Multiobjective Generation Dispatch using Big-Bang and Big-Crunch (BB-BC) Optimization

Transcription

1 Internatonal Journal of Electrcal Engneerng. ISSN Volume 4, Number 5 (20), pp Internatonal Research Publcaton House Multobjectve Generaton Dspatch usng Bg-Bang and Bg-Crunch (BB-BC) Optmzaton C.V. Gopala Krshna Rao and 2 G. Yesuratnam Assocate Professor, E.E.E. Dept., M.V.S.R. Engneerng College, Hyderabad, Inda E-mal: 2 Assocate Professor, EE Dept, Unversty College of Engneerng, Osmana Unversty, Hyderabad, Inda E-mal: ratnamgy2003@yahoo.co.n Abstract Bg Bang and Bg Crunch (BB-BC) optmzaton method for multoptmzaton of economc load dspatch and emsson dspatch of thermal power generatng statons s presented usng weghted sum method, trade-off solutons are obtaned assumng decson makers (DM) requrements by solvng optmal power flows. BB-BC optmzaton whch s a non- dervatve, populaton search optmzaton procedure developed based on Unversal Evoluton, s smple to code and fast optmzaton method trade-off (pareto) soluton obtaned by BB-BC s compared wth well researched most popular PARTICLE SWARM OPTIMIZATION for two typcal power system networks, IEEE-25(Gangour) and IEEE-30 bus systems. Outcome of comparson suggests that BB-BC optmzaton solutons are as superor as PSO Keywords: multobjectve, Bg-Bang-Crunch optmzaton trade-off (Pareto). Introducton Multobjectve optmzaton n power systems s to obtan multple Non-nferor solutons for steady state and dynamc operatng condtons of power system so that Decson Maker (DM) can select one of the best optons for power system operaton. Multobjectve optmzaton n power system can be obtaned for cost mnmzaton of thermal emsson, voltage devatons, power lne thermal lmts etc. Mnmzaton of these functons has to be done by satsfyng varous equalty, n-equalty and operatonal constrants of the power system. Real and reactve power optmzaton are

2 556 C.V. Gopala Krshna Rao and G. Yesuratnam carred out by energy computng centres to arrve at best settngs of generator out puts, specfed voltages of the generators, ratngs and control settngs of power system compensatng devces[,2]. Now-a-days, due to ncreased publc awareness of envronmental protecton and strngent rules by polluton control boards, many researchers explored the optmzaton of power system network for combned economc/ emsson power dspatch [ 3]. Assumng thermal power generaton (whch contrbute major electrc power to the grd), creates thermal emsson such as N OX,S OX and C O2, ncluson of these emssons along wth cost functon leads to multobjectve optmzaton. It s well known fact that there exsts a trade-off between cost of power generaton and Emsson of polluton. DM of energy control system has to depend on multobjectve optmzaton to decde the settngs of the power generator. Many researchers nvestgated varous tradtonal [4] and non-tradtonal optmzaton methods to obtan relable and fast trade-off solutons. Non-tradtonal optmzaton methods such as Genetc Algorthms and ther varants [5], Smulated Annealng [6], Partcle Swarm Optmzaton (PSO) [7], etc wth varous modfcatons are successfully appled to obtan trade-off solutons. These methods may be consdered as non-classcal, unorthodox and stochastc search optmzaton algorthms, known as Evolutonary Algorthms (EA). EA use a populaton of solutons n each teraton nstead of a sngle soluton. If an optmzaton problem has multple optmal solutons, EA can be used to capture multple optmal solutons n ts fnal populaton. Ths ablty of populaton based optmzaton methods to fnd multple optmal solutons n one sngle smulaton run makes these methods unque n solvng multobjectve optmzaton problems. Evolutonary Methods are partcularly requred when functon to be optmzed s dscontnuous such as ramp cost functons wth valve pont loadng [8]. Among stochastc search methods, PSO has ganed utmost popularty for many optmzaton problems of power system due ts smple code and ablty to tune both for local and global search n soluton space[9 ].To the lst of non-tradtonal methods, the recent contrbuton of Optmzaton method s Bg Bang- and Bg-Crunch(BB- BC), developed by Erol and Eksn [ 0]. Ths optmzaton s based on the concept of unversal evoluton. Accordng to ths theory, n the Bg Bang phase, energy dsspaton produces dsorder and randomness. In the Bg Crunch phase, randomly dstrbuted partcles are drawn nto an order. The Bg Bang Bg Crunch (BB BC) Optmzaton method smlarly generates random ponts n the Bg Bang phase and shrnks (draw) these ponts to a sngle representatve pont va a centre of mass n the Bg Crunch phase. After a number of sequental Bg Bangs and Bg Crunches where the dstrbuton of randomness wthn the search space durng the Bg Bang becomes smaller and smaller about the average pont computed durng the Bg Crunch, the algorthm converges to a soluton. The BB BC method has been shown to outperform the enhanced classcal Genetc Algorthm for many benchmark test functons []. In ths paper an effort s made to compare trade -off soluton between economc power dspatch and emsson dspatch (multobjectve optmzaton) by thermal power statons by BB-BC and PSO. To compare the two optmzaton methods two practcal power system networks IEEE-25bus test system (Gangour ), standard IEEE-30 bus test system are consdered

