 6. Check for functonal operatng constrants, for any volaton of these constrants, actvate penaltes and Evaluate FF. set Row select=row select +1, FF=FF+1, return to 5, tll FF count=np. 7. Store current best soluton and ts correspondng control varables. Check for stoppng crtera, f met dsplay current best soluton, else go to step Update control varables n accordance wth update Equaton of respectve optmzaton method. Ths step may result n volaton of control varable lmts. Those volated control varables should be made equal to ther respectve volated lmt. 9. Set k=k+1.return to step 3 tll k=maxgenratons. In ths paper, to satsfy power balance equatons (step 5), FDLF s used [10]. Durng ntal generatons of optmzaton algorthm, FDLF may not converge even though control varables are wthn the range. For such cases, an addtonal large penalty term proportonal to maxmum real and reactve power msmatch s added to FF.FDLF maxmum teratons and power balance msmatch tolerance are set to 8 and 0.001pu respectvely. In step 8, control varables can be updated for BB-BC usng equaton 10 and for FFO usng equaton 12. Whle applyng BB-BC, the mnmum of FF value n each generaton s consdered as u CM. Convergence crtera may be number of generatons or dfference between best functon value of k th and (k+1) th generaton less than a specfed tolerance. The above steps are mplemented for the two test systems mentoned n ths paper. The requred code s wrtten n MATLAB-7.0, as m-fles usng lbrary routnes of MATLAB soft ware. Code s executed on a 2.1 GHz, Pentum IV PC. The choce of optmzaton parameters namely NP (populaton sze), alpha (step sze), γ (absorpton co-effcent), ntal attracton n FFO are presented along wth test- case results. 6. Test Results and Dscussons To test the effectveness and qualty solutons of optmzaton methods of ths paper, OPF smulatons are carred on Ward-Hale- 6 bus and Modfed IEEE-30 bus power system networks. Requred data for the two systems for cost coeffcents of generators, control varable lmts, bus and transmsson lne data are taken from [13]. In both systems, frst bus s slack bus and ts real power lmt s dealt n OPF usng quadratc penalty. Generator voltages of slack bus for both systems are also ncluded as control varables. Total system load consdered for ward- Hale s (1.3500pu +j pu) and for IEEE-30 bus system total base case load s (2.834pu+j1.2620pu). The lower and upper magntudes of all load bus voltages are 0.95 pu and 1.05 pu respectvely. The transformer tap settng s consdered as (0.9+tap_ poston*0.005), where tap_ poston can take 41 dscrete steps n the range 0 to 40 nteger values. The tap_ poston 0 ndcates mnmum tap 0.9 and tap_ poston 40 ndcates maxmum tap of 1.1. Swtchable shunt consdered as (step _val*0.01), where step_ val can take 6 dscrete steps n the range of 0 to 5 nteger values, a step_ val 0 ndcates 0.00pu capactve shunt and 5 ndcates capactve shunt of 0.05pu(on 100MVA base).test results for Ward-Hale and IEEE- 30 bus systems are presented n table 1 and table 2 respectvely. Each test case s run ntally for base load (wthout optmzaton) wth control varables as gven n second column of the tables 1and 2. Cost after optmzaton by FF0 and BB-BC along wth control varables and slack bus power s ndcated n column 3 and column 4 of table 1, 2. Upon close observaton of table 1, 2 optmal cost of real power generaton obtaned by BB-BC and FFO are almost same wth a small edge for FF0. 579

6 C. V. Gopala Krshna Rao, et al. Table 1. Varables for Ward-Hale 6 bus system Varables Base case FFO BB-BC P g1 (pu) P g2 (pu) V g1 (pu) V g2 (pu) Q sh4 (pu) Q sh6 (pu) t 1(6-5) t 2(4-3) Total real power generaton(pu) Total real power losses (pu) cost($/hr) Table 2. Varables for IEEE 30- bus system Varables Base case FFO BB-BC P g1 (pu) P g2 (pu) P g5 (pu) P g8 (pu) P g11 (pu) P g13 (pu) V g1 (pu) V g2 (pu) V g5 (pu) Vg11(pu) V g13 (pu) Q sh10 (pu) Q sh12 (pu) Q sh15 (pu) Q sh17 (pu) Q sh20 (pu) Q sh21 (pu) Q sh23 (pu) Q sh24 (pu) Q sh29 (pu) t 1(6-9) t 2(6-10) t 3(4-12) t 4(28-27) Total Real power generaton(pu) Total Real power losses(pu) Cost($/hr) Both optmzaton methods of ths paper have only one common parameter, NP to be chosen by tral. In case of BB-BC, NP s set to 25 and 50 for Ward-Hale and IEEE-30 bus case respectvely. In case of FF0 other parameters of optmzaton update equaton γ and and are set to 1.The number of smulatons carred out by keepng alpha constant for all generatons of optmzaton process at dfferent values n the range of 0.02 to unty, resulted n hgher cost than BB-BC. To mprove the results, alpha s reduced gradually n small steps as optmzaton proceeds number of generatons. Such reducton of alpha s done by lettng alpha=0.975*alpha, wth alpha as unty before start of optmzaton generatons. Trals made for alpha are ndcated n table 3, wth NP=25 n case of Ward-Hale, NP=40 n case of IEEE 30 bus system. 580

7 Bg-Bang and Bg-Crunch (BB-BC) and FreFly Optmzaton (FFO) Table 3. Varaton of alpha vs Cost alpha Ward-Hale IEEE 30 Cost($/hr) Cost($/hr) Smulatons carred out, by varyng γ n the range 0.85 to 1,( set to 1), after selecton of proper step sze alpha, also resulted n the optmal cost as reported n tables 1,2. Hence, from table 3 and smulaton carred wth varaton of γ, selecton of alpha s crtcal n FFO optmal cost. Step sze alpha reduced n small steps n every generaton lead to the local search of objectve functon. A comparatve Convergence value of FF n $/hr s ndcated n table 4. It s clear from table 4 that qualty solutons can be arrved by both optmzaton methods n early generatons of optmzaton. It can also be observed from table, that FF0 attans qualty solutons than BB-BC, n very ntal generatons (20 generatons). The reason can be attrbuted to the fact that FFO updates control varables n each generaton based on dstance norm between best functon value of FF and the rest of functon values among FF. In case of BB-BC, convergence to optmal value s controlled by k of equaton 4. As optmzaton advances number of generatons, BB-BC optmal search wll be local as ndcated n table 4. Table 5 ndcates data statstcs for ten ndependent test runs wth dfferent ntal values, for 200 generatons. Table 4. A comparatve converge values of FF. Generaton number FFO BB-BC Computatonal tme, dfference between Maxmum and Mnmum cost, Mean cost and Standard devatons provded n Table 5 gves better edge to FFO compared to BB-BC. The best cost by applcaton of problem specfc advanced genetc operators[10], and real coded genetc algorthm[11] for same IEEE-30 bus system are $/h and $/h respectvely. Best cost obtaned by both optmzatons of ths paper s less than genetc approaches. However, the proposed optmzaton approaches of ths paper need to be tested for ther robustness for certan complex non-lnear and non-convex optmzaton stuatons lke reactve power dspatch usng recently proposed an Intellgent Water Drop (IWD) algorthm wth target voltage stablty ndex [ 14] and proposed two step-ntalzaton heurstc search algorthm[15] to optmal power flow wth FACTS devces. 581