3 Multobjectve Generaton Dspatch 557 to obtan trade-off Soluton between cost of power generaton mnmzaton and thermal emsson. BB-BC trade off solutons are compared wth PSO trade-off solutons. The outcome of the comparson clearly ndcates the effectveness of BB- BC optmzaton for trade-off solutons of power systems. Ths paper s organzed as follows: secton2 presents an overvew of multobjectve optmzaton. In Secton3, multobjectve formulaton of mnmum cost dspatch and mnmum emsson dspatch along wth weghted sum formulaton of combned functon s presented. In Secton 4, optmzaton concept of BB (Bg Bang) phase and BC (Bg Crunch),s explaned. Lke all EA optmzatons, BB-BC s also unconstraned optmzaton; hence, Secton V deals wth tacklng of constrants of power system n optmzaton functon and also explans the BB-BC algorthm. Results are dscussed n secton 6, along wth test cases Consdered for smulaton. Conclusons are made n Secton 7. Multobjectve optmzaton Engneerng desgn often deals wth multple, possbly conflctng, objectve functons or desgn crtera. As an example, one may want to maxmze the performance of a system whle mnmzng ts cost. Such desgn problems are the subject of multobjectve optmzaton and can generally be formulated as: mn J(x,p) subject to g(x,p) 0 h(x,p) = 0 J=[J (x),,..j z (x)] T x=[x,,..x n ]T () g=[g (x),..g m (x)] T h=[h (x)..h m2 (x)] T In equaton (), J s an objectve functon vector, x s a desgn vector, p s a vector of fxed parameters, g s an nequalty constrant vector and h s an equalty constrant vector. There are z objectves, m nequalty constrants and m2 equalty constrants. Compared to sngle objectve problems, multobjectve problems are more dffcult to solve, snce there s no unque soluton. There s a set of acceptable trade-off optmal solutons. Ths set s called Pareto front. The preferred soluton, the one most desrable to the DM, s selected from the Pareto set. The Pareto set allows the decson maker to make an nformed decson by observng a wde range of optons snce t contans the solutons that are optmum from an overall standpont. A vector of decson varable s Pareto Optmal f there s no feasble vector of decson varables x whch wll decrease some crteron wthout causng a smultaneous ncrease n at least one another crteron. One of the most wdely used methods for solvng multobjectve optmzaton problems s to transform a multobjectve problem nto a seres of sngle objectve problems. The weghted sum method s a tradtonal method

4 558 C.V. Gopala Krshna Rao and G. Yesuratnam that parametrcally changes the weghts among the objectve functons to obtan the Pareto front. The weght of an objectve can be chosen n proporton to the objectve s relatve mportance n the problem. ε-constrant method, weghted metrc methods, value functon methods, goal programmng methods are some of the other avalable methods [2]. For the power system studes, presented n ths paper, the weghted sum method for multobjectve optmzaton s used. Formulaton for Power System Studes For the power system studes presented n ths paper, two goals (objectves) are consdered for optmzaton, they.economc Dspatch 2.Mnmum Emsson Levels of thermal generators of Electrcal Power transmsson system. In the multobjectve optmzaton, these two objectves are combned nto a sngle objectve functon usng the weghted sum method. Economc Dspatch The optmum power flow can be expressed as constraned optmzaton Problem [ ] requrng the optmzaton of f=f(x,u) Subject to g(x,u)=0 (2) h(x,u) 0 u mn u u max x mn x x max In the above equatons, f (x, u) s the scalar objectve functon, g(x, u) represents non- lnear equalty constrants (power flow equatons), and h(x,u) s the non-lnear nequalty constrant of vector arguments x and u. The vector x conssts of dependent varables (for example bus voltage magntudes and phase angles). The vector u conssts of control varable (for example, real power generaton). Specfcally, when the objectve s to mnmze the total fuel cost, the objectve functon can be expressed as the sum of the fuel cost for all the avalable generatng unts: ( x u) f, = Ng = a + b P + c P g 2 g where Ng s total number of generators P g are the output of generators a, b, c are the cost coeffcents of the generators Mnmum Emsson of Thermal unts The atmospherc emsson can be represented by a functon that lnks emssons wth (3)

5 Multobjectve Generaton Dspatch 559 the power generated by every unt. Combned S O2 and N O x emsson s a functon of generator output and s expressed as follow. 2 2 ( x u) = ( α + βpg + γ Pg + χ sn( λ Pg)) f, Ng = (4) Whereα, β, γ, χ, λ are emsson coeffcents of the generators The combned optmzaton functon wth weghted objectves s f ( x, u) =wt* f ( x, u) +(-wt)* f 2 ( x, u) (5) Where wt can assume any value n the range (to 0).wt=, the combned Optmzaton performs economc cost dspatch, wt=0, the combned optmzaton performs economc emsson dspatch. For any other value of wt optmzaton s obtaned for trade of soluton (Pareto) between the objectves consdered. Bg-Bang and Bg-Crunch (BB-BC) The BB BC method developed by Erol and Eksn conssts of two phases: a Bg Bang phase, and a Bg Crunch phase. In the Bg Bang phase, canddate solutons are randomly dstrbuted over the search space. Smlar to other evolutonary algorthms, ntal solutons are spread all over the search space n a unform manner n the frst Bg Bang. Erol and Eksn assocated the random nature of the Bg Bang to energy dsspaton or the transformaton from an ordered state (a convergent soluton) to a dsorder or chaos state (new set of soluton canddates).randomness can be seen as equvalent to the energy dsspaton n nature whle convergence to a local or global optmum pont can be vewed as gravtatonal attracton. Snce energy dsspaton creates dsorder from ordered partcles, we wll use randomness as a transformaton from a converged soluton (order) to the brth of totally new soluton canddates (dsorder or chaos).the proposed method s smlar to the Genetc Algorthm n respect to creatng an ntal populaton randomly. The creaton of the ntal populaton randomly s called the Bg Bang phase. In ths phase, the canddate solutons are spread all over the search space n a unform manner.the Bg Bang phase s followed by the Bg Crunch phase. The Bg Crunch s a convergence operator that has many nputs but only one output, whch s named as the centre of mass, snce the only output has been derved by calculatng the centre of mass. Here, the term mass refers to the nverse of the mert functon value. The pont representng the centre of mass that s denoted by u c s calculated accordng to: u c = N f = N f = u / (6) Where u s control varable n n-dmensonal search space generated, f s a ftness functon value of u, N s the populaton sze n Bg Bang phase. After the Bg

6 560 C.V. Gopala Krshna Rao and G. Yesuratnam Crunch phase, the algorthm creates the new solutons to be used as the Bg Bang of the next teraton step, by usng the prevous knowledge (centre of mass).ths can be accomplshed by spreadng new off-sprngs around the centre of mass usng a normal dstrbuton (randn) operaton n every drecton, where the standard devaton of ths normal dstrbuton functon decreases as the number of teratons of the algorthm ncreases: ( u * randn) k unew = uc + lmt / (7) Where, u lmt are the maxmum and mnmum lmts of control varables. randn s random normal number between - and. k = (teraton number+). Applcaton of BB-BC to power system studes BB-BC s an unconstraned optmzaton functon. In general, Optmzaton of power system wth problem specfc control varables u ( P g,e g,taps,etc) has to satsfy equalty g(x,u)(power flow equatons), In-equalty constrants h(x,u) h lmt (Branch flow lmts, generator reactve power lmts, slack bus real power output), x(voltage magntude and Phase angle of the load bus bars). In ths paper, real power output of the Generator except slack bus real power output are control varables. Reactve power lmts of the generator are handled n the power flow algorthm pv-pq bus type swtchng. Real power flows of the transmsson branches and slack bus power.e h(x, u) are satsfed by usng penalty functon method. There fore, the combned objectve functon gets transformed to unconstraned functon ff as follows n 2 b f ( x, u ) + pen ( (, ) ) ( ) 2 b h lmt h x u + pen s PGslmt PGs (8) b = Where pen b s penalty for branch real power flow volatons. Pen s s penalty for slack real power generaton. Operatonal varables (x) for specfed loads are calculated by Newton-Raphson (NR) Power flow algorthm. The pseudo code of the BB-BC appled s gven below Step : Generate ntal canddates n a random manner P g s Step 2: Solve NR Power flow and obtan voltage solutons, Calculate the ftness functon (ff) values of all the canddate solutons. Step 3: Fnd the centre of the mass p gc usng equaton 6. Step 4: Calculate new canddates around the centre of the mass usng equaton -7. Bound the control varables wthn the lmts, f control varables volate the lmts.