8 C. V. Gopala Krshna Rao, et al. Table 5. Mnmum, Maxmum, Mean and standard devaton wth dfferent ntal values Ward-Hale IEEE-30 FFO BB-BC FFO BB-BC Mn ($/hr) Max($/hr) Mean ($/hr) Standard devaton Meantme(S) Concluson Bg-Bang and Bg-Crunch and frefly optmzaton methods are appled to solve complex statc optmal power flow problem wth contnuous and dscrete control varables. Test results and smulatons carred towards establshng relablty confrm promsng nature of the twooptmzaton methods for optmal power flow solutons. Careful selecton of step sze n Frefly optmzaton results n optmal, fast and relable optmal power flow solutons than Bg-Bang and Bg-Crunch optmzaton. Both optmzaton methods are smple to mplement compared to Genetc approaches for optmal power flow solutons. References [1] M. Huneault and F D Galana A survey of the optmal Power Flow Lterature. IEEE trans. vol 6, no, May [2] Bansal R. CDr (2005) optmzaton methods for Electrcal power systems an overvew Internatonal journal of Emergng Electrcal Power Systems vol 2, Iss 1, Artcle [3] N. Snha, R. Chakrabart, P. K. Chattopadhaya, Evolutonary programmng technques for Economc load Dspatch IEEE Trans. Evol. comp(1) (2003), pp [4] N. P. Padhy, Artfcal Intellgence and Intellgent Systems Oxford Unversty press [5] K. Erol Osman, Ibrahm Eksn, New optmzaton method: Bg Bang-Bg Crunch, Elsever, Advances n Engneerng Software 37 (2006), pp [6] Yong X.S (2008), Nature nspred Metaheurstc Algorthms, Lunver press ISBN [7] X. S. Yang, Frefly algorthm, Levy flghts and global optmzaton, n Research and Development n Intellgent Systems XXVI, pp , Sprnger, London, UK, [8] M. vnod Kumar G. Lakshm Phan Combned Economc Dspatch Pareto optmal fonts approach usng FreFly optmzaton Internatonal Journal of computer Applcatons Vol 30 no 12 sep [9] A Postopoulous T,Valchos A(2011) Applcaton of FreFly algorthm for solvng Economc Dspatch, Internatonal Journal of Combnatroncs Artcle ID , do. [10] Anatass G. Balrtzs, N. Bskas, Chrstoforos E. Zoumas and Vaslos Petrds Optmal Power flow by Enhanced Genetc Algorthns IEEE Trans vol 17 no 2, May 2002, pp [11] K. S. Lnga Murthy, K. M. Rao, K. Srkanth, Emsson Constraned Optmal power flow usng Bnary and real coded genetc algorthms Journal of the Insttuton of Engneers (Inda), vol 92, Sep 2011, pp 3-8. [12] D. I. Sun, B. Ashley, B. Brewer, A.Hughes and W. F. Tnney, Optmal power flow by Newton approch IEEE transacton on PAS, vol PAS-103, no 5, 1984,pp [13] K. y. Lee, Y. M Park and J L Ortz A unfed approach to optmal real and reactve power Dspatch,System Research, IEEE transacton on PAS,vol PAS-104, no 5, May 1985, pp [14] K. Lenn, M. Surya kalavath An Intellgent Water Drop Algorthm for solvng Optmal Reactve Power Dspatch Problem Internatonal Journal on Electrcal Engneerng and Informatcs Volume 4, Number 3, October

9 Bg-Bang and Bg-Crunch (BB-BC) and FreFly Optmzaton (FFO) [15] A. V. Naresh Babu and S. Svanagaraju A New Approach for Optmal Power Flow Soluton Based on Two Step Intalzaton wth Mult-Lne FACTS Devce Internatonal Journal on Electrcal Engneerng and Informatcs Volume 4, Number 1, March C. V. Gopala Krshna Rao s Assocate Professor n Electrcal and Electroncs Engneerng n MVSR Engneerng College, Hyderabad. He obtaned hs B.Tech from S.V. Unversty, Trupat and M.E n Hgh Voltage Engneerng from College of Engneerng Gundy, Chenna. He has 22 years of teachng experence at graduate level. He s currently a research Scholar n College of Engneerng, Osmana Unversty. To hs credt he has three nternatonal publcatons. Hs area of nterest ncludes Hgh voltage delectrc studes and Soft computng technques appled to Power Systems. Control. G. Yesuratnam s Assocate Professor and Head of the Electrcal Engneerng Department n College of Engneerng, Osmana Unversty, Hyderabad. He obtaned hs M.Tech degree from NIT, Warangal and Ph.D n Electrcal Engneerng from IISC, Bangalore, Inda. He has 17 years of teachng experence at both under graduate and post Graduate levels. To hs credt, he has publshed 5 nternatonal journals, presented papers n 10 natonal and nternatonal conferences. Hs area of nterest ncludes applcaton of Fuzzy and Expert systems to Power System Operaton and 583