7 Multobjectve Generaton Dspatch 56 Step 5: Untl meetng a stoppng crteron, return to step 2. The above algorthm has to be repeated for all wt s (trade-off) of the DM. Partcle Swarm Optmzaton (PSO) s a heurstc search technque that smulates the movements of a flock of brds whch am to fnd food. The relatve smplcty of PSO and the fact that t s a populaton-based technque have made t a natural canddate to be extended for multobjectve optmzaton. PSO optmzaton algorthm requres Optmzaton parameters constrcton factor (cf) and nerta weght (w) to enhance the local and global search. In ths paper, mplementaton of PSO and selecton of tunng parameters for PSO are as explaned n ref [7]. Dscusson of Smulaton Results To test the BB-BC and PSO for multobjectve optmzaton as per the algorthms explaned above, a MATLAB code s wrtten, and s executed on Intel Pentum V, 2.8 GHz. Test cases consdered for smulaton are as follows. Test case IEEE-25bus system(gangour),35 transmsson lnes,5 generators wth generator as slack generator lne data, bus data, cost and NOx emsson coeffcents of the system s avalable [ 2].Total real power load 7.25 p.u and reactve power load s 2.23 p.u. Test case 2 IEEE-30 bus standard test system, 4 transmsson lnes, wth 6 generatng unts, two fxed shunts, and four transformers wth generator as slack generator. The bus and lne data of ths system s avalable [3], Cost coeffcents and combned N O x, and S O2 coeffcents are taken from [7]. Total real power load p.u and reactve power load s.2620 p.u. Test case : Decson makers choce for trade-off vector s assumed as wt= [, 0.8, , 0]; In combned objectve functon of equaton (5) consdered wt=, optmzes mnmum cost dspatch, where as wt=0, optmzes for mnmum emsson dspatch. As The wt departs from tends towards 0, multobjectve functon gves weghtage towards mnmum emsson dspatch and pcks those generators whch ncreases the Cost of power generaton and reduces the emsson.bb-bc parameters are set to Maxteratons 00, populaton sze 30.To compare wth PSO, Maxteratons 00, populaton sze 30.PSO also requres settng of Φ, cf and nerta weght w n equaton (). After few trals, cf=0.729, nerta weght w s set to.2 n the begnnng of teraton process and gradually the value reduced to w/.5 n successve teratons. Advantage of BB-BC, s that t does not requre trals to select parameters for optmzaton. k n equaton 7, s responsble for local search around Centre of mass of populaton as teraton number ncreases. k s taken as (teraton number+). Fg(),(2) ndcates convergence characterstcs for economc dspatch and emsson dspatch by BB-BC and PSO.BB-BC converges n a stepped manner wthout any oscllatons. It 4 s observed that to converge to an accuracy of 0 tolerance BB-BC needs 4-6 teratons more than PSO. However, both optmzaton methods nearly converge to

8 562 C.V. Gopala Krshna Rao and G. Yesuratnam average mnmum n 0 to 30 teratons. Due to absence of Velocty terms for local and global searches n BB-BC for 00 teratons run take less tme. Both optmzatons are run for 00 ndependent tmes and the average executon tmes are computed whch gves the average tme for BB-BC s and for PSO s respectvely. Fgure : Convergence Characterstcs wt=, test case-. Fgure 2: Convergence Characterstcs wt=0, test case-. Table, 2 shows trade-off (pareteo) soluton and generator outputs (control 4 varables of test case-). Results are shown for a tolerance of 0 for cost and emsson. For all trade off weghts BB-BC soluton s as superor as PSO for cost, emsson dspatch and generator outputs. Table : comparatve trade-off soluton of test case-. Trade -off wt BB-BC BB-BC Cost of generaton Cost of generaton Emsson Emsson $/hr $/hr kg/hr kg/hr

9 Multobjectve Generaton Dspatch Table 2: Comparatve Generator outputs of test case- Trade-off weght(wt).0 BB- BC 0.8 BB- BC 0.5 BB- BC 0.3 BB- BC 0.0 BB- BC Pg() p.u Pg(2) p.u Pg(3) p. u Pg(4) p.u Pg(5) p.u Real power loss (p.u) Test case 2: Optmzaton parameters for ths case are smlar to test case-. Emsson functon consdered s combned S O2 and N O x and hence emsson functon has the same form as equaton 4.For ths case, Wt= [, 0] are only weghts that satsfes the trade-off (pareto) soluton by both PSO and BB-BC. Fg 3, 4 shows the convergence characterstcs of BB-BC and PSO respectvely. For ths case also both convergence characterstcs, mnmum dspatch results and generator outputs provded by BB-BC s as superor as to PSO. Table 3, 4 comparsons of results are shown. Tmes of executons are calculated smlar to that n test case-, for00 teraton average tme of BB-BC s less than PSO. Average tme for BB-BC s and for PSO s respectvely.

10 564 C.V. Gopala Krshna Rao and G. Yesuratnam BB-BC Ftness Functon ff wt=,economc Dspatch Test case Iteratons Fgure 3: convergence Characterstcs wt=, test case 2. Ftness Functn ff wt=0,economc Emsson Dspatch Test case-2 BB-BC PO S Iteratons Fgure 4: Convergence Characterstcs wt=0, test case =2. Table 3: comparatve trade-off soluton test case-2. BB-BC BB-BC Trade-off wt cost of Generaton $/hr Cost of generaton $/hr Emsson kg/hr Emsson kg/hr

11 Multobjectve Generaton Dspatch 565 Table 4: Comparatve Generator outputs of test case-2. Trade-off Weght(wt).0 BB-BC 0.0 BB-BC Pg() p.u Pg(2) p.u Pg(3) p.u Pg(4) p.u Pg(5) p.u Pg(6) p.u Real power loss(p.u) BB-BC Real power loss(p.u) BB-BC Conclusons BB-BC optmzaton algorthm s appled for multobjectve optmzaton to obtan trade-off solutons for economc electrcal real power cost dspatch and emsson dspatch of thermal generatng power transmsson system usng weghted sum method. BB-BC optmzaton s found to be soluton effectve and convergence s relable.most attractve feature s that lke other EA, BB-BC does not requre any tral parameters for optmzaton. A number of multple solutons around centre of mass n frst 0-5 teratons found to arrve at average mnmum for all trade-off weghts (wt) n both Test cases. The effectveness of BB-BC has to be tested for optmal locaton of compensatng devces, Reactve power optmzaton, congeston management etc wth more number of control parameters wth dscrete step szes for large scale Power transmsson system. References [] A. J. Wood and Wollenberg Power Generaton and Control, John Wlley and Sons, New York, 984. [2] D. P. Kothar, J. S. Dhllon, Power System Optmzaton, PHI, New Delh,2004. [3] Jacob Zahav and Lawrence Esenberg, An Applcaton of Economc- Envronmental power dspatch IEEE Trans. On systems, Man, and Cybernetcs, July- 977, pp [4] J. W. Talaq, M. E. Harway A summary of Envronmental /Economc dspatch Algorthms IEEE Trans.on Power System, no 3,994,pp [5] Anastasos G. Bakrtzs, N. Bskas, E. Zoumous, Vaslos Petrds, Optmal power [6] Flow by Enhanced Genetc Algorthms IEEE Trans.on Power System, vol 7, No 2, pp

12 566 C.V. Gopala Krshna Rao and G. Yesuratnam [7] K. P. Wong and Y. W. Wong Thermal Generator Schedulng usng Hybrd Genetc/Smulated Annealng Approach IEE proc. part C, vol 42,no4,July 995, p 372. [8] Papya Dutta, A. K.Snha, Envronmental Economc Dspatch Constraned by Voltage Stablty usng PSO, IEEE -2006, pp [9] H. Yang, P. Yang and A. T. Johns, Evolutonary Programmng Based Economc Dspatch for unts wth non-smooth Fuel Cost Functons IEEE Trans. on pwrs, Vol, no, February 996, pp 2-8. [0] L. S. Ttare, L. D. Arya, A PSO for Improvement of Voltage Stablty by Reactve Power Reserve Management Journal of the Insttuton of Engneers (Inda),vol 87,September 2006,pp3-7. [] K. Erol Osman, Ibrahm Eksn, New optmzaton method: Bg Bang-Bg Crunch, Elsever, Advances n Engneerng Software 37 (2006), pp. 06. [2] K. Deb, Mult-Objectve Optmzaton Usng Evolutonary Algorthms, John Wley & Sons, Ltd., Chchester, England, 200. [3] P. H. Hota, R. Chakrabart, P. K. Chattopadhya Multobjectve Dspatch Usng Goal- Attanment Method Journal of the Insttuton of Engneers(Inda),vol 82,September 200,pp22-27 [4] H. Saadat, Power System Analyss, McGraw-Hll, Inc., 999.